chore: Unify math types, utils and functions (#8389)

Co-authored-by: dwelle <5153846+dwelle@users.noreply.github.com>

packages/math/ga/galines.ts

@@ -0,0 +1,52 @@
+import * as GA from "./ga";
+import type { Line, Point } from "./ga";
+
+/**
+ * A line is stored as an array `[0, c, a, b, 0, 0, 0, 0]` representing:
+ *   c * e0 + a * e1 + b*e2
+ *
+ * This maps to a standard formula `a * x + b * y + c`.
+ *
+ * `(-b, a)` corresponds to a 2D vector parallel to the line. The lines
+ * have a natural orientation, corresponding to that vector.
+ *
+ * The magnitude ("norm") of the line is `sqrt(a ^ 2 + b ^ 2)`.
+ * `c / norm(line)` is the oriented distance from line to origin.
+ */
+
+// Returns line with direction (x, y) through origin
+export const vector = (x: number, y: number): Line =>
+  GA.normalized([0, 0, -y, x, 0, 0, 0, 0]);
+
+// For equation ax + by + c = 0.
+export const equation = (a: number, b: number, c: number): Line =>
+  GA.normalized([0, c, a, b, 0, 0, 0, 0]);
+
+export const through = (from: Point, to: Point): Line =>
+  GA.normalized(GA.join(to, from));
+
+export const orthogonal = (line: Line, point: Point): Line =>
+  GA.dot(line, point);
+
+// Returns a line perpendicular to the line through `against` and `intersection`
+// going through `intersection`.
+export const orthogonalThrough = (against: Point, intersection: Point): Line =>
+  orthogonal(through(against, intersection), intersection);
+
+export const parallel = (line: Line, distance: number): Line => {
+  const result = line.slice();
+  result[1] -= distance;
+  return result as unknown as Line;
+};
+
+export const parallelThrough = (line: Line, point: Point): Line =>
+  orthogonal(orthogonal(point, line), point);
+
+export const distance = (line1: Line, line2: Line): number =>
+  GA.inorm(GA.meet(line1, line2));
+
+export const angle = (line1: Line, line2: Line): number =>
+  Math.acos(GA.dot(line1, line2)[0]);
+
+// The orientation of the line
+export const sign = (line: Line): number => Math.sign(line[1]);