Learning Maths again

Useful GLSL Functions

Basic light shading

thanks to everyone who created the Shader School =)

Contents

• Radians / Degrees
• Calculate side lengths
• Rotate a 2D point
• Linear Distance 2 points
• Linear distance between 2 vectors
• Length of a vector
• Add and substract vectors
• Normalize vector
• Dot product vectors
• Finding angle between 2 points
• Finding angle between 2 vectors
• Cross Product

Radians & Degrees

// JavaScript
var angleInDegrees = 45;
var radians = angleInDegrees * Math.PI / 180;
var backToDegrees = radians * 180 / Math.PI;

// GLSL
float angleInDegrees = 90.0;
float r = radians(angleInDegrees);
float angleInDegrees = degrees(r);

Calculate side lengths

SOHCAHTOA

// Javascript
var hyp = 100;
var angleDegrees = 45;
var angleRadians = angleDegrees * Math.PI / 180;

var opposite = Math.sin( angleRadians ) * hyp;
var adjacent = Math.cos( angleRadians ) * hyp;
var tangent  = opposite / adjacent;

// GLSL
float hyp = 100.;
float angleDegrees = 45.;
float angleRadians = radians(angleDegrees);

float opposite = sin(angleRadians) * hyp;
float adjacent = cos(angleRadians) * hyp;
float tangent  = opposite / adjacent;

Rotate a 2D point

var vec2 = {x: 2, y: 3};
var rotatedVector = rotate2D(vec2, angle);

function rotate2D(vector, angle)
{
	var theta = angle * Math.PI / 180; // radians
	var matrix = [  Math.cos(theta),  Math.sin(theta), 
					-Math.sin(theta), Math.cos(theta)
					];
					
	return { 
		x: matrix[0] * vector.x + matrix[1] * vector.y, 
		y: matrix[2] * vector.x + matrix[3] * vector.y
	};
}

// GLSL
vec2 rotate2D(vec2 position, float theta)
{
    // theta in radians
    mat2 m = mat2( cos(theta), sin(theta), -sin(theta), cos(theta) );
    return m * position;
}

Linear Distance 2 points

// JavaScript
var x1 = 3;
var x2 = 5;
var distance = x2 — x1;

// GLSL
float x1 = 3.0;
float x2 = 5.0;
float distance = x2 — x1;

Linear distance between 2 vectors

a² + b² = c²

// Javascript
var v1 = {x: 4, y: -9};
var v2 = {x: 5, y: 15};

var distance = Math.sqrt( Math.pow((v2.x — v1.x), 2) + Math.pow((v2.y — v1.y), 2) );

// GLSL
vec2 x1 = vec2(1.0, 2.0);
vec2 x2 = vec2(2.0, 1.0);
float distance = distance(vec2(x1), vec2(x2));

Length of a vector

a.k.a Magnitude

// Javascript
// 2D -> hypotenuse
var v = {x: 4, y:-9};
var length = Math.sqrt( (Math.pow(v.x, 2) + Math.pow(v.y, 2)) );

// 3D
var v = {x: 4, y:-9, z: 0.5};
var length = Math.sqrt( (Math.pow(v.x, 2) + Math.pow(v.y, 2) + Math.pow(v.z, 2) ));

// GLSL
vec2 v = vec2(1.0, 2.0);
float l = length(v);

Add and substract vectors

var v1 = {x: 2, y: 3};
var v2 = {x: 2, y: -2};
var addedVec = {x: v1.x + v2.x, y: v1.y + v2.y};
var subVec = {x: v1.x - v2.x, y: v1.y - v2.y};

Normalize vector

// Javascript
// 2D
var v = {x: 4, y:-9};
var length = Math.sqrt( (Math.pow(v.x, 2) + Math.pow(v.y,2)) );
var n = {x: v.x / length, y: v.y / length};

// 3D
var v = {x: 4, y:-9, z: 3};
var length = Math.sqrt( Math.pow(v.x, 2) + Math.pow(v.y,2) + Math.pow(v.z,2) );
var n = {x: v.x / length, y: v.y / length, z: v.z / length};

// GLSL
vec2 v = vec2(4.0, -9.0);
vec2 n = normalize(v);

vec3 v = vec3(4.0, -9.0, 3.0);
vec3 n = normalize(v);

Dot product vectors

// Javascript
var v1 = {x: 4, y: 5, z: 9};
var v2 = {x: 5, y: 9, z: -5};
var dot = (v1.x * v2.x) + (v1.y * v2.y) + (v1.z * v2.z);

// GLSL 
vec3 v1 = vec2(4., 5., 9.);
vec3 v2 = vec2(5., 9., -5.);
float dot = dot(v1, v2);

Finding angle between 2 points

//Javascript
var x = -3;
var y = -2;
var radians = Math.atan2(x, y);
var degrees = radians * 180 / Math.PI;

//GLSL
float x = -3.0;
float y = -2.0;
float radians = atan(x, y);
float degrees = degrees(radians);

Finding angle between 2 vectors

// Javascript
var v1 = {x: 4, y: 5, z: 9};
var v2 = {x: 5, y: 9, z: -5};
var dot = (v1.x * v2.x) + (v1.y * v2.y) + (v1.z * v2.z);

var lengthv1 = length(v1); // see length
var lengthv2 = length(v2); // see length

var radians = Math.acos(dot / (lengthv1 * lengthv2));
var angle = radians * 180 / Math.PI;

// GLSL 
vec3 v1 = vec2(4., 5., 9.);
vec3 v2 = vec2(5., 9., -5.);
float dot = dot(v1, v2);
float l1 = length(v1);
float l2 = length(v2);

float radians = acos(dot / (l1 * l2));
float angle = degrees(radians);

Cross Product

// Javascript
var v1 = {x: 1, y: 2, z: 3};
var v2 = {x: 3, y: 2, z: 1};
var cross = {
	x: v1.y*v2.z - v1.z*v2.y, 
	y: v1.z*v2.x - v1.x*v2.z, 
	z: v1.x*v2.y - v1.y*v2.x
};

// GLSL
vec3 v1 = vec3(1.0, 2.0, 3.0);
vec3 v2 = vec3(2.0, 2.0, 1.0);
vec3 cross = cross(v1, v2);

Projection of point B over A

// Javascript
// ||b -> a|| 
var b = {x: 1, y: 3};
var theta = 45;
var projection = normalize(b) * Math.cos(theta); // see Normalize Vector

// GLSL
vec2 b = vec2(1, 3);
float theta = 45;
float projection = normalize(b) * cos(theta);