## 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.