Projective geometric algebra
Frames, coordinates, and change of basis in PGA 2D
PGA motors fit the same transform vocabulary used in graphics systems: local space, world space, parent-child transforms, model transforms, and view transforms. The useful part is that the same sandwich operation applies to points, lines, and shapes.
The previous pages used motors as active motions: a point or shape moved because a motor acted on it. A frame can use the same motor differently. If the frame names local space inside world space, its reverse extracts the local expression of a fixed world object.
In passive mode the world point stays fixed while its local PGA representative changes. In active mode the same frame motor moves a local shape into world space, like a model transform in a scene graph.
A Frame Is A Motor
A local frame can be built from a translation and a rotation:
frame = translator * rotor
inverseFrame = reverse(frame)Read frame as local-to-world. A local point becomes a world point by direct sandwich action:
P_world = frame >>> P_localIn the active model-transform reading, the geometric point being acted on changes place in world space.
Passive Coordinate Change
For a passive frame change, the world object stays fixed. We ask for the same object expressed in the local frame:
P_local = reverse(frame) >>> P_worldMoving or rotating the frame changes P_local because the basis changed. Applying the frame again reconstructs the same world object:
P_world = frame >>> P_localThe calculation is ordinary frame-relative graphics math written with PGA motors instead of matrices.
Graphics Vocabulary
A model transform maps local object geometry into world space. A parent-child transform composes one local frame with another. A view transform is the inverse of a camera frame: world geometry is expressed in camera-local coordinates before projection.
The same active/passive distinction is present in all of those names. Active motion changes the object. Passive frame change changes the coordinates used to describe the object.
Homogeneous 1-Up Coordinates
The same 2D primitives can be read in the homogeneous 1-up space for Algebra(2,0,1). The local frame axes, points, and shape are still 2D PGA objects; the extra coordinate is the projective scale behind their homogeneous representatives.