Projective transformations primarily preserve collinearity, meaning points that lie on a line before the transformation will still lie on a line after the transformation.
A projective transformation, also known as a homography, is a significant concept in geometry and computer vision. As stated in the provided reference (15.2.), it is a non-linear transformation often used to map image points between different camera views, assuming a planar scene.
While they are powerful for relating different perspectives, it's crucial to understand what properties remain invariant (preserved) under these transformations and what properties do not.
Key Properties Preserved by Projective Transformations
Based on the reference and geometric principles, the fundamental property preserved by a projective transformation is:
- Collinearity: Points that lie on a straight line are mapped to points that also lie on a straight line. This is a defining characteristic and essential for many applications.
Properties Not Preserved
Unlike simpler transformations like affine transformations or rigid transformations (like translation, rotation, scaling), projective transformations do not preserve several properties that we commonly associate with Euclidean geometry. The reference specifically highlights that it does not preserve parallelism, lengths, and angles of lines.
Here's a breakdown of properties that are not preserved:
- Parallelism: Lines that are parallel in the original space may not remain parallel after a projective transformation. They can converge at a finite point.
- Lengths: The distances between points are not preserved. Segments can become longer or shorter.
- Angles: The angles between intersecting lines are generally not preserved.
- Ratios of Lengths on Parallel Lines: While affine transformations preserve ratios of lengths along parallel lines, projective transformations do not.
- Midpoints: The midpoint of a line segment is generally not mapped to the midpoint of the transformed segment.
Summary of Preservation
We can summarize the preservation properties using a simple table:
| Property | Preserved? |
|---|---|
| Collinearity | Yes |
| Parallelism | No |
| Lengths | No |
| Angles | No |
Practical Implications
In applications like computer vision, where projective transformations map points between different camera perspectives of a planar surface (like a wall or the ground), the preservation of collinearity is vital. It ensures that straight edges in the real world appear as straight edges in the image, even if the camera is tilted or at an angle. However, because lengths, angles, and parallelism are not preserved, objects can appear distorted, lines can converge (vanishing points), and relative sizes can change depending on their position in the scene and the camera's viewpoint.
