The Eye Is the Centre of Projection

Spend an hour under a camera lucida and you stop thinking of it as a drawing aid and start thinking of it as a coordinate transform you can look through. The prism drops the far wall of the room onto your paper, and your pencil traces where each point lands. Move your head a centimetre and everything slides, because the projection has exactly one privileged point: your eye. That is the centre of projection. Everything the device does, it does relative to that one hole in space.

Which is the pinhole camera model, drawn by hand. A point at world depth z, viewed from a paper plane at distance d, doesn’t arrive at its true (x, y) — it arrives scaled by d / (d + z). Far things shrink toward a vanishing point; near things spread. The whole magic of the lucida is that it performs that division optically, for free, while your slow biological hand copies the result.

Here’s the divide itself, nothing hidden:

(define (project pt d)
  (let ((x (car pt)) (y (cadr pt)) (z (caddr pt)))
    (let ((w (/ d (+ d z))))          ; the perspective divide
      (list (* x w) (* y w)))))       ; where it lands on paper

(for-each
  (lambda (p) (display p) (display " -> ") (display (project p 10)) (newline))
  '((4 4 0) (4 4 10) (4 4 30)))

(4 4 0)  -> (4 4)
(4 4 10) -> (2 2)
(4 4 30) -> (1 1)

One corner of the room, held at the same height and width, marching away from you — and on paper it walks straight toward the centre. That convergence is the thing a lucida forces onto your retina whether your intuition agrees or not.

By the 1980s this had hardened into the graphics pipeline: stuff w into a fourth coordinate, let the hardware divide by it after the matrix multiply. Same arithmetic, wearing a homogeneous coat:

public class Lucida {
    static double[] project(double x, double y, double z, double d) {
        double w = z + d;                       // homogeneous w
        return new double[]{ x * d / w, y * d / w };
    }
    public static void main(String[] a) {
        double[][] pts = {{4,4,0},{4,4,10},{4,4,30}};
        for (double[] p : pts)
            System.out.printf("(%.2f, %.2f)%n",
                project(p[0],p[1],p[2],10)[0], project(p[0],p[1],p[2],10)[1]);
    }
}

The Renaissance draughtsman under a prism and a Silicon Graphics box in 1985 are running the same instruction. One divides in glass and photons; the other divides in silicon and floating point. My pencil, tonight, is just the world’s slowest rasterizer.