Core¶
Math core module: ezdxf.math
These are the core math functions and classes which should be imported from
ezdxf.math
.
Utility Functions¶
Returns the counterclockwise angle span from start to end in degrees. 

Returns the counterclockwise angle span from start to end in radians. 

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

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

Returns the area of a polygon, returns the projected area in the xyplane for any vertices (zaxis will be ignored). 

Returns the closest point to a give base point. 

Returns the counterclockwise params span of an elliptic arc from start to end param. 

Returns 

Returns 

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

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

Returns the count of required knotvalues for a Bspline of order and count control points. 

Returns an uniform knot vector for a Bspline of order and count control points. 

Extended rounding function. 
 ezdxf.math.closest_point(base: UVec, points: Iterable[UVec]) Vec3 ¶
Returns the closest point to a give base point.
 ezdxf.math.uniform_knot_vector(count: int, order: int, normalize=False) list[float] ¶
Returns an uniform knot vector for a Bspline 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 Bspline 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 knotvalues for a Bspline of order and count control points.
 Parameters
count – count of control points, in textbooks referred as “n + 1”
order – order of BSpline, in textbooks referred as “k”
Relationship:
“p” is the degree of the Bspline, textbook notation.
k = p + 1
2 ≤ k ≤ n + 1
 ezdxf.math.xround(value: float, rounding: float = 0.) float ¶
Extended rounding function.
The 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
1.0
round to a multiple of 1: remove fraction
2.0
round to a multiple of 2: xxx2, xxx4, xxx6 …
5.0
round to a multiple of 5: xxx5 or xxx0
10.0
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[UVec]) float ¶
Returns the area of a polygon, returns the projected area in the xyplane for any vertices (zaxis will be ignored).
 ezdxf.math.arc_angle_span_deg(start: float, end: float) float ¶
Returns the counterclockwise 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.
 ezdxf.math.arc_angle_span_rad(start: float, end: float) float ¶
Returns the counterclockwise 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
radius – arc radius
angle – angle span of the arc in radians
sagitta – max. distance from the center of an arc segment to the center of its chord
 ezdxf.math.arc_chord_length(radius: float, sagitta: float) float ¶
Returns the chord length for an arc defined by radius and the sagitta.
 Parameters
radius – arc radius
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 counterclockwise 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 nonuniform xyscaling. Uniform scaling is not stretching in this context.Does not check if the target system is a cartesian coordinate system, use the
Matrix44
propertyis_cartesian
for that.
 ezdxf.math.has_matrix_3d_stretching(m: Matrix44) bool ¶
Returns
True
if matrix m performs a nonuniform xyzscaling. Uniform scaling is not stretching in this context.Does not check if the target system is a cartesian coordinate system, use the
Matrix44
propertyis_cartesian
for that.
2D Graphic Functions¶
Returns the 2D convex hull of given points. 

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

Returns the intersection points for two polylines as list of 

Compute the intersection of two lines in the xyplane. 

Returns 

Test if point is inside polygon. 

Returns 

Returns 

Yields vertices of the offset line to the shape defined by vertices. 

Returns 

The Rytz’s axis construction is a basic method of descriptive Geometry to find the axes, the semimajor axis and semiminor axis, starting from two conjugated halfdiameters. 
 ezdxf.math.convex_hull_2d(points: Iterable[UVec]) list[ezdxf.math._vector.Vec2] ¶
Returns the 2D convex hull of given points.
Returns a closed polyline, first vertex is equal to the last vertex.
 Parameters
points – iterable of points, zaxis 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=1e10) list[ezdxf.math._vector.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 byresult.sort()
to introduce an order.
 ezdxf.math.intersection_line_line_2d(line1: Sequence[Vec2], line2: Sequence[Vec2], virtual=True, abs_tol=TOLERANCE) Optional[Vec2] ¶
Compute the intersection of two lines in the xyplane.
 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 asVec2
 ezdxf.math.is_convex_polygon_2d(polygon: list[ezdxf.math._vector.Vec2], *, strict=False, epsilon=1e6) bool ¶
Returns
True
if the 2D polygon is convex. This function works with open and closed polygons and clockwise or counterclockwise vertex orientation. Coincident vertices will always be skipped and if argument strict isTrue
, polygons with collinear vertices are not considered as convex.This solution works only for simple nonselfintersecting polygons!
 ezdxf.math.is_point_in_polygon_2d(point: Vec2, polygon: Sequence[Vec2], abs_tol=TOLERANCE) int ¶
Test if point is inside polygon. Returns
1
(for outside) if the polygon is degenerated, no exception will be raised.
 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 isTrue
.
 ezdxf.math.is_point_on_line_2d(point: Vec2, start: Vec2, end: Vec2, ray=True, abs_tol=TOLERANCE) bool ¶
Returns
True
if point is on line.
 ezdxf.math.offset_vertices_2d(vertices: Iterable[UVec], 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 xyplane, the zaxis 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 (Uturn), 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 segmentclosed –
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=TOLERANCE) int ¶
Returns
1
if point is left line,+1
if point is right of line and0
if point is on the line. The line is defined by two vertices given as arguments start and end.
 ezdxf.math.rytz_axis_construction(d1: Vec3, d2: Vec3) tuple[ezdxf.math._vector.Vec3, ezdxf.math._vector.Vec3, float] ¶
The Rytz’s axis construction is a basic method of descriptive Geometry to find the axes, the semimajor axis and semiminor axis, starting from two conjugated halfdiameters.
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.
3D Graphic Functions¶
Returns a combined transformation matrix for translation, scaling and rotation about the zaxis. 

Returns the "best fit" normal for a plane defined by three or more vertices. 

Convert multiple quadratic or cubic Bèzier curves into a single cubic Bspline. 

Creates a closed uniform (periodic) Bspline curve (open curve). 

Returns the 

Returns a cubic Bèzier curve 

Returns an approximation for a circular 2D arc by multiple cubic Béziercurves. 

Returns an approximation for an elliptic arc by multiple cubic Béziercurves. 

Returns an interpolation curve for given data points as multiple cubic Béziercurves. 

Returns the normal distance from a point to a 3D line. 

Estimate tangent magnitude of start and end tangents. 

Estimate tangents for curve defined by given fit points. 

Returns a cubic 

Returns a cubic 

Bspline interpolation by the Global Curve Interpolation. 

Return 

Returns the intersection points for two polylines as list of 

Returns the intersection point of two 3D lines, returns 

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

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

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

Returns 

Returns count evenly spaced vertices from start to end. 

Bspline interpolation by 'Local Cubic Curve Interpolation', which creates Bspline from fit points and estimated tangent direction at start, end and passing points. 

Returns normal vector for 3 points, which is the normalized cross product for: 

Creates an open uniform (periodic) Bspline curve (open curve). 

Returns the 

Returns a quadratic Bèzier curve 

Convert quadratic Bèzier curves ( 

Returns a rational Bsplines for a circular 2D arc. 

Returns a rational Bsplines for an elliptic arc. 

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

Calculate the spherical envelope for the given points. 

Split a Bèzier curve at parameter t. 

Split a convex polygon by the given plane. 

Subdivides faces by subdividing edges and adding a center vertex. 

Subdivides faces into triangles by adding a center vertex. 
See also
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: UVec = (0, 0, 0), scale: UVec = (1, 1, 1), z_rotation: float = 0) Matrix44 ¶
Returns a combined transformation matrix for translation, scaling and rotation about the zaxis.
 Parameters
move – translation vector
scale – x, y and zaxis scaling as float triplet, e.g. (2, 2, 1)
z_rotation – rotation angle about the zaxis in radians
 ezdxf.math.best_fit_normal(vertices: Iterable[UVec]) 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]]) BSpline ¶
Convert multiple quadratic or cubic Bèzier curves into a single cubic 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.
 ezdxf.math.closed_uniform_bspline(control_points: Iterable[UVec], order: int = 4, weights: Optional[Iterable[float]] = None) BSpline ¶
Creates a closed uniform (periodic) Bspline curve (open curve).
This Bspline does not pass any of the control points.
 Parameters
control_points – iterable of control points as
Vec3
compatible objectsorder – spline order (degree + 1)
weights – iterable of weight values
 ezdxf.math.cubic_bezier_bbox(curve: Bezier4P, *, abs_tol=1e12) BoundingBox ¶
Returns the
BoundingBox
of a cubic Bézier curve of typeBezier4P
.
 ezdxf.math.cubic_bezier_from_3p(p1: UVec, p2: UVec, p3: UVec) Bezier4P ¶
Returns a cubic Bèzier curve
Bezier4P
from three points. The curve starts at p1, goes through p2 and ends at p3. (source: pomax2)
 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 Béziercurves.
 Parameters
center – circle center as
Vec3
compatible objectradius – circle radius
start_angle – start angle in degrees
end_angle – end angle in degrees
segments – count of Bèziercurve 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 Béziercurves.
 Parameters
ellipse – ellipse parameters as
ConstructionEllipse
objectsegments – count of Bèziercurve segments, at least one segment for each quarter (π/2), 1 for as few as possible.
 ezdxf.math.cubic_bezier_interpolation(points: Iterable[UVec]) Iterable[Bezier4P] ¶
Returns an interpolation curve for given data points as multiple cubic Béziercurves. Returns n1 cubic Béziercurves 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 a point to a 3D line.
 Parameters
point – point to test
start – start point of the 3D line
end – end point of the 3D line
 ezdxf.math.estimate_end_tangent_magnitude(points: list[ezdxf.math._vector.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
“beziern”: 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[ezdxf.math._vector.Vec3], method: str = '5points', normalize=True) list[ezdxf.math._vector.Vec3] ¶
Estimate tangents for curve defined by given fit points. Calculated tangents are normalized (unitvectors).
Available tangent estimation methods:
“3points”: 3 point interpolation
“5points”: 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[UVec], tangents: Optional[Iterable[UVec]] = None) BSpline ¶
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:
CAD applications use the global curve interpolation with start and end derivatives if the end tangents are defined otherwise the equation system will be completed by setting the second derivatives of the start and end point to 0, for more information read this answer on stackoverflow: https://stackoverflow.com/a/74863330/6162864
The degree of the Bspline is always 3 regardless which degree is stored in the SPLINE entity, this is only valid for Bsplines defined by fit points
Knot parametrization method is “chord”
Knot distribution is “natural”
 Parameters
fit_points – points the spline is passing through
tangents – start and end tangent, default is autodetect
 ezdxf.math.fit_points_to_cubic_bezier(fit_points: Iterable[UVec]) BSpline ¶
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 Bspline, 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
 ezdxf.math.global_bspline_interpolation(fit_points: Iterable[UVec], degree: int = 3, tangents: Optional[Iterable[UVec]] = None, method: str = 'chord') BSpline ¶
Bspline interpolation by the Global Curve Interpolation. Given are the fit points and the degree of the Bspline. 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 Bspline, as list of
Vec3
compatible objectstangents – 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 Bspline
method – calculation method for parameter vector t
 Returns
 ezdxf.math.have_bezier_curves_g1_continuity(b1: Union[Bezier3P, Bezier4P], b2: Union[Bezier3P, Bezier4P], g1_tol: float = 1e4) bool ¶
Return
True
if the given adjacent Bézier curves have G1 continuity.
 ezdxf.math.intersect_polylines_3d(p1: Sequence[Vec3], p2: Sequence[Vec3], abs_tol=1e10) list[ezdxf.math._vector.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 byresult.sort()
to introduce an order.
 ezdxf.math.intersection_line_line_3d(line1: Sequence[Vec3], line2: Sequence[Vec3], virtual: bool = True, abs_tol: float = 1e10) Optional[Vec3] ¶
Returns the intersection point of two 3D lines, returns
None
if lines do not intersect.
 ezdxf.math.intersection_line_polygon_3d(start: Vec3, end: Vec3, polygon: Iterable[Vec3], *, coplanar=True, boundary=True, abs_tol=PLANE_EPSILON) 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 validboundary – if
True
an intersection point at the polygon boundary line is validabs_tol – absolute tolerance for comparisons
 ezdxf.math.intersection_ray_polygon_3d(origin: Vec3, direction: Vec3, polygon: Iterable[Vec3], *, boundary=True, abs_tol=PLANE_EPSILON) Optional[Vec3] ¶
Returns the intersection point of the infinite 3D ray defined by origin and the direction vector and the given polygon.
 ezdxf.math.intersection_ray_ray_3d(ray1: Sequence[Vec3], ray2: Sequence[Vec3], abs_tol=TOLERANCE) Sequence[Vec3] ¶
Calculate intersection of two 3D rays, returns a 0tuple for parallel rays, a 1tuple for intersecting rays and a 2tuple for not intersecting and not parallel rays with points of the closest approach on each ray.
 ezdxf.math.is_planar_face(face: Sequence[Vec3], abs_tol=1e9) bool ¶
Returns
True
if sequence of vectors is a planar face. Parameters
face – sequence of
Vec3
objectsabs_tol – tolerance for normals check
 ezdxf.math.linear_vertex_spacing(start: Vec3, end: Vec3, count: int) list[ezdxf.math._vector.Vec3] ¶
Returns count evenly spaced vertices from start to end.
 ezdxf.math.local_cubic_bspline_interpolation(fit_points: Iterable[UVec], method: str = '5points', tangents: Optional[Iterable[UVec]] = None) BSpline ¶
Bspline interpolation by ‘Local Cubic Curve Interpolation’, which creates Bspline 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:
“3points”: 3 point interpolation
“5points”: 5 point interpolation
“bezier”: cubic bezier curve interpolation
“diff”: finite difference
or pass precalculated tangents, which overrides tangent estimation.
 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[UVec], order: int = 4, weights: Optional[Iterable[float]] = None) BSpline ¶
Creates an open uniform (periodic) Bspline 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 objectsorder – spline order (degree + 1)
weights – iterable of weight values
 ezdxf.math.quadratic_bezier_bbox(curve: Bezier3P, *, abs_tol=1e12) BoundingBox ¶
Returns the
BoundingBox
of a quadratic Bézier curve of typeBezier3P
.
 ezdxf.math.quadratic_bezier_from_3p(p1: UVec, p2: UVec, p3: UVec) Bezier3P ¶
Returns a quadratic Bèzier curve
Bezier3P
from three points. The curve starts at p1, goes through p2 and ends at p3. (source: pomax2)
 ezdxf.math.quadratic_to_cubic_bezier(curve: Bezier3P) Bezier4P ¶
Convert quadratic Bèzier curves (
ezdxf.math.Bezier3P
) into cubic Bèzier curves (ezdxf.math.Bezier4P
).
 ezdxf.math.rational_bspline_from_arc(center: Vec3 = (0, 0), radius: float = 1, start_angle: float = 0, end_angle: float = 360, segments: int = 1) BSpline ¶
Returns a rational Bsplines for a circular 2D arc.
 Parameters
center – circle center as
Vec3
compatible objectradius – circle radius
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) BSpline ¶
Returns a rational Bsplines for an elliptic arc.
 Parameters
ellipse – ellipse parameters as
ConstructionEllipse
objectsegments – 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[UVec]) tuple[ezdxf.math._vector.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!
 ezdxf.math.split_bezier(control_points: Sequence[T], t: float) tuple[list[T], list[T]] ¶
Split a Bèzier curve at parameter t.
Returns the control points for two new Bèzier curves of the same degree and type as the input curve. (source: pomax1)
 ezdxf.math.split_polygon_by_plane(polygon: Iterable[Vec3], plane: Plane, *, coplanar=True, abs_tol=PLANE_EPSILON) tuple[Sequence[ezdxf.math._vector.Vec3], Sequence[ezdxf.math._vector.Vec3]] ¶
Split a convex polygon by the given plane.
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.
 ezdxf.math.subdivide_face(face: Sequence[Union[Vec2, Vec3]], quads: bool = True) Iterable[tuple[ezdxf.math._vector.Vec3, ...]] ¶
Subdivides faces by subdividing edges and adding a center vertex.
Transformation Classes¶
An optimized 4x4 transformation matrix. 

Establish an OCS for a given extrusion vector. 

Establish a user coordinate system (UCS). 
OCS Class¶
 class ezdxf.math.OCS(extrusion: UVec = Z_AXIS)¶
Establish an OCS for a given extrusion vector.
 Parameters
extrusion – extrusion vector.
 ux¶
xaxis unit vector
 uy¶
yaxis unit vector
 uz¶
zaxis unit vector
 points_from_wcs(points: Iterable[UVec]) Iterable[UVec] ¶
Returns iterable of OCS vectors from WCS points.
 points_to_wcs(points: Iterable[UVec]) Iterable[UVec] ¶
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: UVec = (0, 0, 0), ux: Optional[UVec] = None, uy: Optional[UVec] = None, uz: Optional[UVec] = None)¶
Establish a user coordinate system (UCS). The UCS is defined by the origin and two unit vectors for the x, y or zaxis, all axis in WCS. The missing axis is the cross product of the given axis.
If x and yaxis 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
 ux¶
xaxis unit vector
 uy¶
yaxis unit vector
 uz¶
zaxis unit vector
 is_cartesian¶
Returns
True
if cartesian coordinate system.
 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.
 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.
 points_to_ocs(points: Iterable[Vec3]) Iterable[Vec3] ¶
Returns iterable of OCS vectors for UCS points.
The
OCS
is defined by the zaxis of theUCS
. 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: UVec, 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 zaxis 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 xaxis.
 Parameters
angle – rotation angle in radians
 rotate_local_y(angle: float) UCS ¶
Returns a new rotated UCS, rotation axis is the local yaxis.
 Parameters
angle – rotation angle in radians
 rotate_local_z(angle: float) UCS ¶
Returns a new rotated UCS, rotation axis is the local zaxis.
 Parameters
angle – rotation angle in radians
 shift(delta: UVec) UCS ¶
Shifts current UCS by delta vector and returns self.
 Parameters
delta – shifting vector
 moveto(location: UVec) 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: UVec, axis: UVec, point: UVec) UCS ¶
Returns a new
UCS
defined by the origin, the xaxis vector and an arbitrary point in the xyplane.
 static from_x_axis_and_point_in_xz(origin: UVec, axis: UVec, point: UVec) UCS ¶
Returns a new
UCS
defined by the origin, the xaxis vector and an arbitrary point in the xzplane.
 static from_y_axis_and_point_in_xy(origin: UVec, axis: UVec, point: UVec) UCS ¶
Returns a new
UCS
defined by the origin, the yaxis vector and an arbitrary point in the xyplane.
 static from_y_axis_and_point_in_yz(origin: UVec, axis: UVec, point: UVec) UCS ¶
Returns a new
UCS
defined by the origin, the yaxis vector and an arbitrary point in the yzplane.
 static from_z_axis_and_point_in_xz(origin: UVec, axis: UVec, point: UVec) UCS ¶
Returns a new
UCS
defined by the origin, the zaxis vector and an arbitrary point in the xzplane.
 static from_z_axis_and_point_in_yz(origin: UVec, axis: UVec, point: UVec) UCS ¶
Returns a new
UCS
defined by the origin, the zaxis vector and an arbitrary point in the yzplane.
 render_axis(layout: BaseLayout, length: float = 1, colors: RGB = (1, 3, 5))¶
Render axis as 3D lines into a layout.
Matrix44¶
 class ezdxf.math.Matrix44(*args)¶
An optimized 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
 classmethod scale(sx: float, sy: Optional[float] = None, sz: Optional[float] = None) Matrix44 ¶
Returns a scaling transformation matrix. If sy is
None
, sy = sx, and if sz isNone
sz = sx.
 classmethod translate(dx: float, dy: float, dz: float) Matrix44 ¶
Returns a translation matrix for translation vector (dx, dy, dz).
 classmethod x_rotate(angle: float) Matrix44 ¶
Returns a rotation matrix about the xaxis.
 Parameters
angle – rotation angle in radians
 classmethod y_rotate(angle: float) Matrix44 ¶
Returns a rotation matrix about the yaxis.
 Parameters
angle – rotation angle in radians
 classmethod z_rotate(angle: float) Matrix44 ¶
Returns a rotation matrix about the zaxis.
 Parameters
angle – rotation angle in radians
 classmethod axis_rotate(axis: UVec, angle: float) Matrix44 ¶
Returns a rotation matrix about an arbitrary axis.
 Parameters
axis – rotation axis as
(x, y, z)
tuple orVec3
objectangle – rotation angle in radians
 classmethod xyz_rotate(angle_x: float, angle_y: float, angle_z: float) Matrix44 ¶
Returns a rotation matrix for rotation about each axis.
 Parameters
angle_x – rotation angle about xaxis in radians
angle_y – rotation angle about yaxis in radians
angle_z – rotation angle about zaxis in radians
 classmethod shear_xy(angle_x: float = 0, angle_y: float = 0) Matrix44 ¶
Returns a translation matrix for shear mapping (visually similar to slanting) in the xyplane.
 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) Matrix44 ¶
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) Matrix44 ¶
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) Matrix44 ¶
Compose a transformation matrix from one or more matrices.
 static ucs(ux: Vec3 = X_AXIS, uy: Vec3 = Y_AXIS, uz: Vec3 = Z_AXIS, origin: Vec3 = NULLVEC) Matrix44 ¶
Returns a matrix for coordinate transformation from WCS to UCS. For transformation from UCS to WCS, transpose the returned matrix.
 Parameters
ux – xaxis for UCS as unit vector
uy – yaxis for UCS as unit vector
uz – zaxis 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 4tuples.
 columns() Iterable[tuple[float, ...]] ¶
Iterate over columns as 4tuples.
 __mul__(other: Matrix44) Matrix44 ¶
Returns a new matrix as result of the matrix multiplication with another matrix.
 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¶
3D bounding box. 

2D bounding box. 

Construction tool for 2D arcs. 

Construction tool for 2D rectangles. 

Construction tool for 2D circles. 

Construction tool for 3D ellipsis. 

Construction tool for 2D lines. 

Construction tool for 3D polylines. 

Construction tool for infinite 2D rays. 

Construction tool for 3D planes. 

Construction tools for 2D shapes. 

Immutable 2D vector class. 

Immutable 3D vector class. 
Vec3¶
 class ezdxf.math.Vec3(*args)¶
Immutable 3D vector class.
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()
, returnsVec3(0, 0, 0)
Vec3((x, y))
, returnsVec3(x, y, 0)
Vec3((x, y, z))
, returnsVec3(x, y, z)
Vec3(x, y)
, returnsVec3(x, y, 0)
Vec3(x, y, z)
, returnsVec3(x, y, z)
Addition, subtraction, scalar multiplication and scalar division left and righthanded 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¶
xaxis value
 y¶
yaxis value
 z¶
zaxis value
 xy¶
Vec3 as
(x, y, 0)
, projected on the xyplane.
 xyz¶
Vec3 as
(x, y, z)
tuple.
 magnitude¶
Length of vector.
 magnitude_xy¶
Length of vector in the xyplane.
 magnitude_square¶
Square length of vector.
 is_null¶
Vec3(0, 0, 0)
. Has a fixed absolute testing tolerance of 1e12! Type
True
if all components are close to zero
 angle¶
Angle between vector and xaxis in the xyplane in radians.
 angle_deg¶
Returns angle of vector and xaxis in the xyplane in degrees.
 spatial_angle¶
Spatial angle between vector and xaxis in radians.
 spatial_angle_deg¶
Spatial angle between vector and xaxis 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
anddict
.
 __getitem__(index: int) float ¶
Support for indexing:
v[0] is v.x
v[1] is v.y
v[2] is v.z
 __iter__() Iterator[float] ¶
Returns iterable of x, y and zaxis.
 __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 zaxis.
 classmethod list(items: Iterable[UVec]) list[ezdxf.math._vector.Vec3] ¶
Returns a list of
Vec3
objects.
 classmethod from_angle(angle: float, length: float = 1.0) Vec3 ¶
Returns a
Vec3
object from angle in radians in the xyplane, zaxis =0
.
 classmethod from_deg_angle(angle: float, length: float = 1.0) Vec3 ¶
Returns a
Vec3
object from angle in degrees in the xyplane, zaxis =0
.
 orthogonal(ccw: bool = True) Vec3 ¶
Returns orthogonal 2D vector, zaxis is unchanged.
 Parameters
ccw – counterclockwise 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 objectfactor – interpolation factor (0 = self, 1 = other, 0.5 = mid point)
 is_parallel(other: Vec3, *, rel_tol: float = 1e9, abs_tol: float = 1e12) bool ¶
Returns
True
if self and other are parallel to vectors.
 isclose(other: UVec, *, rel_tol: float = 1e9, abs_tol: float = 1e12) bool ¶
Returns
True
if self is close to other. Usesmath.isclose()
to compare all axis.Learn more about the
math.isclose()
function in PEP 485.
 __bool__() bool ¶
Returns
True
if vector is not (0, 0, 0).
 angle_about(base: UVec, target: UVec) float ¶
Returns counterclockwise 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 zaxis.
 Parameters
angle – angle in radians
 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¶
Plane¶
 class ezdxf.math.Plane(normal: Vec3, distance: float)¶
Construction tool for 3D planes.
Represents a plane in 3D space as a normal vector and the perpendicular distance from the 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.
 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.
 is_coplanar_vertex(v: Vec3, abs_tol=1e9) bool ¶
Returns
True
if vertex v is coplanar, distance from plane to vertex v is 0.
 is_coplanar_plane(p: Plane, abs_tol=1e9) 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=PLANE_EPSILON) 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 isFalse
the start or end point of the line are ignored as intersection points.
BoundingBox¶
 class ezdxf.math.BoundingBox(vertices: Optional[Iterable[UVec]] = None)¶
3D bounding box.
 Parameters
vertices – iterable of
(x, y, z)
tuples orVec3
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.
 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: UVec) bool ¶
Returns
True
if vertex is inside this bounding box.Vertices at the box border are inside!
 any_inside(vertices: Iterable[UVec]) bool ¶
Returns
True
if any vertex is inside this bounding box.Vertices at the box border are inside!
 all_inside(vertices: Iterable[UVec]) 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 alsohas_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 tohas_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 this bounding box.
 extend(vertices: Iterable[UVec]) 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) BoundingBox ¶
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() Sequence[Vec2] ¶
Returns the corners of the bounding box in the xyplane as
Vec2
objects.
 grow(value: float) None ¶
Grow or shrink the bounding box by an uniform value in x, y and zaxis. 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[UVec]] = None)¶
2D bounding box.
 Parameters
vertices – iterable of
(x, y[, z])
tuples orVec3
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: UVec) bool ¶
Returns
True
if vertex is inside this bounding box.Vertices at the box border are inside!
 any_inside(vertices: Iterable[UVec]) bool ¶
Returns
True
if any vertex is inside this bounding box.Vertices at the box border are inside!
 all_inside(vertices: Iterable[UVec]) 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 alsohas_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 tohas_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
 contains(other: AbstractBoundingBox) bool ¶
Returns
True
if the other bounding box is completely inside this bounding box.
 extend(vertices: Iterable[UVec]) 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) BoundingBox2d ¶
Returns the bounding box of the intersection rectangle of both 2D bounding boxes. Returns an empty bounding box if the intersection area is 0.
ConstructionRay¶
 class ezdxf.math.ConstructionRay(p1: UVec, p2: Optional[UVec] = None, angle: Optional[float] = None)¶
Construction tool for infinite 2D rays.
 Parameters
p1 – definition point 1
p2 – ray direction as 2nd point or
None
angle – ray direction as angle in radians or
None
 slope¶
Slope of ray or
None
if vertical.
 angle¶
Angle between xaxis and ray in radians.
 angle_deg¶
Angle between xaxis and ray in degrees.
 is_vertical¶
True
if ray is vertical (parallel to yaxis).
 is_horizontal¶
True
if ray is horizontal (parallel to xaxis).
 __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: UVec) ConstructionRay ¶
Returns orthogonal ray at location.
 bisectrix(other: ConstructionRay) ConstructionRay ¶
Bisectrix between self and other.
 yof(x: float) float ¶
Returns yvalue of ray for x location.
 Raises
ArithmeticError – for vertical rays
 xof(y: float) float ¶
Returns xvalue of ray for y location.
 Raises
ArithmeticError – for horizontal rays
ConstructionLine¶
 class ezdxf.math.ConstructionLine(start: UVec, end: UVec)¶
Construction tool for 2D lines.
The
ConstructionLine
class is similar toConstructionRay
, 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
 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 ¶
Move line about dx in xaxis and about dy in yaxis.
 Parameters
dx – translation in xaxis
dy – translation in yaxis
 length() float ¶
Returns length of line.
 intersect(other: ConstructionLine, abs_tol: float = TOLERANCE) Optional[Vec2] ¶
Returns the intersection point of to lines or
None
if they have no intersection point. Parameters
other – other
ConstructionLine
abs_tol – tolerance for distance check
 has_intersection(other: ConstructionLine, abs_tol: float = TOLERANCE) bool ¶
Returns
True
if has intersection with other line.
ConstructionCircle¶
 class ezdxf.math.ConstructionCircle(center: UVec, radius: float = 1.0)¶
Construction tool for 2D circles.
 Parameters
center – center point as
Vec2
compatible objectradius – circle radius > 0
 radius¶
radius as float
 bounding_box¶
2D bounding box of circle as
BoundingBox2d
object.
 static from_3p(p1: UVec, p2: UVec, p3: UVec) ConstructionCircle ¶
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 ¶
Move circle about dx in xaxis and about dy in yaxis.
 Parameters
dx – translation in xaxis
dy – translation in yaxis
 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 zaxis, xaxis = 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 counterclockwise around the zaxis, xaxis = 0 deg.
 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!
 tangent(angle: float) ConstructionRay ¶
Returns tangent to circle at angle as
ConstructionRay
object. Parameters
angle – angle in radians
 intersect_ray(ray: ConstructionRay, abs_tol: float = 1e10) 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
objectstuple 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 = 1e10) 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
objectstuple size
Description
0
no intersection
1
line intersects or touches the circle at one point
2
line intersects the circle at two points
 intersect_circle(other: ConstructionCircle, abs_tol: float = 1e10) 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
objectstuple 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: UVec = (0, 0), radius: float = 1.0, start_angle: float = 0.0, end_angle: float = 360.0, is_counter_clockwise: Optional[bool] = True)¶
Construction tool for 2D arcs.
ConstructionArc
represents a 2D arc in the xyplane, use anUCS
to place a DXFArc
entity in 3D space, see methodadd_to_layout()
.Implements the 2D transformation tools:
translate()
,scale_uniform()
androtate_z()
 Parameters
center – center point as
Vec2
compatible objectradius – radius
start_angle – start angle in degrees
end_angle – end angle in degrees
is_counter_clockwise – swaps start and end angle if
False
 radius¶
radius as float
 start_angle¶
start angle in degrees
 end_angle¶
end angle in degrees
 angle_span¶
Returns angle span of arc from start to end param.
 start_angle_rad¶
Returns the start angle in radians.
 end_angle_rad¶
Returns the end angle in radians.
 bounding_box¶
bounding box of arc as
BoundingBox2d
.
 angles(num: int) Iterable[float] ¶
Returns num angles from start to end angle in degrees in counterclockwise 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 zaxis, WCS xaxis = 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 counterclockwise around the zaxis, WCS xaxis = 0 deg.
 translate(dx: float, dy: float) ConstructionArc ¶
Move arc about dx in xaxis and about dy in yaxis, returns self (floating interface).
 Parameters
dx – translation in xaxis
dy – translation in yaxis
 scale_uniform(s: float) ConstructionArc ¶
Scale arc inplace uniform about s in x and yaxis, returns self (floating interface).
 rotate_z(angle: float) ConstructionArc ¶
Rotate arc inplace about zaxis, returns self (floating interface).
 Parameters
angle – rotation angle in degrees
 classmethod from_2p_angle(start_point: UVec, end_point: UVec, angle: float, ccw: bool = True) ConstructionArc ¶
Create arc from two points and enclosing angle. Additional precondition: arc goes by default in counterclockwise orientation from start_point to end_point, can be changed by ccw =
False
.
 classmethod from_2p_radius(start_point: UVec, end_point: UVec, radius: float, ccw: bool = True, center_is_left: bool = True) ConstructionArc ¶
Create arc from two points and arc radius. Additional precondition: arc goes by default in counterclockwise 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 ofcenter_is_left
.
 classmethod from_3p(start_point: UVec, end_point: UVec, def_point: UVec, ccw: bool = True) ConstructionArc ¶
Create arc from three points. Additional precondition: arc goes in counterclockwise orientation from start_point to end_point.
 add_to_layout(layout: BaseLayout, ucs: Optional[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 xyplane, but by using an arbitrary UCS, the arc can be placed in 3D space, automatically OCS transformation included. Parameters
layout – destination layout as
BaseLayout
objectucs – place arc in 3D space by
UCS
objectdxfattribs – additional DXF attributes for the ARC entity
 intersect_ray(ray: ConstructionRay, abs_tol: float = 1e10) 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
objectstuple size
Description
0
no intersection
1
line intersects or touches the arc at one point
2
line intersects the arc at two points
 intersect_line(line: ConstructionLine, abs_tol: float = 1e10) 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
objectstuple size
Description
0
no intersection
1
line intersects or touches the arc at one point
2
line intersects the arc at two points
 intersect_circle(circle: ConstructionCircle, abs_tol: float = 1e10) 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
objectstuple size
Description
0
no intersection
1
circle intersects or touches the arc at one point
2
circle intersects the arc at two points
 intersect_arc(other: ConstructionArc, abs_tol: float = 1e10) 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
objectstuple 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
ConstructionEllipse¶
 class ezdxf.math.ConstructionEllipse(center: UVec = NULLVEC, major_axis: UVec = X_AXIS, extrusion: UVec = Z_AXIS, ratio: float = 1, start_param: float = 0, end_param: float = math.tau, ccw: bool = True)¶
Construction tool for 3D ellipsis.
 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 counterclockwise flag  swaps start and end param if
False
 minor_axis¶
minor axis as
Vec3
, automatically calculated frommajor_axis
andextrusion
.
 ratio¶
ratio of minor axis to major axis (float)
 start¶
start param in radians (float)
 end¶
end param in radians (float)
 start_point¶
Returns start point of ellipse as Vec3.
 end_point¶
Returns end point of ellipse as Vec3.
 property param_span: float¶
Returns the counterclockwise params span from start to end param, see also
ezdxf.math.ellipse_param_span()
for more information.
 to_ocs() ConstructionEllipse ¶
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 counterclockwise 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 counterclockwise around the extrusion vector, major_axis = local xaxis = 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
 params_from_vertices(vertices: Iterable[UVec]) Iterable[float] ¶
Yields ellipse params for all given vertices.
The vertex don’t have 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[str, Any] ¶
Returns required DXF attributes to build an ELLIPSE entity.
Entity ELLIPSE has always a ratio in range from 1e6 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: UVec = NULLVEC, radius: float = 1, extrusion: UVec = Z_AXIS, start_angle: float = 0, end_angle: float = 360, ccw: bool = True) ConstructionEllipse ¶
Returns
ConstructionEllipse
from arc or circle.Arc and Circle parameters defined in OCS.
 Parameters
center – center in OCS
radius – arc or circle radius
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
 swap_axis() None ¶
Swap axis and adjust start and end parameter.
 add_to_layout(layout: BaseLayout, dxfattribs=None) Ellipse ¶
Add ellipse as DXF
Ellipse
entity to a layout. Parameters
layout – destination layout as
BaseLayout
objectdxfattribs – additional DXF attributes for the ELLIPSE entity
ConstructionBox¶
 class ezdxf.math.ConstructionBox(center: UVec = (0, 0), width: float = 1, height: float = 1, angle: float = 0)¶
Construction tool for 2D 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
 bounding_box¶
 incircle_radius¶
incircle radius
 circumcircle_radius¶
circum circle radius
 __repr__() str ¶
Returns string representation of box as
ConstructionBox(center, width, height, angle)
 classmethod from_points(p1: UVec, p2: UVec) ConstructionBox ¶
Creates a box from two opposite corners, box sides are parallel to x and yaxis.
 translate(dx: float, dy: float) None ¶
Move box about dx in xaxis and about dy in yaxis.
 Parameters
dx – translation in xaxis
dy – translation in yaxis
 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_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[ezdxf.math._vector.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[UVec], close: bool = False, rel_tol: float = REL_TOL)¶
Construction tool for 3D polylines.
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 doc = ezdxf.readfile("your.dxf") 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))
 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, ezdxf.math._vector.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.
Shape2d¶
 class ezdxf.math.Shape2d(vertices: Optional[Iterable[UVec]] = None)¶
Construction tools for 2D shapes.
A 2D geometry object as list of
Vec2
objects, vertices can be moved, rotated and scaled. Parameters
vertices – iterable of
Vec2
compatible objects.
 bounding_box¶
 __len__() int ¶
Returns count of vertices.
 append(vertex: UVec) 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
 scale(sx: float = 1.0, sy: float = 1.0) None ¶
Scale shape about sx in xaxis and sy in yaxis.
 scale_uniform(scale: float) None ¶
Scale shape uniform about scale in x and yaxis.
 rotate(angle: float, center: Optional[UVec] = None) None ¶
Rotate shape around rotation center about angle in degrees.
 rotate_rad(angle: float, center: Optional[UVec] = None) None ¶
Rotate shape around rotation center about angle in radians.
 offset(offset: float, closed: bool = False) Shape2d ¶
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 segmentclosed –
True
to handle as closed shape
Curves¶
Approximation tool for parametrized curves. 

Bspline construction tool. 

Generic Bézier curve of any degree. 

Implements an optimized quadratic Bézier curve for exact 3 control points. 

Implements an optimized cubic Bézier curve for exact 4 control points. 



This class represents an euler spiral (clothoid) for curvature (Radius of curvature). 
BSpline¶
 class ezdxf.math.BSpline(control_points: Iterable[UVec], order: int = 4, knots: Optional[Iterable[float]] = None, weights: Optional[Iterable[float]] = None)¶
Bspline construction tool.
Internal representation of a Bspline 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 objectsorder – spline order (degree + 1)
knots – iterable of knot values
weights – iterable of weight values
 property count: int¶
Count of control points, (n + 1 in text book notation).
 property order: int¶
Order (k) of Bspline = p + 1
 property degree: int¶
Degree (p) of Bspline = order  1
 property is_rational¶
Returns
True
if curve is a rational Bspline. (has weights)
 property is_clamped¶
Returns
True
if curve is a clamped (open) Bspline.
 knots() Sequence[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() Sequence[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.
 transform(m: Matrix44) BSpline ¶
Returns a new
BSpline
object transformed by aMatrix44
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
 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[ezdxf.math._vector.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[ezdxf.math._vector.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) BSpline ¶
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]) BSpline ¶
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) BSpline ¶
Returns the ellipse as
BSpline
of 2nd degree with as few control points as possible.
 static from_arc(arc: ConstructionArc) BSpline ¶
Returns the arc as
BSpline
of 2nd degree with as few control points as possible.
 static from_fit_points(points: Iterable[UVec], degree=3, method='chord') BSpline ¶
Returns
BSpline
defined by fit points.
 static arc_approximation(arc: ConstructionArc, num: int = 16) BSpline ¶
Returns an arc approximation as
BSpline
with num control points.
 static ellipse_approximation(ellipse: ConstructionEllipse, num: int = 16) BSpline ¶
Returns an ellipse approximation as
BSpline
with num control points.
 bezier_decomposition() Iterable[list[ezdxf.math._vector.Vec3]] ¶
Decompose a nonrational Bspline into multiple Bézier curves.
This is the preferred method to represent the most common nonrational Bsplines 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. Bspline of 3rd degree yields cubic Bézier curves of 4 control points.
 cubic_bezier_approximation(level: int = 3, segments: Optional[int] = None) Iterable[Bezier4P] ¶
Approximate arbitrary Bsplines (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 nonrational Bsplines, which is maybe the most common used Bspline, is
bezier_decomposition()
the better choice.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 Bspline, every additional level is a subdivision of the previous level.
E.g. a BSpline 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.
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[UVec])¶
Generic Bézier curve of any degree.
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
andBezier3P
classes.Objects are immutable.
 Parameters
defpoints – iterable of definition points as
Vec3
compatible objects.
 params(segments: int) Iterable[float] ¶
Yield evenly spaced parameters from 0 to 1 for given segment count.
 transform(m: Matrix44) Bezier ¶
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
 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[ezdxf.math._vector.Vec3, ezdxf.math._vector.Vec3, ezdxf.math._vector.Vec3] ¶
Returns (point, 1st derivative, 2nd derivative) tuple for parameter t in range [0, 1] as
Vec3
objects.
 derivatives(t: Iterable[float]) Iterable[tuple[ezdxf.math._vector.Vec3, ezdxf.math._vector.Vec3, ezdxf.math._vector.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])¶
Implements an optimized cubic Bézier curve for exact 4 control points.
A Bézier curve is a parametric curve, parameter t goes from 0 to 1, where 0 is the first control point and 1 is the fourth control point.
Special behavior:
 transform(m: Matrix44) Bezier4P ¶
General transformation interface, returns a new
Bezier4p
curve as 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[Union[Vec3, Vec2]] ¶
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
 approximated_length(segments: int = 128) float ¶
Returns estimated length of Bèziercurve as approximation by line segments.
 point(t: float) AnyVec ¶
Returns point for location t` at the Bèziercurve.
 Parameters
t – curve position in the range
[0, 1]
 tangent(t: float) AnyVec ¶
Returns direction vector of tangent for location t at the Bèziercurve.
 Parameters
t – curve position in the range
[0, 1]
Bezier3P¶
 class ezdxf.math.Bezier3P(defpoints: Sequence[UVec])¶
Implements an optimized quadratic Bézier curve for exact 3 control points.
Special behavior:
 transform(m: Matrix44) Bezier3P ¶
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 Bèziercurve as approximation by line segments.
 point(t: float) AnyVec ¶
Returns point for location t` at the Bèziercurve.
 Parameters
t – curve position in the range
[0, 1]
 tangent(t: float) AnyVec ¶
Returns direction vector of tangent for location t at the Bèziercurve.
 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, likeBezier4P
,Bezier3P
andBSpline
.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
 property max_t: float¶
 property 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[UVec]])¶
BezierSurface
defines a mesh of m x n control points. This is a parametric surface, which means the mdimension goes from0
to1
as parameter u and the ndimension goes from0
to1
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 (mdimension)
 ncols¶
count of columns (ndimension)
 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[ezdxf.math._vector.Vec3]] ¶
Approximate surface as grid of
(x, y, z)
Vec3
. Parameters
usegs – count of segments in udirection (mdimension)
vsegs – count of segments in vdirection (ndimension)
 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
curvature – radius of curvature
 radius(t: float) float ¶
Get radius of circle at distance t.
 distance(radius: float) float ¶
Get distance L from origin for radius.
 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') BSpline ¶
Approximate euler spiral as Bspline.
 Parameters
length – length of euler spiral
segments – count of fit points for Bspline calculation
degree – degree of BSpline
method – calculation method for parameter vector t
 Returns