---
name: pixijs-math
description: "Use this skill when working with coordinates, vectors, matrices, shapes, hit testing, or layout rectangles in PixiJS v8. Covers Point/ObservablePoint, Matrix (2D affine, decompose, apply, applyInverse), shapes (Rectangle, Circle, Ellipse, Polygon, RoundedRectangle, Triangle), Rectangle layout helpers (pad, fit, enlarge, ceil, scale, getBounds), strokeContains hit tests, Polygon isClockwise/containsPolygon, toGlobal/toLocal, PointData/PointLike/Size types, DEG_TO_RAD, and pixi.js/math-extras vector and intersection helpers. Triggers on: Point, ObservablePoint, Matrix, Rectangle, Circle, Polygon, Triangle, RoundedRectangle, toGlobal, toLocal, hitArea, strokeContains, pad, fit, enlarge, ceil, getBounds, containsRect, intersects, isClockwise, math-extras, lineIntersection, segmentIntersection, DEG_TO_RAD, PointData."
license: MIT
---

PixiJS exposes lightweight math primitives (Point, Matrix, shape classes) used throughout the library for transforms, hit testing, and coordinate conversion. Import `pixi.js/math-extras` to add vector operations (add, dot, magnitude, reflect) and Rectangle intersection/union helpers.

## Quick Start

```ts
const parent = new Container();
parent.position.set(100, 100);
parent.scale.set(2);
app.stage.addChild(parent);

const child = new Container();
child.position.set(50, 50);
parent.addChild(child);

const globalPt = child.toGlobal(new Point(0, 0));

const m = new Matrix()
  .translate(100, 50)
  .rotate(Math.PI / 4)
  .scale(2, 2);
const world = m.apply(new Point(10, 20));

const hitArea = new Rectangle(0, 0, 200, 100);
console.log(hitArea.contains(50, 50));
```

**Related skills:** `pixijs-scene-container` (transform properties), `pixijs-events` (hitArea usage), `pixijs-scene-core-concepts` (culling with Rectangle).

## Core Patterns

### Point and ObservablePoint

Point is a simple {x, y} value type. ObservablePoint fires a callback when x or y changes; it is used internally by Container's position, scale, pivot, origin, and skew.

```ts
import { Point } from "pixi.js";

const p = new Point(10, 20);
p.set(30, 40); // set both
p.set(50); // x=50, y=50

const clone = p.clone();
console.log(p.equals(clone)); // true

p.copyFrom({ x: 1, y: 2 }); // accepts any PointData

// Point.shared: temporary point, reset to (0,0) on each access
const temp = Point.shared;
temp.set(100, 200);
// do not store a reference to Point.shared
```

Container properties like `position`, `scale`, `pivot`, `origin`, and `skew` are ObservablePoints. Setting `.x` or `.y` on them triggers transform recalculation automatically.

```ts
import { Container } from "pixi.js";

const obj = new Container();
obj.position.set(100, 200); // triggers observer -> marks transform dirty
obj.position.x = 150; // also triggers observer
```

### Matrix (2D affine transform)

Matrix represents a 3x3 affine transform: `| a c tx | b d ty | 0 0 1 |`. It supports translate, scale, rotate, append, prepend, invert, and decompose.

```ts
import { Matrix, Point } from "pixi.js";

// Build a transform
const m = new Matrix()
  .translate(100, 50)
  .rotate(Math.PI / 4)
  .scale(2, 2);

// Transform a point (local -> parent space)
const local = new Point(10, 20);
const world = m.apply(local);

// Inverse transform (parent -> local space)
const backToLocal = m.applyInverse(world);

// Combine matrices
const a = new Matrix().translate(50, 0);
const b = new Matrix().rotate(Math.PI / 2);
a.append(b); // a = a * b

// Decompose into position/scale/rotation/skew
const transform = {
  position: new Point(),
  scale: new Point(),
  pivot: new Point(),
  skew: new Point(),
  rotation: 0,
};
m.decompose(transform);
console.log(transform.rotation); // ~0.785 (PI/4)

// Shared temporary matrix (reset on each access)
const temp = Matrix.shared;
// IDENTITY is read-only reference
const isDefault = m.equals(Matrix.IDENTITY);
```

### Coordinate transforms via Container

Containers provide `toGlobal`, `toLocal`, and `getGlobalPosition` for coordinate conversion.

```ts
import { Container, Point } from "pixi.js";

const parent = new Container();
parent.position.set(100, 100);
parent.scale.set(2);

const child = new Container();
child.position.set(50, 50);
parent.addChild(child);

// Local point in child's space -> global (world) space
const globalPt = child.toGlobal(new Point(0, 0));
// globalPt = { x: 200, y: 200 } (100 + 50*2, 100 + 50*2)

// Global point -> child's local space
const localPt = child.toLocal(new Point(200, 200));
// localPt = { x: 0, y: 0 }

// Convert between two containers
const other = new Container();
other.position.set(300, 300);
const ptInOther = child.toLocal(new Point(10, 10), other);
```

### Shapes and hit testing

Rectangle, Circle, Ellipse, Polygon, RoundedRectangle, and Triangle all implement `contains(x, y)` for point-in-shape tests, plus `getBounds(out?)` and `strokeContains(x, y, width, alignment?)`. They can be used as `hitArea` on containers for custom interaction regions.

```ts
import { Rectangle, Circle, Polygon, Container } from "pixi.js";

const rect = new Rectangle(0, 0, 200, 100);
rect.contains(50, 50); // true
rect.contains(300, 50); // false
rect.left; // 0
rect.right; // 200
rect.top; // 0
rect.bottom; // 100
rect.isEmpty(); // false (Rectangle.EMPTY returns a fresh empty rect)

// Native Rectangle-to-Rectangle methods (no math-extras needed)
const other = new Rectangle(50, 50, 100, 100);
rect.containsRect(other); // true if `other` is fully inside `rect`
rect.intersects(other); // boolean: do they overlap at all?
rect.intersects(other, matrix); // overlap after transforming `other`

// Stroke hit testing (alignment: 1 = inner, 0.5 = centered, 0 = outer)
rect.strokeContains(0, 50, 4); // true if (0,50) lies on a 4px centered stroke
const circle = new Circle(100, 100, 50);
circle.strokeContains(150, 100, 4, 1); // inner-aligned stroke check

// getBounds works on every shape (returns a Rectangle, accepts an out param)
const bounds = circle.getBounds();
const reused = new Rectangle();
new Polygon([0, 0, 100, 0, 50, 100]).getBounds(reused);

// Use as hit area for interaction
const button = new Container();
button.hitArea = new Rectangle(0, 0, 200, 50);
button.eventMode = "static";
button.on("pointerdown", () => {
  /* clicked */
});
```

Do not confuse native `Rectangle.intersects(other)` (returns `boolean`) with math-extras `intersection(other)` (returns a `Rectangle` describing the overlap area).

### Rectangle layout helpers

Rectangle ships with mutating helpers used heavily in UI/layout, bounds aggregation, and pixel snapping. All return `this` for chaining.

```ts
import { Rectangle } from "pixi.js";

const r = new Rectangle(10, 10, 100, 50);

r.pad(5); // grow on all sides: x=5, y=5, w=110, h=60
r.pad(10, 4); // separate horizontal/vertical padding
r.scale(2); // multiply x, y, width, height by 2

// fit shrinks `this` to lie inside another rect (clipping)
const viewport = new Rectangle(0, 0, 200, 200);
new Rectangle(150, 150, 200, 200).fit(viewport); // -> 150, 150, 50, 50

// enlarge expands `this` to include another rect (bounds aggregation)
const total = new Rectangle();
items.forEach((item) =>
  total.enlarge(new Rectangle().copyFromBounds(item.getBounds())),
);

// ceil snaps to a pixel grid (resolution: 1 = whole pixels, 2 = half pixels)
new Rectangle(10.2, 10.6, 100.8, 100.4).ceil();

// copy a Container/Mesh bounds object straight into a Rectangle
new Rectangle().copyFromBounds(container.getBounds());
```

### Polygon

Polygon accepts four constructor formats: a flat number array, an array of point-like objects, or either passed as spread arguments.

```ts
import { Polygon, Point } from "pixi.js";

new Polygon([0, 0, 100, 0, 50, 100]); // flat numbers
new Polygon([new Point(0, 0), new Point(100, 0), new Point(50, 100)]); // PointData[]
new Polygon(0, 0, 100, 0, 50, 100); // spread numbers
new Polygon(new Point(0, 0), new Point(100, 0), new Point(50, 100)); // spread points

const poly = new Polygon([0, 0, 100, 0, 100, 100, 0, 100]);
poly.points; // [0, 0, 100, 0, 100, 100, 0, 100] (mutable flat array)
poly.closePath; // true by default; false produces an open path
poly.startX; // 0  - first vertex
poly.lastX; // 0  - last vertex (lastY for y)
poly.isClockwise(); // shoelace winding test (useful for SVG hole detection)

// Polygon-in-polygon containment for hole detection
const outer = new Polygon([0, 0, 100, 0, 100, 100, 0, 100]);
const hole = new Polygon([25, 25, 75, 25, 75, 75, 25, 75]);
outer.containsPolygon(hole); // true
```

### Constants

```ts
import { DEG_TO_RAD, RAD_TO_DEG, PI_2 } from "pixi.js";

const angle = 45 * DEG_TO_RAD; // 0.785...
const degrees = angle * RAD_TO_DEG; // 45
const fullCircle = PI_2; // Math.PI * 2
```

### Types

- `PointData` - minimal `{x, y}` interface accepted by most APIs. Use it when typing parameters that only need to read coordinates.
- `PointLike` - extends `PointData` with `set()`, `copyFrom()`, `copyTo()`, `equals()`. Implemented by both `Point` and `ObservablePoint`.
- `Size` - `{ width, height }` interface used by renderer/canvas APIs.
- `SHAPE_PRIMITIVE` - string literal union: `'rectangle' | 'circle' | 'ellipse' | 'polygon' | 'roundedRectangle' | 'triangle'`. Every shape exposes `type` so you can branch without `instanceof`.

```ts
import type { PointData } from "pixi.js";

function distance(a: PointData, b: PointData): number {
  const dx = a.x - b.x;
  const dy = a.y - b.y;
  return Math.sqrt(dx * dx + dy * dy);
}
```

### math-extras (side-effect import)

`import 'pixi.js/math-extras'` adds methods to Point, ObservablePoint, and Rectangle via prototype extension. Not included in the default bundle.

```ts
import "pixi.js/math-extras";
import { Point } from "pixi.js";
```

#### Point / ObservablePoint vector methods

All methods accept an optional `out` parameter to avoid allocations. Without `out`, a new Point is returned.

```ts
const a = new Point(3, 4);
const b = new Point(1, 2);

// Arithmetic
const sum = a.add(b); // Point(4, 6)
const diff = a.subtract(b); // Point(2, 2)
const prod = a.multiply(b); // Point(3, 8) - component-wise
const scaled = a.multiplyScalar(2); // Point(6, 8)

// Dot and cross product
const dot = a.dot(b); // 11
const cross = a.cross(b); // 2 (z-component of 3D cross)

// Length
const len = a.magnitude(); // 5
const lenSq = a.magnitudeSquared(); // 25 (faster for comparisons)

// Normalize to unit vector
const unit = a.normalize(); // Point(0.6, 0.8)

// Projection and reflection
const proj = a.project(b); // project a onto b
const refl = a.reflect(new Point(0, 1)); // reflect across normal

// Rotation
const rotated = a.rotate(Math.PI / 2); // rotate 90 degrees

// Reuse existing point to avoid allocation
const out = new Point();
a.add(b, out); // result written to out
```

#### Rectangle extended methods

`containsRect` and `intersects` are native `Rectangle` methods (see above). math-extras adds `equals`, `intersection` (returns the overlap rect), and `union`:

```ts
import "pixi.js/math-extras";
import { Rectangle } from "pixi.js";

const r1 = new Rectangle(0, 0, 100, 100);
const r2 = new Rectangle(50, 50, 100, 100);

r1.equals(r2); // false

const overlap = r1.intersection(r2); // Rectangle(50, 50, 50, 50)
const envelope = r1.union(r2); // Rectangle(0, 0, 150, 150)

// Optional out parameter
const out = new Rectangle();
r1.intersection(r2, out);
```

#### Geometry utility functions

These functions are exported from `pixi.js/math-extras`, not the main `pixi.js` entry.

```ts
import {
  floatEqual,
  lineIntersection,
  segmentIntersection,
} from "pixi.js/math-extras";
import { Point } from "pixi.js";

// Epsilon-based float comparison (default epsilon: Number.EPSILON)
floatEqual(0.1 + 0.2, 0.3, 1e-10); // true with reasonable epsilon
floatEqual(1.0, 1.001, 0.01); // true (custom epsilon)

// Unbounded line intersection (returns {x: NaN, y: NaN} if parallel)
const hit = lineIntersection(
  new Point(0, 0),
  new Point(10, 10), // line A
  new Point(10, 0),
  new Point(0, 10), // line B
); // Point(5, 5)
if (isNaN(hit.x)) {
  /* lines are parallel */
}

// Bounded segment intersection (returns {x: NaN, y: NaN} if segments don't cross)
const segHit = segmentIntersection(
  new Point(0, 0),
  new Point(10, 10),
  new Point(10, 0),
  new Point(0, 10),
); // Point(5, 5)
if (isNaN(segHit.x)) {
  /* segments don't intersect */
}
```

## Common Mistakes

### HIGH: Importing from @pixi/math

Wrong:

```ts
import { Point } from "@pixi/math";
```

Correct:

```ts
import { Point } from "pixi.js";
```

v8 uses a single `pixi.js` package. All sub-packages like `@pixi/math`, `@pixi/core`, etc. were removed.


### MEDIUM: Mutating ObservablePoint without triggering observer

Wrong:

```ts
// Replacing the reference loses observation
let pos = container.position;
pos = new Point(100, 200); // container.position unchanged
```

Correct:

```ts
// Mutate in place to trigger the observer
container.position.set(100, 200);
// or
container.position.x = 100;
container.position.y = 200;
// or copy from another point
container.position.copyFrom(new Point(100, 200));
```

Container's position, scale, pivot, origin, and skew are ObservablePoints. Setting `.x` or `.y` on them triggers the container's transform update. Reassigning the variable reference does not modify the container. Always mutate the existing ObservablePoint via `.set()`, `.copyFrom()`, or direct property assignment on the original object.


### MEDIUM: Not importing math-extras for extended methods

Wrong:

```ts
import { Point } from "pixi.js";
const p = new Point(1, 2);
p.add(new Point(3, 4)); // TypeError: p.add is not a function
```

Correct:

```ts
import "pixi.js/math-extras";
import { Point } from "pixi.js";
const p = new Point(1, 2);
const sum = p.add(new Point(3, 4)); // works
```

Extended math utilities (add, subtract, multiply, magnitude, normalize, dot, cross, etc. on Point; intersection methods on shapes) require an explicit `import 'pixi.js/math-extras'`. These are not included in the default bundle.


### MEDIUM: Storing references to shared/temporary objects

Wrong:

```ts
const myPoint = Point.shared;
myPoint.set(100, 200);
// ... later ...
console.log(myPoint.x); // 0 (reset on next access)
```

Correct:

```ts
const myPoint = new Point();
myPoint.copyFrom(Point.shared.set(100, 200));
// or just
const myPoint = new Point(100, 200);
```

`Point.shared` and `Matrix.shared` are reset to zero/identity every time they are accessed. They exist for one-off calculations within a single expression. Never store a reference to them.


## API Reference

- [Point](https://pixijs.download/release/docs/maths.Point.html.md)
- [ObservablePoint](https://pixijs.download/release/docs/maths.ObservablePoint.html.md)
- [Matrix](https://pixijs.download/release/docs/maths.Matrix.html.md)
- [Rectangle](https://pixijs.download/release/docs/maths.Rectangle.html.md)
- [Circle](https://pixijs.download/release/docs/maths.Circle.html.md)
- [Ellipse](https://pixijs.download/release/docs/maths.Ellipse.html.md)
- [Polygon](https://pixijs.download/release/docs/maths.Polygon.html.md)
- [RoundedRectangle](https://pixijs.download/release/docs/maths.RoundedRectangle.html.md)
- [Triangle](https://pixijs.download/release/docs/maths.Triangle.html.md)
- [DEG_TO_RAD](https://pixijs.download/release/docs/maths.DEG_TO_RAD.html.md)