No, perspective projection is fundamentally non-linear.
Understanding Perspective Projection
Perspective projection is a method used in computer graphics and vision to simulate how objects appear smaller the farther away they are from the viewer. It mimics the way we see the world, where parallel lines seem to converge at a vanishing point on the horizon.
Think about standing on a long, straight road; the edges of the road appear to meet in the distance. This effect is captured by perspective projection.
Why Perspective Projection is Non-Linear
The core reason perspective projection is non-linear lies in the calculation used to project a 3D point onto a 2D plane. The coordinates of the projected point depend not just on the point's position but also on its depth (distance from the viewpoint).
Specifically, the transformation involves division by the depth coordinate (often denoted as 'z'). This division makes the transformation non-linear.
As the reference states: "Perspective projections are non-linear, so they can't be represented by matrices...".
Implications of Non-Linearity
Because perspective projection is non-linear, it cannot be represented by a simple 4x4 matrix multiplication like linear transformations such as translation, rotation, or scaling can be.
This is why, in graphics pipelines, perspective projection is often handled separately or represented using homogeneous coordinates, which allow the non-linear division by depth to be incorporated into a matrix operation by performing it as the final step (the 'perspective divide'). Even then, the transformation before the division step is linear in homogeneous space, but the overall mapping from 3D object space to 2D screen space after the divide is non-linear.
Here are some key implications:
- Matrix Representation: Cannot be represented by a standard linear transformation matrix in 3D Euclidean space.
- Preservation of Properties: Does not preserve ratios of lengths or angles (unlike orthogonal projection, which is linear). Parallel lines that are not parallel to the projection plane converge.
- Implementation: Requires special handling in graphics pipelines, often involving homogeneous coordinates and a perspective divide step.
In summary, the division by depth is the defining characteristic that makes perspective projection a non-linear transformation.
