Rotating Vectors Without Floating Point

To mark the hour lines on a horizontal sundial, you need to compute the angle H for each hour from noon:

tan(H) = tan(15° × hours) × sin(latitude)

For my latitude in Alberta — about 53.5°N — that means evaluating tangent and sine. On paper, you’d look them up in tables. In 1624, Edmund Gunter published a seven-place table of logarithms specifically so navigators and surveyors could avoid multiplication.

Three hundred years later, Jack Volder at Convair was designing navigation computers for B-58 bombers. He needed trig functions in real-time, on hardware that could barely add. No lookup tables — too much memory. No Taylor series — too many multiplications. His solution, published in 1959: rotate a vector toward the target angle using only shifts and adds.

The trick is that arctan(2^-i) forms a predictable sequence. Rotating by those angles means multiplying coordinates by powers of two — which is just bit-shifting. You converge on any angle through a sequence of “rotate clockwise” or “rotate anticlockwise” decisions.

const ATAN = [0.7854, 0.4636, 0.2449, 0.1244, 0.0624, 0.0312];
const K = 0.6073;

function cordic(theta: number): [number, number] {
  let [x, y, a] = [K, 0, 0];
  for (let i = 0; i < ATAN.length; i++) {
    const d = theta > a ? 1 : -1;
    [x, y] = [x - d * y * 2 ** -i, y + d * x * 2 ** -i];
    a += d * ATAN[i];
  }
  return [x, y]; // [cos(θ), sin(θ)]
}
console.log(cordic(Math.PI / 4)); // ≈ [0.707, 0.707]

let atan = [|0.7854; 0.4636; 0.2449; 0.1244; 0.0624; 0.0312|]
let k = 0.6073

let cordic theta =
  let rec go i x y a =
    if i >= Array.length atan then (x, y) else
    let d = if theta > a then 1. else -1. in
    let s = 2. ** float (-i) in
    go (i+1) (x -. d *. y *. s) (y +. d *. x *. s) (a +. d *. atan.(i))
  in go 0 k 0. 0.

let () = let c, s = cordic (Float.pi /. 4.) in
  Printf.printf "cos: %.4f, sin: %.4f\n" c s

Six iterations, twelve additions, no multiplications — and you get three decimal places of precision. HP used CORDIC in their scientific calculators from 1972 onward. The algorithm handles sine, cosine, arctangent, and even hyperbolic functions by changing the rotation direction logic.

There’s a pleasing symmetry here. Sundials predate mechanical clocks by millennia, yet designing one sends you straight to trigonometry. And computing that trigonometry without modern hardware sends you to an algorithm from the same decade that launched humans toward the moon.