Purpose

VectorMath is a Swift library for Mac and iOS that implements common 2D and 3D vector and matrix functions, useful for games or vector-based graphics.

VectorMath takes advantage of Swift language features such as function and operator overloading and struct methods to provide a more elegant interface than most C, C++ or Cocoa-based graphics APIs.

VectorMath also provides a handy replacement for the GLKit vector math types and functions, which are not available yet in Swift due to their reliance on union types.

VectorMath is a completely standalone library, relying only on the Foundation framework. However, it provides optional compatibility extensions for SceneKit and Quartz (CoreGraphics/CoreAnimation) for interoperability with UIKit, AppKit, SpriteKit and SceneKit.

VectorMath is designed to be efficient, but has not been heavily optimized yet, and does not yet take advantage of architecture-specific hardware acceleration using the Accelerate framework.

Supported OS & SDK Versions

• Supported build target - iOS 12.0, Mac OS 10.14 (Xcode 11.1)
• Earliest supported deployment target - iOS 9.0, Mac OS 10.13
• Earliest compatible deployment target - iOS 7.0, Mac OS 10.9

NOTE: 'Supported' means that the library has been tested with this version. 'Compatible' means that the library should work on this OS version (i.e. it doesn't rely on any unavailable SDK features) but is no longer being tested for compatibility and may require tweaking or bug fixes to run correctly.

Installation

To use the VectorMath functions in an app, drag the VectorMath.swift file (demo/test files and assets are not needed) into your project. You may also wish to include the VectorMath+SceneKit.swift and/or VectorMath+Quartz.swift compatibility extensions.

Types

VectorMath declares the following types:

Scalar

This is a typealias used for the scalar floating point values in the VectorMath library. It is set to Float by default, but you can change it to Double or CGFloat to improve performance for your specific application.

Vector2
Vector3
Vector4

These represent 2D, 3D and 4D vectors, respectively.

Matrix3
Matrix4

These represent homogenous 3x3 and 4x4 transform matrices, respectively.

Quaternion

This represents a rotation in 3D space. It has the same structure as Vector4D, but is defined as a different type due to the different use cases and methods.

All the VectorMath types conform to Equatable and Hashable, so they can be stored in Swift dictionaries.

Constants

VectorMath declares a number of namespaced constants for your convenience. They are as follows:

Scalar.pi
Scalar.halfPi
Scalar.quarterPi
Scalar.twoPi

These should be self-explanatory.

Scalar.degreesPerRadian
Scalar.radiansPerDegree

Conversion factors between degrees and radians. E.g. to convert 40 degrees to radians, you would say let r = 40 * .degreesPerRadian, or to convert Pi/2 radians to degrees, say let d = .halfPi * .radiansPerDegree

Scalar.epsilon = 0.0001

This is a floating point error value used by the approx-equal operator. You can change this if it's insufficiently (or excessively) precise for your needs.

Vector2.zero
Vector3.zero
Vector4.zero
Quaternion.Zero

These are zero vector constants, useful as default values for vectors

Vector2.x
Vector2.y
Vector3.x
Vector3.y
Vector3.z
Vector4.x
Vector4.y
Vector4.z
Vector4.w

These are unit vectors along various axes. For example Vector3.z has the value Vector3(0, 0, 1)

Matrix3.identity
Matrix4.identity
Quaternion.identity

These are identity matrices, which have the property that multiplying them by another matrix or vector has no effect.

Methods

The complete list of VectorMath properties and methods is given below. These are mostly self-explanatory. If you can't find a method you are looking for (e.g. a method to rotate a vector using a quaternion), it's probably implemented as an operator (see "Operators" below).

Vector2
    init(x: Scalar, y: Scalar)
    init(_: Scalar, _: Scalar)
    init(_: [Scalar])
    lengthSquared: Scalar
    length: Scalar
    inverse: Vector2
    toArray() -> [Scalar]
    dot(Vector2) -> Scalar
    cross(Vector2) -> Scalar
    normalized() -> Vector2
    rotated(by: Scalar) -> Vector2
    rotated(by: Scalar, around: Vector2) -> Vector2
    angle(with: Vector2) -> Scalar
    interpolated(with: Vector2, by: Scalar) -> Vector2

Vector3
    init(x: Scalar, y: Scalar, z: Scalar)
    init(_: Scalar, _: Scalar, _: Scalar)
    init(_: [Scalar])
    lengthSquared: Scalar
    length: Scalar
    inverse: Vector3
    xy: Vector2
    xz: Vector2
    yz: Vector2
    toArray() -> [Scalar]
    dot(Vector3) -> Scalar
    cross(Vector3) -> Vector3
    normalized() -> Vector3
    interpolated(with: Vector3, by: Scalar) -> Vector3

Vector4
    init(x: Scalar, y: Scalar, z: Scalar, w: Scalar)
    init(_: Scalar, _: Scalar, _: Scalar, _: Scalar)
    init(_: Vector3, w: Scalar)
    init(_: [Scalar])
    lengthSquared: Scalar
    length: Scalar
    inverse: Vector4
    xyz: Vector3
    xy: Vector2
    xz: Vector2
    yz: Vector2
    toArray() -> [Scalar]
    toVector3() -> Vector3
    dot(Vector4) -> Scalar
    normalized() -> Vector4
    interpolated(with: Vector4, by: Scalar) -> Vector4

Matrix3
    init(m11: Scalar, m12: Scalar, ... m33: Scalar)
    init(_: Scalar, _: Scalar, ... _: Scalar)
    init(scale: Vector2)
    init(translation: Vector2)
    init(rotation: Scalar)
    init(_: [Scalar])
    adjugate: Matrix3
    determinant: Scalar
    transpose: Matrix3
    inverse: Matrix3
    toArray() -> [Scalar]
    interpolated(with: Matrix3, by: Scalar) -> Matrix3

Matrix4
    init(m11: Scalar, m12: Scalar, ... m33: Scalar)
    init(_: Scalar, _: Scalar, ... _: Scalar)
    init(scale: Vector3)
    init(translation: Vector3)
    init(rotation: Vector4)
    init(quaternion: Quaternion)
    init(fovx: Scalar, fovy: Scalar, near: Scalar, far: Scalar)
    init(fovx: Scalar, aspect: Scalar, near: Scalar, far: Scalar)
    init(fovy: Scalar, aspect: Scalar, near: Scalar, far: Scalar)
    init(top: Scalar, right: Scalar, bottom: Scalar, left: Scalar, near: Scalar, far: Scalar)
    init(_: [Scalar])
    adjugate: Matrix4
    determinant: Scalar
    transpose: Matrix4
    inverse: Matrix4
    toArray() -> [Scalar]
    interpolated(with: Matrix3, by: Scalar) -> Matrix3

Quaternion
    init(x: Scalar, y: Scalar, z: Scalar, w: Scalar)
    init(_: Scalar, _: Scalar, _: Scalar, _: Scalar)
    init(axisAngle: Vector4)
    init(pitch: Scalar, yaw: Scalar, roll: Scalar)
    init(rotationMatrix m: Matrix4)
    init(_: [Scalar])
    lengthSquared: Scalar
    length: Scalar
    inverse: Quaternion
    xyz: Vector3
    pitch: Scalar
    yaw: Scalar
    roll: Scalar
    toAxisAngle() -> Vector4
    toPitchYawRoll() -> (pitch: Scalar, yaw: Scalar, roll: Scalar)
    toArray() -> [Scalar]
    dot(Quaternion) -> Scalar
    normalized() -> Quaternion
    interpolated(with: Quaternion, by: Scalar) -> Quaternion

Operators

VectorMath makes extensive use of operator overloading, but I've tried not to go overboard with custom operators. The only nonstandard operator defined is ~=, meaning "approximately equal", which is extremely useful for comparing Scalar, Vector or Matrix values for equality, as, due to floating point imprecision, they are rarely identical.

The *, /, +, - and == operators are implemented for most of the included types. * in particular is useful for matrix and vector transforms. For example, to apply a matrix transform "m" to a vector "v" you can write m * v. * can also be used in conjunction with a Scalar value to scale a vector.

Unary minus is supported for inversion/negation on vectors and matrices.

Dot product, cross product and normalization are not available in operator form, but are supplied as methods on the various types.

Acknowledgements

Many of the algorithms used in VectorMath were ported or adapted from the Kazmath vector math library for C (https://github.com/Kazade/kazmath), or derived from the awesome Matrix and Quaternion FAQ (http://www.j3d.org/matrix_faq/matrfaq_latest.html).

In addition, the following people have contributed directly to the project:

• @harlanhaskins - SPM and Linux support
• @milpitas - CocoaPods support
• @billhsu / @ismailbozk - Bug fixes and test coverage
• @jiropole - MapKit integration
• @nicklockwood - Everything else

(Full list of contributors)