# 1.1. Frame and Transformation

Frame and Transformation are two basic classes in the COMPAS framework and can be used to describe position/orientation and coordinate systems. The Frame consists of a point and and two orthonormal base vectors (xaxis, yaxis). Transformation is the base class for transformations like Rotation, Translation, Scale, Reflection, Projection and Shear.

Here is a simple example of how to use a frame as a coordinate system: Starting from a point P in the local (user-defined, relative) coordinate system of frame F, i.e. its position is relative to the origin and orientation of F, we want to get the position of P in the global (world, absolute) coordinate system.

from compas.geometry import Point
from compas.geometry import Vector
from compas.geometry import Frame

point = Point(146.00, 150.00, 161.50)
xaxis = Vector(0.9767, 0.0010, -0.214)
yaxis = Vector(0.1002, 0.8818, 0.4609)

# coordinate system F
F = Frame(point, xaxis, yaxis)

# point in F (local coordinates)
P = Point(35., 35., 35.)
# point in global (world) coordinates
P_ = F.represent_point_in_global_coordinates(P)


Industrial robots do not have a common way of describing the pose orientation. The frame provides methods to specify the orientation in various conventions.

from compas.geometry import Point
from compas.geometry import Vector
from compas.geometry import Frame

point = Point(146.00, 150.00, 161.50)
xaxis = Vector(0.9767, 0.0010, -0.214)
yaxis = Vector(0.1002, 0.8818, 0.4609)

F = Frame(point, xaxis, yaxis)

print(F.quaternion) # ABB
print(F.euler_angles(static=False, axes='xyz')) # Staubli
print(F.euler_angles(static=False, axes='zyx')) # KUKA
print(F.axis_angle_vector) # UR


From the orientation (Rotation) of the frame, several other representations of rotation can be derived, such as Euler angles, axis-angle representation, and quaternion.

from compas.geometry import Frame
from compas.geometry import Rotation

F1 = Frame([0, 0, 0], [0.68, 0.68, 0.27], [-0.67, 0.73, -0.15])

# euler angles
args = False, 'xyz'
alpha, beta, gamma = F1.euler_angles(*args)

# check if angles are correct
xaxis, yaxis, zaxis = [1, 0, 0], [0, 1, 0], [0, 0, 1]
Rx = Rotation.from_axis_and_angle(xaxis, alpha)
Ry = Rotation.from_axis_and_angle(yaxis, beta)
Rz = Rotation.from_axis_and_angle(zaxis, gamma)
F2 = Frame.worldXY()
print('Are equal?', F1 == F2.transformed(Rx * Ry * Rz))

# quaternion
q = F1.quaternion
F2 = Frame.from_quaternion(q)
print('Are equal?', F1 == F2)

# axis-angle
ax = F1.axis_angle_vector
F2 = Frame.from_axis_angle_vector(ax)
print('Are equal?', F1 == F2)