# Core¶

Math core module: `ezdxf.math`

These are the core math functions and classes which should be imported from `ezdxf.math`.

## Functions¶

ezdxf.math.closest_point(base: Union[Sequence[float], Vec2, Vec3], points: Iterable[Union[Sequence[float], Vec2, Vec3]]) Vec3

Returns the closest point to base.

Parameters
ezdxf.math.uniform_knot_vector(count: int, order: int, normalize=False) List[float]

Returns an uniform knot vector for a B-spline of order and count control points.

order = degree + 1

Parameters
• count – count of control points

• order – spline order

• normalize – normalize values in range [0, 1] if `True`

ezdxf.math.open_uniform_knot_vector(count: int, order: int, normalize=False) List[float]

Returns an open (clamped) uniform knot vector for a B-spline of order and count control points.

order = degree + 1

Parameters
• count – count of control points

• order – spline order

• normalize – normalize values in range [0, 1] if `True`

ezdxf.math.required_knot_values(count: int, order: int) int

Returns the count of required knot values for a B-spline of order and count control points.

Parameters
• count – count of control points, in text-books referred as “n + 1”

• order – order of B-Spline, in text-books referred as “k”

Relationship:

“p” is the degree of the B-spline, text-book notation.

• k = p + 1

• 2 ≤ k ≤ n + 1

ezdxf.math.xround(value: float, rounding: float = 0.0) float

Extended rounding function, argument rounding defines the rounding limit:

 0 remove fraction 0.1 round next to x.1, x.2, … x.0 0.25 round next to x.25, x.50, x.75 or x.00 0.5 round next to x.5 or x.0 1 round to a multiple of 1: remove fraction 2 round to a multiple of 2: xxx2, xxx4, xxx6 … 5 round to a multiple of 5: xxx5 or xxx0 10 round to a multiple of 10: xx10, xx20, …
Parameters
• value – float value to round

• rounding – rounding limit

ezdxf.math.linspace(start: float, stop: float, num: int, endpoint=True) Iterable[float]

Return evenly spaced numbers over a specified interval, like numpy.linspace().

Returns num evenly spaced samples, calculated over the interval [start, stop]. The endpoint of the interval can optionally be excluded.

ezdxf.math.area(vertices: Iterable[Union[Sequence[float], Vec2, Vec3]]) float

Returns the area of a polygon, returns the projected area in the xy-plane for any vertices (z-axis will be ignored).

ezdxf.math.arc_angle_span_deg(start: float, end: float) float

Returns the counter-clockwise angle span from start to end in degrees.

Returns the angle span in the range of [0, 360], 360 is a full circle. Full circle handling is a special case, because normalization of angles which describe a full circle would return 0 if treated as regular angles. e.g. (0, 360) → 360, (0, -360) → 360, (180, -180) → 360. Input angles with the same value always return 0 by definition: (0, 0) → 0, (-180, -180) → 0, (360, 360) → 0.

Returns the counter-clockwise angle span from start to end in radians.

Returns the angle span in the range of [0, 2π], 2π is a full circle. Full circle handling is a special case, because normalization of angles which describe a full circle would return 0 if treated as regular angles. e.g. (0, 2π) → 2π, (0, -2π) → 2π, (π, -π) → 2π. Input angles with the same value always return 0 by definition: (0, 0) → 0, (-π, -π) → 0, (2π, 2π) → 0.

ezdxf.math.arc_segment_count(radius: float, angle: float, sagitta: float) int

Returns the count of required segments for the approximation of an arc for a given maximum sagitta.

Parameters

• angle – angle span of the arc in radians

• sagitta – max. distance from the center of an arc segment to the center of its chord

Returns the chord length for an arc defined by radius and the sagitta.

Parameters

• sagitta – distance from the center of the arc to the center of its base

ezdxf.math.ellipse_param_span(start_param: float, end_param: float) float

Returns the counter-clockwise params span of an elliptic arc from start- to end param.

Returns the param span in the range [0, 2π], 2π is a full ellipse. Full ellipse handling is a special case, because normalization of params which describe a full ellipse would return 0 if treated as regular params. e.g. (0, 2π) → 2π, (0, -2π) → 2π, (π, -π) → 2π. Input params with the same value always return 0 by definition: (0, 0) → 0, (-π, -π) → 0, (2π, 2π) → 0.

Alias to function: `ezdxf.math.arc_angle_span_rad()`

ezdxf.math.has_matrix_2d_stretching(m: Matrix44) bool

Returns `True` if matrix m performs a non-uniform xy-scaling. Uniform scaling is not stretching in this context.

Does not check if the target system is a cartesian coordinate system, use the `Matrix44` property `is_cartesian` for that.

ezdxf.math.has_matrix_3d_stretching(m: Matrix44) bool

Returns `True` if matrix m performs a non-uniform xyz-scaling. Uniform scaling is not stretching in this context.

Does not check if the target system is a cartesian coordinate system, use the `Matrix44` property `is_cartesian` for that.

## 2D Graphic Functions¶

ezdxf.math.convex_hull_2d(points: Iterable[Union[Sequence[float], Vec2, Vec3]]) List[Vec2]

Returns 2D convex hull for points as list of `Vec2`. Returns a closed polyline, first vertex == last vertex.

Parameters

points – iterable of points, z-axis is ignored

ezdxf.math.distance_point_line_2d(point: Vec2, start: Vec2, end: Vec2) float

Returns the normal distance from point to 2D line defined by start- and end point.

ezdxf.math.intersect_polylines_2d(p1: Sequence[Vec2], p2: Sequence[Vec2], abs_tol=1e-10) List[Vec2]

Returns the intersection points for two polylines as list of `Vec2` objects, the list is empty if no intersection points exist. Does not return self intersection points of p1 or p2. Duplicate intersection points are removed from the result list, but the list does not have a particular order! You can sort the result list by `result.sort()` to introduce an order.

Parameters

New in version 0.17.2.

ezdxf.math.intersection_line_line_2d(line1: Sequence[Vec2], line2: Sequence[Vec2], virtual=True, abs_tol=1e-10) Optional[Vec2]

Compute the intersection of two lines in the xy-plane.

Parameters
• line1 – start- and end point of first line to test e.g. ((x1, y1), (x2, y2)).

• line2 – start- and end point of second line to test e.g. ((x3, y3), (x4, y4)).

• virtual`True` returns any intersection point, `False` returns only real intersection points.

• abs_tol – tolerance for intersection test.

Returns

`None` if there is no intersection point (parallel lines) or intersection point as `Vec2`

ezdxf.math.is_convex_polygon_2d(polygon: List[Vec2], *, strict=False, epsilon=1e-06) bool

Returns `True` if the 2D polygon is convex. This function works with open and closed polygons and clockwise or counter-clockwise vertex orientation. Coincident vertices will always be skipped and if argument strict is `True`, polygons with collinear vertices are not considered as convex.

This solution works only for simple non-self-intersecting polygons!

ezdxf.math.is_point_in_polygon_2d(point: Vec2, polygon: Sequence[Vec2], abs_tol=1e-10) int

Test if point is inside polygon. Returns `-1` (for outside) if the polygon is degenerated, no exception will be raised.

Parameters
Returns

`+1` for inside, `0` for on boundary line, `-1` for outside

ezdxf.math.is_point_left_of_line(point: Vec2, start: Vec2, end: Vec2, colinear=False) bool

Returns `True` if point is “left of line” defined by start- and end point, a colinear point is also “left of line” if argument colinear is `True`.

Parameters
ezdxf.math.is_point_on_line_2d(point: Vec2, start: Vec2, end: Vec2, ray=True, abs_tol=1e-10) bool

Returns `True` if point is on line.

Parameters
• point – 2D point to test as `Vec2`

• start – line definition point as `Vec2`

• end – line definition point as `Vec2`

• ray – if `True` point has to be on the infinite ray, if `False` point has to be on the line segment

• abs_tol – tolerance for on the line test

ezdxf.math.offset_vertices_2d(vertices: Iterable[Union[Sequence[float], Vec2, Vec3]], offset: float, closed: bool = False) Iterable[Vec2]

Yields vertices of the offset line to the shape defined by vertices. The source shape consist of straight segments and is located in the xy-plane, the z-axis of input vertices is ignored. Takes closed shapes into account if argument closed is `True`, which yields intersection of first and last offset segment as first vertex for a closed shape. For closed shapes the first and last vertex can be equal, else an implicit closing segment from last to first vertex is added. A shape with equal first and last vertex is not handled automatically as closed shape.

Warning

Adjacent collinear segments in opposite directions, same as a turn by 180 degree (U-turn), leads to unexpected results.

Parameters
• vertices – source shape defined by vertices

• offset – line offset perpendicular to direction of shape segments defined by vertices order, offset > `0` is ‘left’ of line segment, offset < `0` is ‘right’ of line segment

• closed`True` to handle as closed shape

```source = [(0, 0), (3, 0), (3, 3), (0, 3)]
result = list(offset_vertices_2d(source, offset=0.5, closed=True))
``` Example for a closed collinear shape, which creates 2 additional vertices and the first one has an unexpected location:

```source = [(0, 0), (0, 1), (0, 2), (0, 3)]
result = list(offset_vertices_2d(source, offset=0.5, closed=True))
``` ezdxf.math.point_to_line_relation(point: Vec2, start: Vec2, end: Vec2, abs_tol=1e-10) int

Returns `-1` if point is left line, `+1` if point is right of line and `0` if point is on the line. The line is defined by two vertices given as arguments start and end.

Parameters
ezdxf.math.rytz_axis_construction(d1: Vec3, d2: Vec3) Tuple[Vec3, Vec3, float]

The Rytz’s axis construction is a basic method of descriptive Geometry to find the axes, the semi-major axis and semi-minor axis, starting from two conjugated half-diameters.

Source: Wikipedia

Given conjugated diameter d1 is the vector from center C to point P and the given conjugated diameter d2 is the vector from center C to point Q. Center of ellipse is always `(0, 0, 0)`. This algorithm works for 2D/3D vectors.

Parameters
Returns

Tuple of (major axis, minor axis, ratio)

## 3D Graphic Functions¶

The free online book 3D Math Primer for Graphics and Game Development is a very good resource for learning vector math and other graphic related topics, it is easy to read for beginners and especially targeted to programmers.

ezdxf.math.basic_transformation(move: Union[Sequence[float], Vec2, Vec3] = (0, 0, 0), scale: Union[Sequence[float], Vec2, Vec3] = (1, 1, 1), z_rotation: float = 0)

Returns a combined transformation matrix for translation, scaling and rotation about the z-axis.

Parameters
• move – translation vector

• scale – x-, y- and z-axis scaling as float triplet, e.g. (2, 2, 1)

ezdxf.math.best_fit_normal(vertices: Iterable[Union[Sequence[float], Vec2, Vec3]]) Vec3

Returns the “best fit” normal for a plane defined by three or more vertices. This function tolerates imperfect plane vertices. Safe function to detect the extrusion vector of flat arbitrary polygons.

ezdxf.math.bezier_to_bspline(curves: Iterable[Union[Bezier3P, Bezier4P]])

Convert multiple quadratic or cubic Bèzier curves into a single cubic B-spline (`ezdxf.math.BSpline`). For good results the curves must be lined up seamlessly, i.e. the starting point of the following curve must be the same as the end point of the previous curve. G1 continuity or better at the connection points of the Bézier curves is required to get best results.

New in version 0.16.

ezdxf.math.closed_uniform_bspline(control_points: Iterable[Union[Sequence[float], Vec2, Vec3]], order: int = 4, weights: Optional[Iterable[float]] = None)

Creates an closed uniform (periodic) B-spline curve (open curve).

This B-spline does not pass any of the control points.

Parameters
• control_points – iterable of control points as `Vec3` compatible objects

• order – spline order (degree + 1)

• weights – iterable of weight values

ezdxf.math.cubic_bezier_bbox(curve: Bezier4P, *, abs_tol=1e-12)

Returns the `BoundingBox` of a cubic Bézier curve of type `Bezier4P`.

New in version 0.18.

ezdxf.math.cubic_bezier_from_3p(p1: Union[Sequence[float], Vec2, Vec3], p2: Union[Sequence[float], Vec2, Vec3], p3: Union[Sequence[float], Vec2, Vec3])

Returns a cubic Bèzier curve `Bezier4P` from three points. The curve starts at p1, goes through p2 and ends at p3. (source: pomax-2)

New in version 0.17.2.

ezdxf.math.cubic_bezier_from_arc(center: UVec = (0, 0, 0), radius: float = 1, start_angle: float = 0, end_angle: float = 360, segments: int = 1) Iterable[Bezier4P]

Returns an approximation for a circular 2D arc by multiple cubic Bézier-curves.

Parameters
• center – circle center as `Vec3` compatible object

• start_angle – start angle in degrees

• end_angle – end angle in degrees

• segments – count of Bèzier-curve segments, at least one segment for each quarter (90 deg), 1 for as few as possible.

ezdxf.math.cubic_bezier_from_ellipse(ellipse: ConstructionEllipse, segments: int = 1) Iterable[Bezier4P]

Returns an approximation for an elliptic arc by multiple cubic Bézier-curves.

Parameters
• ellipse – ellipse parameters as `ConstructionEllipse` object

• segments – count of Bèzier-curve segments, at least one segment for each quarter (π/2), 1 for as few as possible.

ezdxf.math.cubic_bezier_interpolation(points: Iterable[Union[Sequence[float], Vec2, Vec3]]) Iterable[Bezier4P]

Returns an interpolation curve for given data points as multiple cubic Bézier-curves. Returns n-1 cubic Bézier-curves for n given data points, curve i goes from point[i] to point[i+1].

Parameters

points – data points

ezdxf.math.distance_point_line_3d(point: Vec3, start: Vec3, end: Vec3) float

Returns the normal distance from point to 3D line defined by start- and end point.

ezdxf.math.estimate_end_tangent_magnitude(points: List[Vec3], method: str = 'chord') Tuple[float, float]

Estimate tangent magnitude of start- and end tangents.

Available estimation methods:

• “chord”: total chord length, curve approximation by straight segments

• “arc”: total arc length, curve approximation by arcs

• “bezier-n”: total length from cubic bezier curve approximation, n segments per section

Parameters
• points – start-, end- and passing points of curve

• method – tangent magnitude estimation method

ezdxf.math.estimate_tangents(points: List[Vec3], method: str = '5-points', normalize=True) List[Vec3]

Estimate tangents for curve defined by given fit points. Calculated tangents are normalized (unit-vectors).

Available tangent estimation methods:

• “3-points”: 3 point interpolation

• “5-points”: 5 point interpolation

• “bezier”: tangents from an interpolated cubic bezier curve

• “diff”: finite difference

Parameters
• points – start-, end- and passing points of curve

• method – tangent estimation method

• normalize – normalize tangents if `True`

Returns

tangents as list of `Vec3` objects

ezdxf.math.fit_points_to_cad_cv(fit_points: Iterable[Union[Sequence[float], Vec2, Vec3]], tangents: Optional[Iterable[Union[Sequence[float], Vec2, Vec3]]] = None, estimate: str = '5-p')

Returns a cubic `BSpline` from fit points as close as possible to common CAD applications like BricsCAD.

There exist infinite numerical correct solution for this setup, but some facts are known:

• Global curve interpolation with start- and end derivatives, e.g. 6 fit points creates 8 control vertices in BricsCAD

• Degree of B-spline is always 3, the stored degree is ignored, this is only valid for B-splines defined by fit points

• Knot parametrization method is “chord”

• Knot distribution is “natural”

The last missing parameter is the start- and end tangents estimation method used by BricsCAD, if these tangents are stored in the DXF file provide them as argument tangents as 2-tuple (start, end) and the interpolated control vertices will match the BricsCAD calculation, except for floating point imprecision.

If the end tangents are not given, the start- and ent tangent directions will be estimated. The argument estimate lets choose from different estimation methods (first 3 letters are significant):

• “3-points”: 3 point interpolation

• “5-points”: 5 point interpolation

• “bezier”: tangents from an interpolated cubic bezier curve

• “diff”: finite difference

The estimation method “5-p” yields the closest match to the BricsCAD rendering, but sometimes “bez” creates a better result.

If I figure out how BricsCAD estimates the end tangents directions, the argument estimate gets an additional value for that case. The existing estimation methods will perform the same way as now, except for bug fixes. But the default value may change, therefore set argument estimate to specific value to always get the same result in the future.

Parameters
• fit_points – points the spline is passing through

• tangents – start- and end tangent, default is autodetect

• estimate – tangent direction estimation method

Changed in version 0.16: removed unused arguments degree and method

ezdxf.math.fit_points_to_cubic_bezier(fit_points: Iterable[Union[Sequence[float], Vec2, Vec3]])

Returns a cubic `BSpline` from fit points without end tangents.

This function uses the cubic Bèzier interpolation to create multiple Bèzier curves and combine them into a single B-spline, this works for short simple splines better than the `fit_points_to_cad_cv()`, but is worse for longer and more complex splines.

Parameters

fit_points – points the spline is passing through

New in version 0.16.

ezdxf.math.global_bspline_interpolation(fit_points: Iterable[Union[Sequence[float], Vec2, Vec3]], degree: int = 3, tangents: Optional[Iterable[Union[Sequence[float], Vec2, Vec3]]] = None, method: str = 'chord')

B-spline interpolation by the Global Curve Interpolation. Given are the fit points and the degree of the B-spline. The function provides 3 methods for generating the parameter vector t:

• “uniform”: creates a uniform t vector, from 0 to 1 evenly spaced, see uniform method

• “chord”, “distance”: creates a t vector with values proportional to the fit point distances, see chord length method

• “centripetal”, “sqrt_chord”: creates a t vector with values proportional to the fit point sqrt(distances), see centripetal method

• “arc”: creates a t vector with values proportional to the arc length between fit points.

It is possible to constraint the curve by tangents, by start- and end tangent if only two tangents are given or by one tangent for each fit point.

If tangents are given, they represent 1st derivatives and should be scaled if they are unit vectors, if only start- and end tangents given the function `estimate_end_tangent_magnitude()` helps with an educated guess, if all tangents are given, scaling by chord length is a reasonable choice (Piegl & Tiller).

Parameters
• fit_points – fit points of B-spline, as list of `Vec3` compatible objects

• tangents – if only two vectors are given, take the first and the last vector as start- and end tangent constraints or if for all fit points a tangent is given use all tangents as interpolation constraints (optional)

• degree – degree of B-spline

• method – calculation method for parameter vector t

Returns

`BSpline`

ezdxf.math.have_bezier_curves_g1_continuity(b1: Union[Bezier3P, Bezier4P], b2: Union[Bezier3P, Bezier4P], g1_tol: float = 0.0001) bool

Return `True` if the given adjacent bezier curves have G1 continuity.

New in version 0.16.

ezdxf.math.intersect_polylines_3d(p1: Sequence[Vec3], p2: Sequence[Vec3], abs_tol=1e-10) List[Vec3]

Returns the intersection points for two polylines as list of `Vec3` objects, the list is empty if no intersection points exist. Does not return self intersection points of p1 or p2. Duplicate intersection points are removed from the result list, but the list does not have a particular order! You can sort the result list by `result.sort()` to introduce an order.

Parameters

New in version 0.17.2.

ezdxf.math.intersection_line_line_3d(line1: Sequence[Vec3], line2: Sequence[Vec3], virtual: bool = True, abs_tol: float = 1e-10) Optional[Vec3]

Returns the intersection point of two 3D lines, returns `None` if lines do not intersect.

Parameters
• line1 – first line as tuple of two points as `Vec3` objects

• line2 – second line as tuple of two points as `Vec3` objects

• virtual`True` returns any intersection point, `False` returns only real intersection points

• abs_tol – absolute tolerance for comparisons

New in version 0.17.2.

ezdxf.math.intersection_line_polygon_3d(start: Vec3, end: Vec3, polygon: Iterable[Vec3], *, coplanar=True, boundary=True, abs_tol=1e-09) Optional[Vec3]

Returns the intersection point of the 3D line form start to end and the given polygon.

Parameters
• start – start point of 3D line as `Vec3`

• end – end point of 3D line as `Vec3`

• polygon – 3D polygon as iterable of `Vec3`

• coplanar – if `True` a coplanar start- or end point as intersection point is valid

• boundary – if `True` an intersection point at the polygon boundary line is valid

• abs_tol – absolute tolerance for comparisons

New in version 0.18.

ezdxf.math.intersection_ray_polygon_3d(origin: Vec3, direction: Vec3, polygon: Iterable[Vec3], *, boundary=True, abs_tol=1e-09) Optional[Vec3]

Returns the intersection point of the infinite 3D ray defined by origin and the direction vector and the given polygon.

Parameters
• origin – origin point of the 3D ray as `Vec3`

• direction – direction vector of the 3D ray as `Vec3`

• polygon – 3D polygon as iterable of `Vec3`

• boundary – if `True` intersection points at the polygon boundary line are valid

• abs_tol – absolute tolerance for comparisons

New in version 0.18.

ezdxf.math.intersection_ray_ray_3d(ray1: Sequence[Vec3], ray2: Sequence[Vec3], abs_tol=1e-10) Sequence[Vec3]

Calculate intersection of two 3D rays, returns a 0-tuple for parallel rays, a 1-tuple for intersecting rays and a 2-tuple for not intersecting and not parallel rays with points of the closest approach on each ray.

Parameters
• ray1 – first ray as tuple of two points as `Vec3` objects

• ray2 – second ray as tuple of two points as `Vec3` objects

• abs_tol – absolute tolerance for comparisons

ezdxf.math.is_planar_face(face: Sequence[Vec3], abs_tol=1e-09) bool

Returns `True` if sequence of vectors is a planar face.

Parameters
ezdxf.math.linear_vertex_spacing(start: Vec3, end: Vec3, count: int) List[Vec3]

Returns count evenly spaced vertices from start to end.

ezdxf.math.local_cubic_bspline_interpolation(fit_points: Iterable[Union[Sequence[float], Vec2, Vec3]], method: str = '5-points', tangents: Optional[Iterable[Union[Sequence[float], Vec2, Vec3]]] = None)

B-spline interpolation by ‘Local Cubic Curve Interpolation’, which creates B-spline from fit points and estimated tangent direction at start-, end- and passing points.

Source: Piegl & Tiller: “The NURBS Book” - chapter 9.3.4

Available tangent estimation methods:

• “3-points”: 3 point interpolation

• “5-points”: 5 point interpolation

• “bezier”: cubic bezier curve interpolation

• “diff”: finite difference

or pass pre-calculated tangents, which overrides tangent estimation.

Parameters
• fit_points – all B-spline fit points as `Vec3` compatible objects

• method – tangent estimation method

• tangents – tangents as `Vec3` compatible objects (optional)

Returns

`BSpline`

ezdxf.math.normal_vector_3p(a: Vec3, b: Vec3, c: Vec3) Vec3

Returns normal vector for 3 points, which is the normalized cross product for: `a->b x a->c`.

ezdxf.math.open_uniform_bspline(control_points: Iterable[Union[Sequence[float], Vec2, Vec3]], order: int = 4, weights: Optional[Iterable[float]] = None)

Creates an open uniform (periodic) B-spline curve (open curve).

This is an unclamped curve, which means the curve passes none of the control points.

Parameters
• control_points – iterable of control points as `Vec3` compatible objects

• order – spline order (degree + 1)

• weights – iterable of weight values

Returns the `BoundingBox` of a quadratic Bézier curve of type `Bezier3P`.

New in version 0.18.

ezdxf.math.quadratic_bezier_from_3p(p1: Union[Sequence[float], Vec2, Vec3], p2: Union[Sequence[float], Vec2, Vec3], p3: Union[Sequence[float], Vec2, Vec3])

Returns a quadratic Bèzier curve `Bezier3P` from three points. The curve starts at p1, goes through p2 and ends at p3. (source: pomax-2)

New in version 0.17.2.

Convert quadratic Bèzier curves (`ezdxf.math.Bezier3P`) into cubic Bèzier curves (`ezdxf.math.Bezier4P`).

New in version 0.16.

ezdxf.math.rational_bspline_from_arc(center: Vec3 = (0, 0), radius: float = 1, start_angle: float = 0, end_angle: float = 360, segments: int = 1)

Returns a rational B-splines for a circular 2D arc.

Parameters
• center – circle center as `Vec3` compatible object

• start_angle – start angle in degrees

• end_angle – end angle in degrees

• segments – count of spline segments, at least one segment for each quarter (90 deg), default is 1, for as few as needed.

ezdxf.math.rational_bspline_from_ellipse(ellipse: ConstructionEllipse, segments: int = 1)

Returns a rational B-splines for an elliptic arc.

Parameters
• ellipse – ellipse parameters as `ConstructionEllipse` object

• segments – count of spline segments, at least one segment for each quarter (π/2), default is 1, for as few as needed.

ezdxf.math.safe_normal_vector(vertices: Sequence[Vec3]) Vec3

Safe function to detect the normal vector for a face or polygon defined by 3 or more vertices.

ezdxf.math.spherical_envelope(points: Sequence[Union[Sequence[float], Vec2, Vec3]]) Tuple[Vec3, float]

Calculate the spherical envelope for the given points. Returns the centroid (a.k.a. geometric center) and the radius of the enclosing sphere.

Note

The result does not represent the minimal bounding sphere!

New in version 0.18.

ezdxf.math.split_bezier(control_points: Sequence[T], t: float) Tuple[List[T], List[T]]

Split Bèzier curves at parameter t by de Casteljau’s algorithm (source: pomax-1). Returns the control points for two new Bèzier curves of the same degree and type as the input curve.

Parameters
• control_points – of the Bèzier curve as `Vec2` or `Vec3` objects. Requires 3 points for a quadratic curve, 4 points for a cubic curve , …

• t – parameter where to split the curve in the range [0, 1]

New in version 0.17.2.

ezdxf.math.split_polygon_by_plane(polygon: Iterable[Vec3], plane: Plane, *, coplanar=True, abs_tol=1e-09) Tuple[Sequence[Vec3], Sequence[Vec3]]

Split a convex polygon by the given plane if needed. Returns a tuple of front- and back vertices (front, back). Returns also coplanar polygons if the argument coplanar is `True`, the coplanar vertices goes into either front or back depending on their orientation with respect to this plane.

New in version 0.18.

ezdxf.math.subdivide_face(face: Sequence[Union[Vec2, Vec3]], quads: bool = True) Iterable[Tuple[Vec3, ...]]

Yields new subdivided faces. Creates new faces from subdivided edges and the face midpoint by linear interpolation.

Parameters
ezdxf.math.subdivide_ngons(faces: Iterable[Sequence[Union[Vec2, Vec3]]], max_vertex_count=4) Iterable[Sequence[Vec3]]

Yields only triangles or quad faces, subdivides ngons into triangles.

Parameters

## Transformation Classes¶

### OCS Class¶

class ezdxf.math.OCS(extrusion: Union[Sequence[float], Vec2, Vec3] = Vec3(0.0, 0.0, 1.0))

Establish an OCS for a given extrusion vector.

Parameters

extrusion – extrusion vector.

ux

x-axis unit vector

uy

y-axis unit vector

uz

z-axis unit vector

from_wcs(point: Union[Sequence[float], Vec2, Vec3]) Union[Sequence[float], Vec2, Vec3]

Returns OCS vector for WCS point.

points_from_wcs(points: Iterable[Union[Sequence[float], Vec2, Vec3]]) Iterable[Union[Sequence[float], Vec2, Vec3]]

Returns iterable of OCS vectors from WCS points.

to_wcs(point: Union[Sequence[float], Vec2, Vec3]) Union[Sequence[float], Vec2, Vec3]

Returns WCS vector for OCS point.

points_to_wcs(points: Iterable[Union[Sequence[float], Vec2, Vec3]]) Iterable[Union[Sequence[float], Vec2, Vec3]]

Returns iterable of WCS vectors for OCS points.

render_axis(layout: BaseLayout, length: float = 1, colors: RGB = (1, 3, 5)) None

Render axis as 3D lines into a layout.

### UCS Class¶

class ezdxf.math.UCS(origin: Union[Sequence[float], Vec2, Vec3] = (0, 0, 0), ux: Optional[Union[Sequence[float], Vec2, Vec3]] = None, uy: Optional[Union[Sequence[float], Vec2, Vec3]] = None, uz: Optional[Union[Sequence[float], Vec2, Vec3]] = None)

Establish an user coordinate system (UCS). The UCS is defined by the origin and two unit vectors for the x-, y- or z-axis, all axis in WCS. The missing axis is the cross product of the given axis.

If x- and y-axis are `None`: ux = `(1, 0, 0)`, uy = `(0, 1, 0)`, uz = `(0, 0, 1)`.

Unit vectors don’t have to be normalized, normalization is done at initialization, this is also the reason why scaling gets lost by copying or rotating.

Parameters
• origin – defines the UCS origin in world coordinates

• ux – defines the UCS x-axis as vector in WCS

• uy – defines the UCS y-axis as vector in WCS

• uz – defines the UCS z-axis as vector in WCS

ux

x-axis unit vector

uy

y-axis unit vector

uz

z-axis unit vector

is_cartesian

Returns `True` if cartesian coordinate system.

copy() UCS

Returns a copy of this UCS.

to_wcs(point: Vec3) Vec3

Returns WCS point for UCS point.

points_to_wcs(points: Iterable[Vec3]) Iterable[Vec3]

Returns iterable of WCS vectors for UCS points.

direction_to_wcs(vector: Vec3) Vec3

Returns WCS direction for UCS vector without origin adjustment.

from_wcs(point: Vec3) Vec3

Returns UCS point for WCS point.

points_from_wcs(points: Iterable[Vec3]) Iterable[Vec3]

Returns iterable of UCS vectors from WCS points.

direction_from_wcs(vector: Vec3) Vec3

Returns UCS vector for WCS vector without origin adjustment.

to_ocs(point: Vec3) Vec3

Returns OCS vector for UCS point.

The `OCS` is defined by the z-axis of the `UCS`.

points_to_ocs(points: Iterable[Vec3]) Iterable[Vec3]

Returns iterable of OCS vectors for UCS points.

The `OCS` is defined by the z-axis of the `UCS`.

Parameters

points – iterable of UCS vertices

to_ocs_angle_deg(angle: float) float

Transforms angle from current UCS to the parent coordinate system (most likely the WCS) including the transformation to the OCS established by the extrusion vector `UCS.uz`.

Parameters

angle – in UCS in degrees

transform(m: Matrix44) UCS

General inplace transformation interface, returns self (floating interface).

Parameters

m – 4x4 transformation matrix (`ezdxf.math.Matrix44`)

rotate(axis: Union[Sequence[float], Vec2, Vec3], angle: float) UCS

Returns a new rotated UCS, with the same origin as the source UCS. The rotation vector is located in the origin and has WCS coordinates e.g. (0, 0, 1) is the WCS z-axis as rotation vector.

Parameters
• axis – arbitrary rotation axis as vector in WCS

• angle – rotation angle in radians

rotate_local_x(angle: float) UCS

Returns a new rotated UCS, rotation axis is the local x-axis.

Parameters

angle – rotation angle in radians

rotate_local_y(angle: float) UCS

Returns a new rotated UCS, rotation axis is the local y-axis.

Parameters

angle – rotation angle in radians

rotate_local_z(angle: float) UCS

Returns a new rotated UCS, rotation axis is the local z-axis.

Parameters

angle – rotation angle in radians

shift(delta: Union[Sequence[float], Vec2, Vec3]) UCS

Shifts current UCS by delta vector and returns self.

Parameters

delta – shifting vector

moveto(location: Union[Sequence[float], Vec2, Vec3]) UCS

Place current UCS at new origin location and returns self.

Parameters

location – new origin in WCS

static from_x_axis_and_point_in_xy(origin: Union[Sequence[float], Vec2, Vec3], axis: Union[Sequence[float], Vec2, Vec3], point: Union[Sequence[float], Vec2, Vec3]) UCS

Returns an new `UCS` defined by the origin, the x-axis vector and an arbitrary point in the xy-plane.

Parameters
• origin – UCS origin as (x, y, z) tuple in WCS

• axis – x-axis vector as (x, y, z) tuple in WCS

• point – arbitrary point unlike the origin in the xy-plane as (x, y, z) tuple in WCS

static from_x_axis_and_point_in_xz(origin: Union[Sequence[float], Vec2, Vec3], axis: Union[Sequence[float], Vec2, Vec3], point: Union[Sequence[float], Vec2, Vec3]) UCS

Returns an new `UCS` defined by the origin, the x-axis vector and an arbitrary point in the xz-plane.

Parameters
• origin – UCS origin as (x, y, z) tuple in WCS

• axis – x-axis vector as (x, y, z) tuple in WCS

• point – arbitrary point unlike the origin in the xz-plane as (x, y, z) tuple in WCS

static from_y_axis_and_point_in_xy(origin: Union[Sequence[float], Vec2, Vec3], axis: Union[Sequence[float], Vec2, Vec3], point: Union[Sequence[float], Vec2, Vec3]) UCS

Returns an new `UCS` defined by the origin, the y-axis vector and an arbitrary point in the xy-plane.

Parameters
• origin – UCS origin as (x, y, z) tuple in WCS

• axis – y-axis vector as (x, y, z) tuple in WCS

• point – arbitrary point unlike the origin in the xy-plane as (x, y, z) tuple in WCS

static from_y_axis_and_point_in_yz(origin: Union[Sequence[float], Vec2, Vec3], axis: Union[Sequence[float], Vec2, Vec3], point: Union[Sequence[float], Vec2, Vec3]) UCS

Returns an new `UCS` defined by the origin, the y-axis vector and an arbitrary point in the yz-plane.

Parameters
• origin – UCS origin as (x, y, z) tuple in WCS

• axis – y-axis vector as (x, y, z) tuple in WCS

• point – arbitrary point unlike the origin in the yz-plane as (x, y, z) tuple in WCS

static from_z_axis_and_point_in_xz(origin: Union[Sequence[float], Vec2, Vec3], axis: Union[Sequence[float], Vec2, Vec3], point: Union[Sequence[float], Vec2, Vec3]) UCS

Returns an new `UCS` defined by the origin, the z-axis vector and an arbitrary point in the xz-plane.

Parameters
• origin – UCS origin as (x, y, z) tuple in WCS

• axis – z-axis vector as (x, y, z) tuple in WCS

• point – arbitrary point unlike the origin in the xz-plane as (x, y, z) tuple in WCS

static from_z_axis_and_point_in_yz(origin: Union[Sequence[float], Vec2, Vec3], axis: Union[Sequence[float], Vec2, Vec3], point: Union[Sequence[float], Vec2, Vec3]) UCS

Returns an new `UCS` defined by the origin, the z-axis vector and an arbitrary point in the yz-plane.

Parameters
• origin – UCS origin as (x, y, z) tuple in WCS

• axis – z-axis vector as (x, y, z) tuple in WCS

• point – arbitrary point unlike the origin in the yz-plane as (x, y, z) tuple in WCS

render_axis(layout: BaseLayout, length: float = 1, colors: Tuple[int, int, int] = (1, 3, 5))

Render axis as 3D lines into a layout.

### Matrix44¶

class ezdxf.math.Matrix44(*args)

This is a 4x4 transformation matrix.

The utility functions for constructing transformations and transforming vectors and points assumes that vectors are stored as row vectors, meaning when multiplied, transformations are applied left to right (e.g. vAB transforms v by A then by B).

Matrix44 initialization:

• `Matrix44()` returns the identity matrix.

• `Matrix44(values)` values is an iterable with the 16 components of the matrix.

• `Matrix44(row1, row2, row3, row4)` four rows, each row with four values.

__repr__() str

Returns the representation string of the matrix: `Matrix44((col0, col1, col2, col3), (...), (...), (...))`

get_row(row: int) Tuple[float, ...]

Get row as list of four float values.

Parameters

row – row index [0 .. 3]

set_row(row: int, values: Sequence[float]) None

Sets the values in a row.

Parameters
• row – row index [0 .. 3]

• values – iterable of four row values

get_col(col: int) Tuple[float, ...]

Returns a column as a tuple of four floats.

Parameters

col – column index [0 .. 3]

set_col(col: int, values: Sequence[float])

Sets the values in a column.

Parameters
• col – column index [0 .. 3]

• values – iterable of four column values

copy()

Returns a copy of same type.

__copy__()

Returns a copy of same type.

classmethod scale(sx: float, sy: Optional[float] = None, sz: Optional[float] = None)

Returns a scaling transformation matrix. If sy is `None`, sy = sx, and if sz is `None` sz = sx.

classmethod translate(dx: float, dy: float, dz: float)

Returns a translation matrix for translation vector (dx, dy, dz).

classmethod x_rotate(angle: float)

Returns a rotation matrix about the x-axis.

Parameters

angle – rotation angle in radians

classmethod y_rotate(angle: float)

Returns a rotation matrix about the y-axis.

Parameters

angle – rotation angle in radians

classmethod z_rotate(angle: float)

Returns a rotation matrix about the z-axis.

Parameters

angle – rotation angle in radians

classmethod axis_rotate(axis: UVec, angle: float)

Returns a rotation matrix about an arbitrary axis.

Parameters
• axis – rotation axis as `(x, y, z)` tuple or `Vec3` object

• angle – rotation angle in radians

classmethod xyz_rotate(angle_x: float, angle_y: float, angle_z: float)

Returns a rotation matrix for rotation about each axis.

Parameters

classmethod shear_xy(angle_x: float = 0, angle_y: float = 0)

Returns a translation matrix for shear mapping (visually similar to slanting) in the xy-plane.

Parameters
• angle_x – slanting angle in x direction in radians

• angle_y – slanting angle in y direction in radians

classmethod perspective_projection(left: float, right: float, top: float, bottom: float, near: float, far: float)

Returns a matrix for a 2D projection.

Parameters
• left – Coordinate of left of screen

• right – Coordinate of right of screen

• top – Coordinate of the top of the screen

• bottom – Coordinate of the bottom of the screen

• near – Coordinate of the near clipping plane

• far – Coordinate of the far clipping plane

classmethod perspective_projection_fov(fov: float, aspect: float, near: float, far: float)

Returns a matrix for a 2D projection.

Parameters
• fov – The field of view (in radians)

• aspect – The aspect ratio of the screen (width / height)

• near – Coordinate of the near clipping plane

• far – Coordinate of the far clipping plane

static chain(*matrices: Matrix44)

Compose a transformation matrix from one or more matrices.

static ucs(ux: Vec3 = Vec3(1.0, 0.0, 0.0), uy: Vec3 = Vec3(0.0, 1.0, 0.0), uz: Vec3 = Vec3(0.0, 0.0, 1.0), origin: Vec3 = Vec3(0.0, 0.0, 0.0))

Returns a matrix for coordinate transformation from WCS to UCS. For transformation from UCS to WCS, transpose the returned matrix.

Parameters
• ux – x-axis for UCS as unit vector

• uy – y-axis for UCS as unit vector

• uz – z-axis for UCS as unit vector

• origin – UCS origin as location vector

__hash__()

Return hash(self).

__getitem__(index: Tuple[int, int])

Get (row, column) element.

__setitem__(index: Tuple[int, int], value: float)

Set (row, column) element.

__iter__() Iterator[float]

Iterates over all matrix values.

rows() Iterable[Tuple[float, ...]]

Iterate over rows as 4-tuples.

columns() Iterable[Tuple[float, ...]]

Iterate over columns as 4-tuples.

__mul__(other: Matrix44)

Returns a new matrix as result of the matrix multiplication with another matrix.

__imul__(other: Matrix44)

Inplace multiplication with another matrix.

transform(vector: UVec) Vec3

Returns a transformed vertex.

transform_direction(vector: UVec, normalize=False) Vec3

Returns a transformed direction vector without translation.

transform_vertices(vectors: Iterable[UVec]) Iterable[Vec3]

Returns an iterable of transformed vertices.

transform_directions(vectors: Iterable[UVec], normalize=False) Iterable[Vec3]

Returns an iterable of transformed direction vectors without translation.

transpose() None

Swaps the rows for columns inplace.

determinant() float

Returns determinant.

inverse() None

Calculates the inverse of the matrix.

Raises

ZeroDivisionError – if matrix has no inverse.

property is_cartesian: bool

Returns `True` if target coordinate system is a right handed orthogonal coordinate system.

property is_orthogonal: bool

Returns `True` if target coordinate system has orthogonal axis.

Does not check for left- or right handed orientation, any orientation of the axis valid.

## Basic Construction Classes¶

### UVec¶

class ezdxf.math.UVec

Type alias for `Union[Sequence[float], Vec2, Vec3]`

### Vec3¶

class ezdxf.math.Vec3(*args)

This is an immutable universal 3D vector object. This class is optimized for universality not for speed. Immutable means you can’t change (x, y, z) components after initialization:

```v1 = Vec3(1, 2, 3)
v2 = v1
v2.z = 7  # this is not possible, raises AttributeError
v2 = Vec3(v2.x, v2.y, 7)  # this creates a new Vec3() object
assert v1.z == 3  # and v1 remains unchanged
```

`Vec3` initialization:

• `Vec3()`, returns `Vec3(0, 0, 0)`

• `Vec3((x, y))`, returns `Vec3(x, y, 0)`

• `Vec3((x, y, z))`, returns `Vec3(x, y, z)`

• `Vec3(x, y)`, returns `Vec3(x, y, 0)`

• `Vec3(x, y, z)`, returns `Vec3(x, y, z)`

Addition, subtraction, scalar multiplication and scalar division left and right handed are supported:

```v = Vec3(1, 2, 3)
v + (1, 2, 3) == Vec3(2, 4, 6)
(1, 2, 3) + v == Vec3(2, 4, 6)
v - (1, 2, 3) == Vec3(0, 0, 0)
(1, 2, 3) - v == Vec3(0, 0, 0)
v * 3 == Vec3(3, 6, 9)
3 * v == Vec3(3, 6, 9)
Vec3(3, 6, 9) / 3 == Vec3(1, 2, 3)
-Vec3(1, 2, 3) == (-1, -2, -3)
```

Comparison between vectors and vectors or tuples is supported:

```Vec3(1, 2, 3) < Vec3 (2, 2, 2)
(1, 2, 3) < tuple(Vec3(2, 2, 2))  # conversion necessary
Vec3(1, 2, 3) == (1, 2, 3)

bool(Vec3(1, 2, 3)) is True
bool(Vec3(0, 0, 0)) is False
```
x

x-axis value

y

y-axis value

z

z-axis value

xy

Vec3 as `(x, y, 0)`, projected on the xy-plane.

xyz

Vec3 as `(x, y, z)` tuple.

vec2

Real 2D vector as `Vec2` object.

magnitude

Length of vector.

magnitude_xy

Length of vector in the xy-plane.

magnitude_square

Square length of vector.

is_null

`Vec3(0, 0, 0)`. Has a fixed absolute testing tolerance of 1e-12!

Type

`True` if all components are close to zero

angle

Angle between vector and x-axis in the xy-plane in radians.

angle_deg

Returns angle of vector and x-axis in the xy-plane in degrees.

spatial_angle

Spatial angle between vector and x-axis in radians.

spatial_angle_deg

Spatial angle between vector and x-axis in degrees.

__str__() str

Return `'(x, y, z)'` as string.

__repr__() str

Return `'Vec3(x, y, z)'` as string.

__len__() int

Returns always `3`.

__hash__() int

Returns hash value of vector, enables the usage of vector as key in `set` and `dict`.

copy() Vec3

Returns a copy of vector as `Vec3` object.

__copy__() Vec3

Returns a copy of vector as `Vec3` object.

__deepcopy__(memodict: dict) Vec3

`copy.deepcopy()` support.

__getitem__(index: int) float

Support for indexing:

• v is v.x

• v is v.y

• v is v.z

__iter__() Iterator[float]

Returns iterable of x-, y- and z-axis.

__abs__() float

Returns length (magnitude) of vector.

replace(x: Optional[float] = None, y: Optional[float] = None, z: Optional[float] = None) Vec3

Returns a copy of vector with replaced x-, y- and/or z-axis.

classmethod generate(items: Iterable['UVec']) Iterable['Vec3']

Returns an iterable of `Vec3` objects.

classmethod list(items: Iterable['UVec']) List['Vec3']

Returns a list of `Vec3` objects.

classmethod tuple(items: Iterable['UVec']) Sequence['Vec3']

Returns a tuple of `Vec3` objects.

classmethod from_angle(angle: float, length: float = 1.0) Vec3

Returns a `Vec3` object from angle in radians in the xy-plane, z-axis = `0`.

classmethod from_deg_angle(angle: float, length: float = 1.0) Vec3

Returns a `Vec3` object from angle in degrees in the xy-plane, z-axis = `0`.

orthogonal(ccw: bool = True) Vec3

Returns orthogonal 2D vector, z-axis is unchanged.

Parameters

ccw – counter clockwise if `True` else clockwise

lerp(other: UVec, factor=0.5) Vec3

Returns linear interpolation between self and other.

Parameters
• other – end point as `Vec3` compatible object

• factor – interpolation factor (`0` = self, `1` = other, `0.5` = mid point)

is_parallel(other: Vec3, *, rel_tol: float = 1e-09, abs_tol: float = 1e-12) bool

Returns `True` if self and other are parallel to vectors.

project(other: UVec) Vec3

Returns projected vector of other onto self.

normalize(length: float = 1.0) Vec3

Returns normalized vector, optional scaled by length.

reversed() Vec3

Returns negated vector (-self).

isclose(other: UVec, *, rel_tol: float = 1e-09, abs_tol: float = 1e-12) bool

Returns `True` if self is close to other. Uses `math.isclose()` to compare all axis.

Learn more about the `math.isclose()` function in PEP 485.

__neg__() Vec3

Returns negated vector (-self).

__bool__() bool

Returns `True` if vector is not `(0, 0, 0)`.

__eq__(other: UVec) bool

Equal operator.

Parameters

other`Vec3` compatible object

__lt__(other: UVec) bool

Lower than operator.

Parameters

other`Vec3` compatible object

Add `Vec3` operator: self + other.

RAdd `Vec3` operator: other + self.

__sub__(other: UVec) Vec3

Sub `Vec3` operator: self - other.

__rsub__(other: UVec) Vec3

RSub `Vec3` operator: other - self.

__mul__(other: float) Vec3

Scalar Mul operator: self * other.

__rmul__(other: float) Vec3

Scalar RMul operator: other * self.

__truediv__(other: float) Vec3

Scalar Div operator: self / other.

dot(other: UVec) float

Dot operator: self . other

Parameters

other`Vec3` compatible object

cross(other: UVec) Vec3

Dot operator: self x other

Parameters

other`Vec3` compatible object

distance(other: UVec) float

Returns distance between self and other vector.

Returns counter clockwise angle in radians about self from base to target when projected onto the plane defined by self as the normal vector.

Parameters
• base – base vector, defines angle 0

• target – target vector

angle_between(other: UVec) float

Returns angle between self and other in radians. +angle is counter clockwise orientation.

Parameters

other`Vec3` compatible object

rotate(angle: float) Vec3

Returns vector rotated about angle around the z-axis.

Parameters

rotate_deg(angle: float) Vec3

Returns vector rotated about angle around the z-axis.

Parameters

angle – angle in degrees

static sum(items: Iterable['UVec']) Vec3

ezdxf.math.X_AXIS

`Vec3(1, 0, 0)`

ezdxf.math.Y_AXIS

`Vec3(0, 1, 0)`

ezdxf.math.Z_AXIS

`Vec3(0, 0, 1)`

ezdxf.math.NULLVEC

`Vec3(0, 0, 0)`

### Vec2¶

class ezdxf.math.Vec2(v=(0.0, 0.0), y=None)

`Vec2` represents a special 2D vector `(x, y)`. The `Vec2` class is optimized for speed and not immutable, `iadd()`, `isub()`, `imul()` and `idiv()` modifies the vector itself, the `Vec3` class returns a new object.

`Vec2` initialization accepts float-tuples `(x, y[, z])`, two floats or any object providing `x` and `y` attributes like `Vec2` and `Vec3` objects.

Parameters
• v – vector object with `x` and `y` attributes/properties or a sequence of float `[x, y, ...]` or x-axis as float if argument y is not `None`

• y – second float for `Vec2(x, y)`

`Vec2` implements a subset of `Vec3`.

### Plane¶

class ezdxf.math.Plane(normal: Vec3, distance: float)

Represents a plane in 3D space as normal vector and the perpendicular distance from origin.

normal

Normal vector of the plane.

distance_from_origin

The (perpendicular) distance of the plane from origin (0, 0, 0).

vector

Returns the location vector.

classmethod from_3p(a: Vec3, b: Vec3, c: Vec3)

Returns a new plane from 3 points in space.

classmethod from_vector(vector: Union[Sequence[float], Vec2, Vec3])

Returns a new plane from the given location vector.

copy()

Returns a copy of the plane.

signed_distance_to(v: Vec3) float

Returns signed distance of vertex v to plane, if distance is > 0, v is in ‘front’ of plane, in direction of the normal vector, if distance is < 0, v is at the ‘back’ of the plane, in the opposite direction of the normal vector.

distance_to(v: Vec3) float

Returns absolute (unsigned) distance of vertex v to plane.

is_coplanar_vertex(v: Vec3, abs_tol=1e-09) bool

Returns `True` if vertex v is coplanar, distance from plane to vertex v is 0.

is_coplanar_plane(p: Plane, abs_tol=1e-09) bool

Returns `True` if plane p is coplanar, normal vectors in same or opposite direction.

intersect_line(start: Vec3, end: Vec3, *, coplanar=True, abs_tol=1e-09) Optional[Vec3]

Returns the intersection point of the 3D line from start to end and this plane or `None` if there is no intersection. If the argument coplanar is `False` the start- or end point of the line are ignored as intersection points.

New in version 0.18.

intersect_ray(origin: Vec3, direction: Vec3) Optional[Vec3]

Returns the intersection point of the infinite 3D ray defined by origin and the direction vector and this plane or `None` if there is no intersection. A coplanar ray does not intersect the plane!

New in version 0.18.

### BoundingBox¶

class ezdxf.math.BoundingBox(vertices: Optional[Iterable[Union[Sequence[float], Vec2, Vec3]]] = None)

3D bounding box.

Parameters

vertices – iterable of `(x, y, z)` tuples or `Vec3` objects

extmin

“lower left” corner of bounding box

extmax

“upper right” corner of bounding box

property is_empty: bool

Returns `True` if the bounding box is empty or the bounding box has a size of 0 in any or all dimensions or is undefined.

Changed in version 0.18.

property has_data: bool

Returns `True` if the bonding box has known limits.

property size

Returns size of bounding box.

property center

Returns center of bounding box.

inside(vertex: Union[Sequence[float], Vec2, Vec3]) bool

Returns `True` if vertex is inside this bounding box.

Vertices at the box border are inside!

any_inside(vertices: Iterable[Union[Sequence[float], Vec2, Vec3]]) bool

Returns `True` if any vertex is inside this bounding box.

Vertices at the box border are inside!

all_inside(vertices: Iterable[Union[Sequence[float], Vec2, Vec3]]) bool

Returns `True` if all vertices are inside this bounding box.

Vertices at the box border are inside!

has_intersection(other: AbstractBoundingBox) bool

Returns `True` if this bounding box intersects with other but does not include touching bounding boxes, see also `has_overlap()`:

```bbox1 = BoundingBox([(0, 0, 0), (1, 1, 1)])
bbox2 = BoundingBox([(1, 1, 1), (2, 2, 2)])
assert bbox1.has_intersection(bbox2) is False
```
has_overlap(other: AbstractBoundingBox) bool

Returns `True` if this bounding box intersects with other but in contrast to `has_intersection()` includes touching bounding boxes too:

```bbox1 = BoundingBox([(0, 0, 0), (1, 1, 1)])
bbox2 = BoundingBox([(1, 1, 1), (2, 2, 2)])
assert bbox1.has_overlap(bbox2) is True
```
contains(other: AbstractBoundingBox) bool

Returns `True` if the other bounding box is completely inside of this bounding box.

New in version 0.17.2.

extend(vertices: Iterable[Union[Sequence[float], Vec2, Vec3]]) None

Extend bounds by vertices.

Parameters

vertices – iterable of vertices

union(other: AbstractBoundingBox)

Returns a new bounding box as union of this and other bounding box.

intersection(other: AbstractBoundingBox)

Returns the bounding box of the intersection cube of both 3D bounding boxes. Returns an empty bounding box if the intersection volume is 0.

rect_vertices() Tuple[Vec2, ...]

Returns the corners of the bounding box in the xy-plane as `Vec2` objects.

cube_vertices() Tuple[Vec3, ...]

Returns the 3D corners of the bounding box as `Vec3` objects.

grow(value: float) None

Grow or shrink the bounding box by an uniform value in x, y and z-axis. A negative value shrinks the bounding box. Raises `ValueError` for shrinking the size of the bounding box to zero or below in any dimension.

### BoundingBox2d¶

class ezdxf.math.BoundingBox2d(vertices: Optional[Iterable[Union[Sequence[float], Vec2, Vec3]]] = None)

Optimized 2D bounding box.

Parameters

vertices – iterable of `(x, y[, z])` tuples or `Vec3` objects

extmin

“lower left” corner of bounding box

extmax

“upper right” corner of bounding box

property is_empty: bool

Returns `True` if the bounding box is empty. The bounding box has a size of 0 in any or all dimensions or is undefined.

property has_data: bool

Returns `True` if the bonding box has known limits.

property size

Returns size of bounding box.

property center

Returns center of bounding box.

inside(vertex: Union[Sequence[float], Vec2, Vec3]) bool

Returns `True` if vertex is inside this bounding box.

Vertices at the box border are inside!

any_inside(vertices: Iterable[Union[Sequence[float], Vec2, Vec3]]) bool

Returns `True` if any vertex is inside this bounding box.

Vertices at the box border are inside!

all_inside(vertices: Iterable[Union[Sequence[float], Vec2, Vec3]]) bool

Returns `True` if all vertices are inside this bounding box.

Vertices at the box border are inside!

has_intersection(other: AbstractBoundingBox) bool

Returns `True` if this bounding box intersects with other but does not include touching bounding boxes, see also `has_overlap()`:

```bbox1 = BoundingBox2d([(0, 0), (1, 1)])
bbox2 = BoundingBox2d([(1, 1), (2, 2)])
assert bbox1.has_intersection(bbox2) is False
```
has_overlap(other: AbstractBoundingBox) bool

Returns `True` if this bounding box intersects with other but in contrast to `has_intersection()` includes touching bounding boxes too:

```bbox1 = BoundingBox2d([(0, 0), (1, 1)])
bbox2 = BoundingBox2d([(1, 1), (2, 2)])
assert bbox1.has_overlap(bbox2) is True
```

New in version 0.17.2.

contains(other: AbstractBoundingBox) bool

Returns `True` if the other bounding box is completely inside of this bounding box.

New in version 0.17.2.

extend(vertices: Iterable[Union[Sequence[float], Vec2, Vec3]]) None

Extend bounds by vertices.

Parameters

vertices – iterable of vertices

union(other: AbstractBoundingBox)

Returns a new bounding box as union of this and other bounding box.

intersection(other: AbstractBoundingBox)

Returns the bounding box of the intersection rectangle of both 2D bounding boxes. Returns an empty bounding box if the intersection area is 0.

rect_vertices() Tuple[Vec2, ...]

Returns the corners of the bounding box in the xy-plane as `Vec2` objects.

### ConstructionRay¶

class ezdxf.math.ConstructionRay(p1: Union[Sequence[float], Vec2, Vec3], p2: Optional[Union[Sequence[float], Vec2, Vec3]] = None, angle: Optional[float] = None)

Infinite 2D construction ray as immutable object.

Parameters
• p1 – definition point 1

• p2 – ray direction as 2nd point or `None`

• angle – ray direction as angle in radians or `None`

location

Location vector as `Vec2`.

direction

Direction vector as `Vec2`.

slope

Slope of ray or `None` if vertical.

angle

Angle between x-axis and ray in radians.

angle_deg

Angle between x-axis and ray in degrees.

is_vertical

`True` if ray is vertical (parallel to y-axis).

is_horizontal

`True` if ray is horizontal (parallel to x-axis).

__str__()

Return str(self).

is_parallel(other: ConstructionRay) bool

Returns `True` if rays are parallel.

intersect(other: ConstructionRay) Vec2

Returns the intersection point as `(x, y)` tuple of self and other.

Raises

ParallelRaysError – if rays are parallel

orthogonal(location: Union[Sequence[float], Vec2, Vec3])

Returns orthogonal ray at location.

bisectrix(other: ConstructionRay)

Bisectrix between self and other.

yof(x: float) float

Returns y-value of ray for x location.

Raises

ArithmeticError – for vertical rays

xof(y: float) float

Returns x-value of ray for y location.

Raises

ArithmeticError – for horizontal rays

### ConstructionLine¶

class ezdxf.math.ConstructionLine(start: Union[Sequence[float], Vec2, Vec3], end: Union[Sequence[float], Vec2, Vec3])

2D ConstructionLine is similar to `ConstructionRay`, but has a start- and endpoint. The direction of line goes from start- to endpoint, “left of line” is always in relation to this line direction.

Parameters
start

start point as `Vec2`

end

end point as `Vec2`

bounding_box

bounding box of line as `BoundingBox2d` object.

ray

collinear `ConstructionRay`.

is_vertical

`True` if line is vertical.

is_horizontal

`True` if line is horizontal.

__str__()

Return str(self).

translate(dx: float, dy: float) None

Parameters
• dx – translation in x-axis

• dy – translation in y-axis

length() float

Returns length of line.

midpoint() Vec2

Returns mid point of line.

inside_bounding_box(point: Union[Sequence[float], Vec2, Vec3]) bool

Returns `True` if point is inside of line bounding box.

intersect(other: ConstructionLine, abs_tol: float = 1e-10) Optional[Vec2]

Returns the intersection point of to lines or `None` if they have no intersection point.

Parameters
has_intersection(other: ConstructionLine, abs_tol: float = 1e-10) bool

Returns `True` if has intersection with other line.

is_point_left_of_line(point: Union[Sequence[float], Vec2, Vec3], colinear=False) bool

Returns `True` if point is left of construction line in relation to the line direction from start to end.

If colinear is `True`, a colinear point is also left of the line.

### ConstructionCircle¶

class ezdxf.math.ConstructionCircle(center: Union[Sequence[float], Vec2, Vec3], radius: float = 1.0)

Circle construction tool.

Parameters
• center – center point as `Vec2` compatible object

center

center point as `Vec2`

bounding_box

2D bounding box of circle as `BoundingBox2d` object.

static from_3p(p1: Union[Sequence[float], Vec2, Vec3], p2: Union[Sequence[float], Vec2, Vec3], p3: Union[Sequence[float], Vec2, Vec3])

Creates a circle from three points, all points have to be compatible to `Vec2` class.

__str__() str

Returns string representation of circle “ConstructionCircle(center, radius)”.

translate(dx: float, dy: float) None

Parameters
• dx – translation in x-axis

• dy – translation in y-axis

point_at(angle: float) Vec2

Returns point on circle at angle as `Vec2` object.

Parameters

angle – angle in radians, angle goes counter clockwise around the z-axis, x-axis = 0 deg.

vertices(angles: Iterable[float]) Iterable[Vec2]

Yields vertices of the circle for iterable angles.

Parameters

angles – iterable of angles as radians, angle goes counter clockwise around the z-axis, x-axis = 0 deg.

New in version 0.17.1.

flattening(sagitta: float) Iterator[Vec2]

Approximate the circle by vertices, argument sagitta is the max. distance from the center of an arc segment to the center of its chord. Returns a closed polygon where the start vertex is coincident with the end vertex!

New in version 0.17.1.

inside(point: Union[Sequence[float], Vec2, Vec3]) bool

Returns `True` if point is inside circle.

tangent(angle: float)

Returns tangent to circle at angle as `ConstructionRay` object.

Parameters

intersect_ray(ray: ConstructionRay, abs_tol: float = 1e-10) Sequence[Vec2]

Returns intersection points of circle and ray as sequence of `Vec2` objects.

Parameters
• ray – intersection ray

• abs_tol – absolute tolerance for tests (e.g. test for tangents)

Returns

tuple of `Vec2` objects

tuple size

Description

0

no intersection

1

ray is a tangent to circle

2

ray intersects with the circle

intersect_line(line: ConstructionLine, abs_tol: float = 1e-10) Sequence[Vec2]

Returns intersection points of circle and line as sequence of `Vec2` objects.

Parameters
• line – intersection line

• abs_tol – absolute tolerance for tests (e.g. test for tangents)

Returns

tuple of `Vec2` objects

tuple size

Description

0

no intersection

1

line intersects or touches the circle at one point

2

line intersects the circle at two points

New in version 0.17.1.

intersect_circle(other: ConstructionCircle, abs_tol: float = 1e-10) Sequence[Vec2]

Returns intersection points of two circles as sequence of `Vec2` objects.

Parameters
• other – intersection circle

• abs_tol – absolute tolerance for tests

Returns

tuple of `Vec2` objects

tuple size

Description

0

no intersection

1

circle touches the other circle at one point

2

circle intersects with the other circle

### ConstructionArc¶

class ezdxf.math.ConstructionArc(center: Union[Sequence[float], Vec2, Vec3] = (0, 0), radius: float = 1.0, start_angle: float = 0.0, end_angle: float = 360.0, is_counter_clockwise: bool = True)

This is a helper class to create parameters for the DXF `Arc` class.

`ConstructionArc` represents a 2D arc in the xy-plane, use an `UCS` to place arc in 3D space, see method `add_to_layout()`.

Implements the 2D transformation tools: `translate()`, `scale_uniform()` and `rotate_z()`

Parameters
• center – center point as `Vec2` compatible object

• start_angle – start angle in degrees

• end_angle – end angle in degrees

• is_counter_clockwise – swaps start- and end angle if `False`

center

center point as `Vec2`

start_angle

start angle in degrees

end_angle

end angle in degrees

angle_span

Returns angle span of arc from start- to end param.

Returns the start angle in radians.

Returns the end angle in radians.

start_point

start point of arc as `Vec2`.

end_point

end point of arc as `Vec2`.

bounding_box

bounding box of arc as `BoundingBox2d`.

angles(num: int) Iterable[float]

Returns num angles from start- to end angle in degrees in counter-clockwise order.

All angles are normalized in the range from [0, 360).

vertices(a: Iterable[float]) Iterable[Vec2]

Yields vertices on arc for angles in iterable a in WCS as location vectors.

Parameters

a – angles in the range from 0 to 360 in degrees, arc goes counter clockwise around the z-axis, WCS x-axis = 0 deg.

tangents(a: Iterable[float]) Iterable[Vec2]

Yields tangents on arc for angles in iterable a in WCS as direction vectors.

Parameters

a – angles in the range from 0 to 360 in degrees, arc goes counter-clockwise around the z-axis, WCS x-axis = 0 deg.

translate(dx: float, dy: float)

Move arc about dx in x-axis and about dy in y-axis, returns self (floating interface).

Parameters
• dx – translation in x-axis

• dy – translation in y-axis

scale_uniform(s: float)

Scale arc inplace uniform about s in x- and y-axis, returns self (floating interface).

rotate_z(angle: float)

Rotate arc inplace about z-axis, returns self (floating interface).

Parameters

angle – rotation angle in degrees

classmethod from_2p_angle(start_point: Union[Sequence[float], Vec2, Vec3], end_point: Union[Sequence[float], Vec2, Vec3], angle: float, ccw: bool = True)

Create arc from two points and enclosing angle. Additional precondition: arc goes by default in counter-clockwise orientation from start_point to end_point, can be changed by ccw = `False`.

Parameters
• start_point – start point as `Vec2` compatible object

• end_point – end point as `Vec2` compatible object

• angle – enclosing angle in degrees

• ccw – counter-clockwise direction if `True`

classmethod from_2p_radius(start_point: Union[Sequence[float], Vec2, Vec3], end_point: Union[Sequence[float], Vec2, Vec3], radius: float, ccw: bool = True, center_is_left: bool = True)

Create arc from two points and arc radius. Additional precondition: arc goes by default in counter-clockwise orientation from start_point to end_point can be changed by ccw = `False`.

The parameter center_is_left defines if the center of the arc is left or right of the line from start_point to end_point. Parameter ccw = `False` swaps start- and end point, which also inverts the meaning of `center_is_left`.

Parameters
• start_point – start point as `Vec2` compatible object

• end_point – end point as `Vec2` compatible object

• ccw – counter-clockwise direction if `True`

• center_is_left – center point of arc is left of line from start- to end point if `True`

classmethod from_3p(start_point: Union[Sequence[float], Vec2, Vec3], end_point: Union[Sequence[float], Vec2, Vec3], def_point: Union[Sequence[float], Vec2, Vec3], ccw: bool = True)

Create arc from three points. Additional precondition: arc goes in counter-clockwise orientation from start_point to end_point.

Parameters
• start_point – start point as `Vec2` compatible object

• end_point – end point as `Vec2` compatible object

• def_point – additional definition point as `Vec2` compatible object

• ccw – counter-clockwise direction if `True`

add_to_layout(layout: BaseLayout, ucs: UCS = None, dxfattribs=None) Arc

Add arc as DXF `Arc` entity to a layout.

Supports 3D arcs by using an UCS. An `ConstructionArc` is always defined in the xy-plane, but by using an arbitrary UCS, the arc can be placed in 3D space, automatically OCS transformation included.

Parameters
intersect_ray(ray: ConstructionRay, abs_tol: float = 1e-10) Sequence[Vec2]

Returns intersection points of arc and ray as sequence of `Vec2` objects.

Parameters
• ray – intersection ray

• abs_tol – absolute tolerance for tests (e.g. test for tangents)

Returns

tuple of `Vec2` objects

tuple size

Description

0

no intersection

1

line intersects or touches the arc at one point

2

line intersects the arc at two points

New in version 0.17.1.

intersect_line(line: ConstructionLine, abs_tol: float = 1e-10) Sequence[Vec2]

Returns intersection points of arc and line as sequence of `Vec2` objects.

Parameters
• line – intersection line

• abs_tol – absolute tolerance for tests (e.g. test for tangents)

Returns

tuple of `Vec2` objects

tuple size

Description

0

no intersection

1

line intersects or touches the arc at one point

2

line intersects the arc at two points

New in version 0.17.1.

intersect_circle(circle: ConstructionCircle, abs_tol: float = 1e-10) Sequence[Vec2]

Returns intersection points of arc and circle as sequence of `Vec2` objects.

Parameters
• circle – intersection circle

• abs_tol – absolute tolerance for tests

Returns

tuple of `Vec2` objects

tuple size

Description

0

no intersection

1

circle intersects or touches the arc at one point

2

circle intersects the arc at two points

New in version 0.17.1.

intersect_arc(other: ConstructionArc, abs_tol: float = 1e-10) Sequence[Vec2]

Returns intersection points of two arcs as sequence of `Vec2` objects.

Parameters
• other – other intersection arc

• abs_tol – absolute tolerance for tests

Returns

tuple of `Vec2` objects

tuple size

Description

0

no intersection

1

other arc intersects or touches the arc at one point

2

other arc intersects the arc at two points

New in version 0.17.1.

### ConstructionEllipse¶

class ezdxf.math.ConstructionEllipse(center: Union[Sequence[float], Vec2, Vec3] = Vec3(0.0, 0.0, 0.0), major_axis: Union[Sequence[float], Vec2, Vec3] = Vec3(1.0, 0.0, 0.0), extrusion: Union[Sequence[float], Vec2, Vec3] = Vec3(0.0, 0.0, 1.0), ratio: float = 1, start_param: float = 0, end_param: float = 6.283185307179586, ccw: bool = True)

This is a helper class to create parameters for 3D ellipses.

Parameters
• center – 3D center point

• major_axis – major axis as 3D vector

• extrusion – normal vector of ellipse plane

• ratio – ratio of minor axis to major axis

• start_param – start param in radians

• end_param – end param in radians

• ccw – is counter clockwise flag - swaps start- and end param if `False`

center

center point as `Vec3`

major_axis

major axis as `Vec3`

minor_axis

minor axis as `Vec3`, automatically calculated from `major_axis` and `extrusion`.

extrusion

extrusion vector (normal of ellipse plane) as `Vec3`

ratio

ratio of minor axis to major axis (float)

start

end

start_point

Returns start point of ellipse as Vec3.

end_point

Returns end point of ellipse as Vec3.

property param_span: float

Returns the counter clockwise params span from start- to end param, see also `ezdxf.math.ellipse_param_span()` for more information.

to_ocs()

Returns ellipse parameters as OCS representation.

OCS elevation is stored in `center.z`.

params(num: int) Iterable[float]

Returns num params from start- to end param in counter clockwise order.

All params are normalized in the range from [0, 2π).

vertices(params: Iterable[float]) Iterable[Vec3]

Yields vertices on ellipse for iterable params in WCS.

Parameters

params – param values in the range from [0, 2π) in radians, param goes counter clockwise around the extrusion vector, major_axis = local x-axis = 0 rad.

flattening(distance: float, segments: int = 4) Iterable[Vec3]

Adaptive recursive flattening. The argument segments is the minimum count of approximation segments, if the distance from the center of the approximation segment to the curve is bigger than distance the segment will be subdivided. Returns a closed polygon for a full ellipse: start vertex == end vertex.

Parameters
• distance – maximum distance from the projected curve point onto the segment chord.

• segments – minimum segment count

New in version 0.15.

params_from_vertices(vertices: Iterable[Union[Sequence[float], Vec2, Vec3]]) Iterable[float]

Yields ellipse params for all given vertices.

The vertex don’t has to be exact on the ellipse curve or in the range from start- to end param or even in the ellipse plane. Param is calculated from the intersection point of the ray projected on the ellipse plane from the center of the ellipse through the vertex.

Warning

An input for start- and end vertex at param 0 and 2π return unpredictable results because of floating point inaccuracy, sometimes 0 and sometimes 2π.

dxfattribs() Dict

Returns required DXF attributes to build an ELLIPSE entity.

Entity ELLIPSE has always a ratio in range from 1e-6 to 1.

main_axis_points() Iterable[Vec3]

Yields main axis points of ellipse in the range from start- to end param.

classmethod from_arc(center: Union[Sequence[float], Vec2, Vec3] = Vec3(0.0, 0.0, 0.0), radius: float = 1, extrusion: Union[Sequence[float], Vec2, Vec3] = Vec3(0.0, 0.0, 1.0), start_angle: float = 0, end_angle: float = 360, ccw: bool = True)

Returns `ConstructionEllipse` from arc or circle.

Arc and Circle parameters defined in OCS.

Parameters
• center – center in OCS

• extrusion – OCS extrusion vector

• start_angle – start angle in degrees

• end_angle – end angle in degrees

• ccw – arc curve goes counter clockwise from start to end if `True`

transform(m: Matrix44) None

Transform ellipse in place by transformation matrix m.

swap_axis() None

Swap axis and adjust start- and end parameter.

Add ellipse as DXF `Ellipse` entity to a layout.

Parameters
• layout – destination layout as `BaseLayout` object

• dxfattribs – additional DXF attributes for the ELLIPSE entity

### ConstructionBox¶

class ezdxf.math.ConstructionBox(center: Union[Sequence[float], Vec2, Vec3] = (0, 0), width: float = 1, height: float = 1, angle: float = 0)

Helper class to create rectangles.

Parameters
• center – center of rectangle

• width – width of rectangle

• height – height of rectangle

• angle – angle of rectangle in degrees

center

box center

width

box width

height

box height

angle

rotation angle in degrees

corners

box corners as sequence of `Vec2` objects.

bounding_box

`BoundingBox2d`

__iter__() Iterable[Vec2]

Iterable of box corners as `Vec2` objects.

__getitem__(corner) Vec2

Get corner by index corner, `list` like slicing is supported.

__repr__() str

Returns string representation of box as `ConstructionBox(center, width, height, angle)`

classmethod from_points(p1: Union[Sequence[float], Vec2, Vec3], p2: Union[Sequence[float], Vec2, Vec3])

Creates a box from two opposite corners, box sides are parallel to x- and y-axis.

Parameters
translate(dx: float, dy: float) None

Parameters
• dx – translation in x-axis

• dy – translation in y-axis

expand(dw: float, dh: float) None

Expand box: dw expand width, dh expand height.

scale(sw: float, sh: float) None

Scale box: sw scales width, sh scales height.

rotate(angle: float) None

Rotate box by angle in degrees.

is_inside(point: Union[Sequence[float], Vec2, Vec3]) bool

Returns `True` if point is inside of box.

is_any_corner_inside(other: ConstructionBox) bool

Returns `True` if any corner of other box is inside this box.

is_overlapping(other: ConstructionBox) bool

Returns `True` if this box and other box do overlap.

border_lines() Sequence[ConstructionLine]

Returns borderlines of box as sequence of `ConstructionLine`.

intersect(line: ConstructionLine) List[Vec2]

Returns 0, 1 or 2 intersection points between line and box borderlines.

Parameters

line – line to intersect with borderlines

Returns

list of intersection points

list size

Description

0

no intersection

1

line touches box at one corner

2

line intersects with box

### ConstructionPolyline¶

class ezdxf.math.ConstructionPolyline(vertices: Iterable[Union[Sequence[float], Vec2, Vec3]], close: bool = False, rel_tol: float = 1e-09)

A polyline construction tool to measure, interpolate and divide anything that can be approximated or flattened into vertices. This is an immutable data structure which supports the `Sequence` interface.

Parameters
• vertices – iterable of polyline vertices

• close`True` to close the polyline (first vertex == last vertex)

• rel_tol – relative tolerance for floating point comparisons

Example to measure or divide a SPLINE entity:

```import ezdxf
from ezdxf.math import ConstructionPolyline

msp = doc.modelspace()
spline = msp.query("SPLINE").first
if spline is not None:
polyline = ConstructionPolyline(spline.flattening(0.01))
print(f"Entity {spline} has an approximated length of {polyline.length}")
# get dividing points with a distance of 1.0 drawing unit to each other
points = list(polyline.divide_by_length(1.0))
```

New in version 0.18.

property length: float

Returns the overall length of the polyline.

property is_closed: bool

Returns `True` if the polyline is closed (first vertex == last vertex).

data(index: int) Tuple[float, float, Vec3]

Returns the tuple (distance from start, distance from previous vertex, vertex). All distances measured along the polyline.

index_at(distance: float) int

Returns the data index of the exact or next data entry for the given distance. Returns the index of last entry if distance > `length`.

vertex_at(distance: float) Vec3

Returns the interpolated vertex at the given distance from the start of the polyline.

divide(count: int) Iterator[Vec3]

Returns count interpolated vertices along the polyline. Argument count has to be greater than 2 and the start- and end vertices are always included.

divide_by_length(length: float, force_last: bool = False) Iterator[Vec3]

Returns interpolated vertices along the polyline. Each vertex has a fix distance length from its predecessor. Yields the last vertex if argument force_last is `True` even if the last distance is not equal to length.

### Shape2d¶

class ezdxf.math.Shape2d(vertices: Optional[Iterable[Union[Sequence[float], Vec2, Vec3]]] = None)

2D geometry object as list of `Vec2` objects, vertices can be moved, rotated and scaled.

Parameters

vertices – iterable of `Vec2` compatible objects.

vertices

List of `Vec2` objects

bounding_box

`BoundingBox2d`

__len__() int

Returns count of vertices.

__getitem__(item: Union[int, slice]) Vec2

Get vertex by index item, supports `list` like slicing.

append(vertex: Union[Sequence[float], Vec2, Vec3]) None

Append single vertex.

Parameters

vertex – vertex as `Vec2` compatible object

extend(vertices: Iterable) None

Append multiple vertices.

Parameters

vertices – iterable of vertices as `Vec2` compatible objects

translate(vector: Union[Sequence[float], Vec2, Vec3]) None

scale(sx: float = 1.0, sy: float = 1.0) None

Scale shape about sx in x-axis and sy in y-axis.

scale_uniform(scale: float) None

Scale shape uniform about scale in x- and y-axis.

rotate(angle: float, center: Optional[Union[Sequence[float], Vec2, Vec3]] = None) None

Rotate shape around rotation center about angle in degrees.

rotate_rad(angle: float, center: Optional[Union[Sequence[float], Vec2, Vec3]] = None) None

offset(offset: float, closed: bool = False)

Returns a new offset shape, for more information see also `ezdxf.math.offset_vertices_2d()` function.

Parameters
• offset – line offset perpendicular to direction of shape segments defined by vertices order, offset > `0` is ‘left’ of line segment, offset < `0` is ‘right’ of line segment

• closed`True` to handle as closed shape

convex_hull()

Returns convex hull as new shape.

## Curves¶

### BSpline¶

class ezdxf.math.BSpline(control_points: Iterable[Union[Sequence[float], Vec2, Vec3]], order: int = 4, knots: Optional[Iterable[float]] = None, weights: Optional[Iterable[float]] = None)

Representation of a B-spline curve. The default configuration of the knot vector is a uniform open knot vector (“clamped”).

Factory functions:

Parameters
• control_points – iterable of control points as `Vec3` compatible objects

• order – spline order (degree + 1)

• knots – iterable of knot values

• weights – iterable of weight values

property control_points: Sequence[ezdxf.math._vector.Vec3]

Control points as tuple of `Vec3`

property count: int

Count of control points, (n + 1 in text book notation).

property order: int

Order (k) of B-spline = p + 1

property degree: int

Degree (p) of B-spline = order - 1

property max_t: float

Biggest knot value.

property is_rational

Returns `True` if curve is a rational B-spline. (has weights)

property is_clamped

Returns `True` if curve is a clamped (open) B-spline.

knots() Tuple[float, ...]

Returns a tuple of knot values as floats, the knot vector always has order + count values (n + p + 2 in text book notation).

weights() Tuple[float, ...]

Returns a tuple of weights values as floats, one for each control point or an empty tuple.

params(segments: int) Iterable[float]

Yield evenly spaced parameters for given segment count.

reverse()

Returns a new `BSpline` object with reversed control point order.

transform(m: Matrix44)

Returns a new `BSpline` object transformed by a `Matrix44` transformation matrix.

approximate(segments: int = 20) Iterable[Vec3]

Approximates curve by vertices as `Vec3` objects, vertices count = segments + 1.

flattening(distance: float, segments: int = 4) Iterator[Vec3]

Adaptive recursive flattening. The argument segments is the minimum count of approximation segments between two knots, if the distance from the center of the approximation segment to the curve is bigger than distance the segment will be subdivided.

Parameters
• distance – maximum distance from the projected curve point onto the segment chord.

• segments – minimum segment count between two knots

New in version 0.15.

point(t: float) Vec3

Returns point for parameter t.

Parameters

t – parameter in range [0, max_t]

points(t: Iterable[float]) Iterable[Vec3]

Yields points for parameter vector t.

Parameters

t – parameters in range [0, max_t]

derivative(t: float, n: int = 2) List[Vec3]

Return point and derivatives up to n <= degree for parameter t.

e.g. n=1 returns point and 1st derivative.

Parameters
• t – parameter in range [0, max_t]

• n – compute all derivatives up to n <= degree

Returns

n+1 values as `Vec3` objects

derivatives(t: Iterable[float], n: int = 2) Iterable[List[Vec3]]

Yields points and derivatives up to n <= degree for parameter vector t.

e.g. n=1 returns point and 1st derivative.

Parameters
• t – parameters in range [0, max_t]

• n – compute all derivatives up to n <= degree

Returns

List of n+1 values as `Vec3` objects

insert_knot(t: float)

Insert an additional knot, without altering the shape of the curve. Returns a new `BSpline` object.

Parameters

t – position of new knot 0 < t < max_t

knot_refinement(u: Iterable[float])

Insert multiple knots, without altering the shape of the curve. Returns a new `BSpline` object.

Parameters

u – vector of new knots t and for each t: 0 < t < max_t

static from_ellipse(ellipse: ConstructionEllipse)

Returns the ellipse as `BSpline` of 2nd degree with as few control points as possible.

static from_arc(arc: ConstructionArc)

Returns the arc as `BSpline` of 2nd degree with as few control points as possible.

static from_fit_points(points: Iterable[Union[Sequence[float], Vec2, Vec3]], degree=3, method='chord')

Returns `BSpline` defined by fit points.

static arc_approximation(arc: ConstructionArc, num: int = 16)

Returns an arc approximation as `BSpline` with num control points.

static ellipse_approximation(ellipse: ConstructionEllipse, num: int = 16)

Returns an ellipse approximation as `BSpline` with num control points.

bezier_decomposition() Iterable[List[Vec3]]

Decompose a non-rational B-spline into multiple Bézier curves.

This is the preferred method to represent the most common non-rational B-splines of 3rd degree by cubic Bézier curves, which are often supported by render backends.

Returns

Yields control points of Bézier curves, each Bézier segment has degree+1 control points e.g. B-spline of 3rd degree yields cubic Bézier curves of 4 control points.

cubic_bezier_approximation(level: int = 3, segments: int = None) Iterable['Bezier4P']

Approximate arbitrary B-splines (degree != 3 and/or rational) by multiple segments of cubic Bézier curves. The choice of cubic Bézier curves is based on the widely support of this curves by many render backends. For cubic non-rational B-splines, which is maybe the most common used B-spline, is `bezier_decomposition()` the better choice.

1. approximation by level: an educated guess, the first level of approximation segments is based on the count of control points and their distribution along the B-spline, every additional level is a subdivision of the previous level.

E.g. a B-Spline of 8 control points has 7 segments at the first level, 14 at the 2nd level and 28 at the 3rd level, a level >= 3 is recommended.

1. approximation by a given count of evenly distributed approximation segments.

Parameters
• level – subdivision level of approximation segments (ignored if argument segments is not `None`)

• segments – absolute count of approximation segments

Returns

Yields control points of cubic Bézier curves as `Bezier4P` objects

### Bezier¶

class ezdxf.math.Bezier(defpoints: Iterable[Union[Sequence[float], Vec2, Vec3]])

A Bézier curve is a parametric curve used in computer graphics and related fields. Bézier curves are used to model smooth curves that can be scaled indefinitely. “Paths”, as they are commonly referred to in image manipulation programs, are combinations of linked Bézier curves. Paths are not bound by the limits of rasterized images and are intuitive to modify. (Source: Wikipedia)

This is a generic implementation which works with any count of definition points greater than 2, but it is a simple and slow implementation. For more performance look at the specialized `Bezier4P` class.

Objects are immutable.

Parameters

defpoints – iterable of definition points as `Vec3` compatible objects.

control_points

Control points as tuple of `Vec3` objects.

params(segments: int) Iterable[float]

Yield evenly spaced parameters from 0 to 1 for given segment count.

reverse()

Returns a new Bèzier-curve with reversed control point order.

transform(m: Matrix44)

General transformation interface, returns a new `Bezier` curve.

Parameters

m – 4x4 transformation matrix (`ezdxf.math.Matrix44`)

approximate(segments: int = 20) Iterable[Vec3]

Approximates curve by vertices as `Vec3` objects, vertices count = segments + 1.

flattening(distance: float, segments: int = 4) Iterable[Vec3]

Adaptive recursive flattening. The argument segments is the minimum count of approximation segments, if the distance from the center of the approximation segment to the curve is bigger than distance the segment will be subdivided.

Parameters
• distance – maximum distance from the center of the curve (Cn) to the center of the linear (C1) curve between two approximation points to determine if a segment should be subdivided.

• segments – minimum segment count

New in version 0.15.

point(t: float) Vec3

Returns a point for parameter t in range [0, 1] as `Vec3` object.

points(t: Iterable[float]) Iterable[Vec3]

Yields multiple points for parameters in vector t as `Vec3` objects. Parameters have to be in range [0, 1].

derivative(t: float) Tuple[Vec3, Vec3, Vec3]

Returns (point, 1st derivative, 2nd derivative) tuple for parameter t in range [0, 1] as `Vec3` objects.

derivatives(t: Iterable[float]) Iterable[Tuple[Vec3, Vec3, Vec3]]

Returns multiple (point, 1st derivative, 2nd derivative) tuples for parameter vector t as `Vec3` objects. Parameters in range [0, 1]

### Bezier4P¶

class ezdxf.math.Bezier4P(defpoints: Sequence[UVec])
control_points

Control points as tuple of `Vec3` or `Vec2` objects.

reverse()

Returns a new Bèzier-curve with reversed control point order.

transform(m: Matrix44)

General transformation interface, returns a new `Bezier4p` curve and it is always a 3D curve.

Parameters

m – 4x4 transformation matrix (`ezdxf.math.Matrix44`)

New in version 0.14.

approximate(segments: int) Iterable[AnyVec]

Approximate Bézier curve by vertices, yields segments + 1 vertices as `(x, y[, z])` tuples.

Parameters

segments – count of segments for approximation

flattening(distance: float, segments: int = 4) Iterable[Union[Vec2, Vec3]]

Adaptive recursive flattening. The argument segments is the minimum count of approximation segments, if the distance from the center of the approximation segment to the curve is bigger than distance the segment will be subdivided.

Parameters
• distance – maximum distance from the center of the cubic (C3) curve to the center of the linear (C1) curve between two approximation points to determine if a segment should be subdivided.

• segments – minimum segment count

New in version 0.15.

approximated_length(segments: int = 128) float

Returns estimated length of Bèzier-curve as approximation by line segments.

point(t: float) AnyVec

Returns point for location t` at the Bèzier-curve.

Parameters

t – curve position in the range `[0, 1]`

tangent(t: float) AnyVec

Returns direction vector of tangent for location t at the Bèzier-curve.

Parameters

t – curve position in the range `[0, 1]`

### Bezier3P¶

class ezdxf.math.Bezier3P(defpoints: Sequence[UVec])
control_points

Control points as tuple of `Vec3` or `Vec2` objects.

reverse()

Returns a new Bèzier-curve with reversed control point order.

transform(m: Matrix44)

General transformation interface, returns a new `Bezier3P` curve and it is always a 3D curve.

Parameters

m – 4x4 transformation matrix (`ezdxf.math.Matrix44`)

approximate(segments: int) Iterable['AnyVec']

Approximate Bézier curve by vertices, yields segments + 1 vertices as `(x, y[, z])` tuples.

Parameters

segments – count of segments for approximation

flattening(distance: float, segments: int = 4) Iterable['AnyVec']

Adaptive recursive flattening. The argument segments is the minimum count of approximation segments, if the distance from the center of the approximation segment to the curve is bigger than distance the segment will be subdivided.

Parameters
• distance – maximum distance from the center of the quadratic (C2) curve to the center of the linear (C1) curve between two approximation points to determine if a segment should be subdivided.

• segments – minimum segment count

approximated_length(segments: int = 128) float

Returns estimated length of Bèzier-curve as approximation by line segments.

point(t: float) AnyVec

Returns point for location t` at the Bèzier-curve.

Parameters

t – curve position in the range `[0, 1]`

tangent(t: float) AnyVec

Returns direction vector of tangent for location t at the Bèzier-curve.

Parameters

t – curve position in the range `[0, 1]`

### ApproxParamT¶

class ezdxf.math.ApproxParamT(curve, *, max_t: float = 1.0, segments: int = 100)

Approximation tool for parametrized curves.

• approximate parameter t for a given distance from the start of the curve

• approximate the distance for a given parameter t from the start of the curve

These approximations can be applied to all parametrized curves which provide a `point()` method, like `Bezier4P`, `Bezier3P` and `BSpline`.

The approximation is based on equally spaced parameters from 0 to max_t for a given segment count. The `flattening()` method can not be used for the curve approximation, because the required parameter t is not logged by the flattening process.

Parameters
• curve – curve object, requires a method `point()`

• max_t – the max. parameter value

• segments – count of approximation segments

New in version 0.18.

property max_t: float
property polyline: ezdxf.math.polyline.ConstructionPolyline
param_t(distance: float)

Approximate parameter t for the given distance from the start of the curve.

distance(t: float) float

Approximate the distance from the start of the curve to the point t on the curve.

### BezierSurface¶

class ezdxf.math.BezierSurface(defpoints: List[List[Union[Sequence[float], Vec2, Vec3]]])

`BezierSurface` defines a mesh of m x n control points. This is a parametric surface, which means the m-dimension goes from `0` to `1` as parameter u and the n-dimension goes from `0` to `1` as parameter v.

Parameters

defpoints – matrix (list of lists) of m rows and n columns: [ [m1n1, m1n2, … ], [m2n1, m2n2, …] … ] each element is a 3D location as `(x, y, z)` tuple.

nrows

count of rows (m-dimension)

ncols

count of columns (n-dimension)

point(u: float, v: float) Vec3

Returns a point for location (u, v) at the Bézier surface as `(x, y, z)` tuple, parameters u and v in the range of `[0, 1]`.

approximate(usegs: int, vsegs: int) List[List[Vec3]]

Approximate surface as grid of `(x, y, z)` `Vec3`.

Parameters
• usegs – count of segments in u-direction (m-dimension)

• vsegs – count of segments in v-direction (n-dimension)

Returns

list of usegs + 1 rows, each row is a list of vsegs + 1 vertices as `Vec3`.

### EulerSpiral¶

class ezdxf.math.EulerSpiral(curvature: float = 1.0)

This class represents an euler spiral (clothoid) for curvature (Radius of curvature).

This is a parametric curve, which always starts at the origin = `(0, 0)`.

Parameters

Get radius of circle at distance t.

tangent(t: float) Vec3

Get tangent at distance t as `Vec3` object.

Get distance L from origin for radius.

point(t: float) Vec3

Get point at distance t as `Vec3`.

circle_center(t: float) Vec3

Get circle center at distance t.

approximate(length: float, segments: int) Iterable[Vec3]

Approximate curve of length with line segments. Generates segments+1 vertices as `Vec3` objects.

bspline(length: float, segments: int = 10, degree: int = 3, method: str = 'uniform')

Approximate euler spiral as B-spline.

Parameters
• length – length of euler spiral

• segments – count of fit points for B-spline calculation

• degree – degree of BSpline

• method – calculation method for parameter vector t

Returns

`BSpline`