The Geometry of Sight
When you look down a long straight road, the parallel edges appear to converge toward a point on the horizon. This is not an optical illusion — it is the fundamental operation of projective geometry. Renaissance artists discovered perspective drawing by formalizing this convergence, and mathematicians later built an entire geometry around it where parallel lines always meet — at infinity.
Invariants Under Projection
Projective geometry strips away distances and angles, keeping only what survives projection: collinearity (points on a line remain on a line) and the cross-ratio of four collinear points. This single numerical invariant, (AC·BD)/(BC·AD), remains unchanged no matter how severely you project or distort the figure. It is the cornerstone of projective measurement.
From Art to Computer Vision
Projective geometry is the mathematical foundation of computer vision. Every camera performs a projective transformation from 3D world to 2D image. Homography matrices describe how planar surfaces transform between views. Structure-from-motion algorithms recover 3D geometry from multiple 2D images using projective invariants — the same mathematics that Brunelleschi used to paint convincing depth in 1420.
Exploring Perspective
This simulation lets you manipulate field of view, tilt, and pan to see how a 3D grid projects onto a 2D canvas. Watch parallel grid lines converge to vanishing points. Increase the field of view to create dramatic wide-angle distortion. The cross-ratio readout confirms that this projective invariant remains constant as you transform the view.