# Object Coordinate System (OCS)¶

The points associated with each entity are expressed in terms of the entity’s own object coordinate system (OCS). The OCS was referred to as ECS in previous releases of AutoCAD.

With OCS, the only additional information needed to describe the entity’s position in 3D space is the 3D vector describing the z-axis of the OCS (often referenced as extrusion vector), and the elevation value, which is the distance of the entity xy-plane to the WCS/OCS origin.

For a given z-axis (extrusion) direction, there are an infinite number of coordinate systems, defined by translating the origin in 3D space and by rotating the x- and y-axis around the z-axis. However, for the same z-axis direction, there is only one OCS. It has the following properties:

• Its origin coincides with the WCS origin.

• The orientation of the x- and y-axis within the xy-plane are calculated in an arbitrary but consistent manner. AutoCAD performs this calculation using the arbitrary axis algorithm (see below).

• Because of the Arbitrary Axis Algorithm the OCS can only represent a right-handed coordinate system!

The following entities do not lie in a particular plane. All points are expressed in world coordinates. Of these entities, only lines and points can be extruded. Their extrusion direction can differ from the world z-axis.

These entities are planar in nature. All points are expressed in object coordinates. All of these entities can be extruded. Their extrusion direction can differ from the world z-axis.

Some of a `Dimension`’s points are expressed in WCS and some in OCS.

## Elevation¶

Elevation group code 38:

Exists only in output from versions prior to R11. Otherwise, Z coordinates are supplied as part of each of the entity’s defining points.

## Arbitrary Axis Algorithm¶

The arbitrary axis algorithm is used by AutoCAD internally to implement the arbitrary but consistent generation of object coordinate systems for all entities that use object coordinates.

Given a unit-length vector to be used as the z-axis of a coordinate system, the arbitrary axis algorithm generates a corresponding x-axis for the coordinate system. The y-axis follows by application of the right-hand rule.

We are looking for the arbitrary x- and y-axis to go with the normal Az (the arbitrary z-axis). They will be called Ax and Ay (using `Vec3`):

```Az = Vec3(entity.dxf.extrusion).normalize()  # normal (extrusion) vector
if (abs(Az.x) < 1/64.) and (abs(Az.y) < 1/64.):
Ax = Vec3(0, 1, 0).cross(Az).normalize()  # the cross-product operator
else:
Ax = Vec3(0, 0, 1).cross(Az).normalize()  # the cross-product operator
Ay = Az.cross(Ax).normalize()
```

## WCS to OCS¶

```def wcs_to_ocs(point):
px, py, pz = Vec3(point)  # point in WCS
x = px * Ax.x + py * Ax.y + pz * Ax.z
y = px * Ay.x + py * Ay.y + pz * Ay.z
z = px * Az.x + py * Az.y + pz * Az.z
return Vec3(x, y, z)
```

## OCS to WCS¶

```Wx = wcs_to_ocs((1, 0, 0))
Wy = wcs_to_ocs((0, 1, 0))
Wz = wcs_to_ocs((0, 0, 1))

def ocs_to_wcs(point):
px, py, pz = Vec3(point)  # point in OCS
x = px * Wx.x + py * Wx.y + pz * Wx.z
y = px * Wy.x + py * Wy.y + pz * Wy.z
z = px * Wz.x + py * Wz.y + pz * Wz.z
return Vec3(x, y, z)
```