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:
ScalarThis 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
Vector4These represent 2D, 3D and 4D vectors, respectively.
Matrix3
Matrix4These represent homogenous 3x3 and 4x4 transform matrices, respectively.
QuaternionThis 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.twoPiThese should be self-explanatory.
Scalar.degreesPerRadian
Scalar.radiansPerDegreeConversion 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 = 0.0001This 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.ZeroThese 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.wThese are unit vectors along various axes. For example Vector3.z has the value Vector3(0, 0, 1)
Matrix3.identity
Matrix4.identity
Quaternion.identityThese 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) -> QuaternionOperators
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