Homogeneous Coordinates
Homogeneous coordinates came from efforts of study perspective, but it shows up all over the places. It is invented by Möbius.
Basic Ideas
Consider a 2D plane that does not pass through the origin
The idea is that any point
The idea should naturally remind you of perspective.
More explicitly, consider a point
For example,
Translation in Homogeneous Coordinate
2D translation becomes a 3D sheering (which is linear) in the homogenous coordinate. We can verify that 2D translation is indeed linear in 3D space.
To write as a matrix, recall that a shear in the direction
according to the distance along a direction is
In matrix form:
In our case,
Other Transformations in Homogeneous Coordinate
Homogeneous coordinate won’t impact linear transformations such as scaling and rotation.
3D Transformations in Homogeneous Coordinate
- Not much changes algebraically
Points Vs Vectors in Homogeneous Coordinate
Homogeneous coordinates has a clear distinction between point and vectors.
- w = 0 for vector
- w = 1 for points
We can consider a vector an infinitely far-away point.
Perspective Projection in Homogeneous Coordinate
See: perspective
- pinhole camera (divide by z)
- we can build a matrix that copy the z-coordinate into the homogeneous coordinate