What is this library?

CGL vectors library is a built-in library of CGL that you can use & utilize for your projects. It performs most of the basic vector & matrix mathematics, and is heavily used in our reference solution.

Why you should use it?

The vectors library contains fully overloaded operators (so you can write vector equations just like normal ones), and it contains a lot of optimizations. These features reduces code size and risk of getting something wrong in the equation. This usually also improves the performance.

References

Data Types

Vector Types

• Vector2D
• Vector3D
• Spectrum is an alias of Vector3D, as RGB vectors are still three-component vectors
• Vector4D

Matrix Types

• Matrix3x3
• Matrix4x4

Constructor

Vector2D() / Vector3D() / Vector4D() Creates a 2D/3D/4D zero vector.

Vector2D(x) / Vector3D(x) / Vector4D(x) Creates a 2D/3D/4D vector containing x: {x, x} / {x, x, x} / {x, x, x, x}

Vector2D(x,y) / Vector3D(x,y,z) / Vector4D(x,y,z,w) Creates a 2D/3D/4D vector: {x, y} / {x, y, z} / {x, y, z, w}

Operations

Vector Indexing

Vector2D v;

x = v.x = v[0]
y = v.y = v[1]


For 3D vectors, as colors and spectrums can also be represented, you can also index 3D vectors / spectrums using r, g, and b

Vector3D v;

x = v.x = v[0] = r = v.r
y = v.y = v[1] = g = v.g
z = v.z = v[2] = b = v.b


For 4D vectors, as colors with transparency also be represented, you can also index 4D vectors / spectrums using r, g, b, and a

Vector4D v;

x = v.x = v[0] = r = v.r
y = v.y = v[1] = g = v.g
z = v.z = v[2] = b = v.b
w = v.w = v[3] = a = v.a


Matrix Indexing

MatrixND m


m[n] is the n-th column of m, in a form of a VectorND m[n].x is the n-th column first row of m m[n][i] is the n-th column i-th row of m

Vector-Scalar Operations

Assume v is a vector, s is a scalar:

VectorND v;
double s;


Vector-scalar multiplication / division v * s Returns {v.x * s, v.y * s, v.z * s} s * v Returns {v.x * s, v.y * s, v.z * s} v / s Returns {v.x / s, v.y / s, v.z / s} s / v Returns {s / v.x, s / v.y, s / v.z}

Vector-Vector Operations

Assume v1 and v2 are vectors of the same size.

Vector-vector division (element-wise division) v1 / v2 returns {v1.x / v2.x, v1.y / v2.y, v1.z / v2.z}

Vector-vector addition (element-wise addition) v1 + v2 returns {v1.x + v2.x, v1.y + v2.y, v1.z + v2.z}

Vector-vector subtraction (element-wise subtraction) v1 - v2 returns {v1.x - v2.x, v1.y - v2.y, v1.z - v2.z}

Dot product dot(v1, v2) returns dot product v1.x * v2.x + v1.y * v2.y + v1.z * v2.z

Cross product cross(v1, v2) returns the cross product of v1 and v2

Outer product outer(v1, v2) retruns a MatrixNxN, the outer product of v1 and v2

Vector Methods v.rcp() returns per-entry reciprocal {1.0 / v.x, 1.0 / v.y, 1.0 / v.z} v.norm() returns euclidean length sqrt(v.x * v.x + v.y * v.y + v.z * v.z) v.norm2() returns square of euclidean length v.x * v.x + v.y * v.y + v.z * v.z v.unit() returns normalized unit vector {v.x / v.norm(), v.y / v.norm(), v.z / v.norm()} v.normalize() normalizes the vector to unit vector. (does not return anything)

For 3D vectors: v.illum() returns the perceived brightness of a spectrum (color) vector v.toColor() returns a Color object from the spectrum object. Vector3D::fromColor(c) returns a Vector3D object construted from Color object c.

Matrix-Matrix Operations

Assume A1 and A2 are MatrixNxN

A1 - A2 returns element-wise subtraction A1 + A2 returns element-wise addition A1 * A2 returns matrix-matrix multiplication

Matrix-Vector Operations

Assume A is MatrixNxN and v is VectorND

A*v returns matrix-vector multiplication. Returns a VectorND

Matrix-Scalar Operations

Assume A is MatrixNxN and s is scalar

A * s or s * A returns {A[0] * s, A[1] * s, A[2] * s}

Matrix Methods

A.det() retruns determinant of A (double) A.norm() returns Frobenius norm of A A.inv() returns the inverse of A

Printing Vectors

You can use std::cout << v << std::endl to print vectors directly.

You may see it printed out as a color with R,G,B channels, as color is just like any other three-component vector. The R/G/B channels correspond to X/Y/Z axes

Printing Matrices

You can use std::cout << A << std::endl to print matrices directly.