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.center
main
h1
| Rendering a shit-tonne of dimentions on a 2D canvas (
a(href="https://github.com/ninivert/N-Dimension-Cube-Renderer" target="_blank") source code
| )
.row
.render
canvas#canvas
.info
p#name
p#info
.controls
.row
span Dimension
small Goes from 2 to 10 (Warning: high dimensions are laggy)
form#dimension
.row
span Scaling
small How large the figure is
form#scaling
.row
span Speed
small How fast the figure is spinning
form#speed
.row
span Distance
small Works only on a non-isometric projection
form#distance
.row
span Isometric projection
small Non-isometric projections need large scaling
form#isometric
.row
span Rotations
small Enable the rotation of planes defined by the axes
form#rotations
$black: #333
$dimgray: #666
$gray: #aaa
$white: #eee
*
box-sizing: border-box
font-family: sans-serif
html,
body
margin: 0
padding: 0
width: 100%
height: 100%
body
background: $black
color: $white
overflow: auto
a
color: $white
.center
margin: auto
padding: 2rem
width: 1200px
max-width: 100%
min-height: 100%
display: flex
justify-content: center
align-items: center
// background: #fff
main
width: 100%
h1
margin: 0
margin-bottom: 2rem
padding-bottom: 1rem
border-bottom: 1px solid $dimgray
font-weight: normal
font-size: 1rem
text-align: center
text-transform: uppercase
letter-spacing: 2px
line-height: 1
.render
margin-bottom: 1rem
canvas
max-width: 100%
margin: 0 auto
display: block
.info
margin-top: 1rem
p
text-align: center
margin: 0
&#name
font-size: 2rem
&#info
font-size: 1rem
color: $gray
.controls
.row
margin-bottom: 1rem
span
small
display: block
color: $gray
font-size: 0.7em
form
margin: 0
width: 100%
input[type='number']
width: 100%
margin: 0
margin-top: 0.5rem
padding: 0
padding-bottom: 0.4rem
border: none
border-bottom: 1px solid $dimgray
background: transparent
color: $white
font-size: 1rem
cursor: n-resize
input[type='checkbox']
display: none
& + label
display: inline-block
color: $dimgray
margin-top: 0.5rem
margin-right: 0.5rem
cursor: pointer
&:checked + label
color: $white
&#isometric
input[type='checkbox'] + label::after
content: 'Nope'
input[type='checkbox']:checked + label::after
content: 'Yup'
.row
width: 100%
margin: auto
display: flex
flex-wrap: wrap
align-items: top
justify-content: space-around
> *
flex: 0 1 400px
display: inline-block
//
//
// MATRIX.JS
//
//
/*
Note:
According to https://jsperf.com/js-nested-vs-flat-array-retrieve-values/
It is faster to use nested 2D arrays than flattened 2D arrays, even with TypedArrays
So I'll be using [Float32Array(n), ..., Float32Array(n)] rather than Float32Array(n*n)
*/
/*
This library is NOT robust, at all
I'm assuming the user did not make any mistakes inputting the array
Normally each layer should have the same length
That way I can read the matrix dimensions from m.length and m[0].length
*/
/*
TODO:
- Make a vector class that ties in neatly with the matrix class
*/
//
// High-perf static matrix library
//
function Matrix(input) { return Matrix.from(input) }
Matrix.add = function(m1, m2) {
const m1height = m1.length, m1width = m1[0].length
const m2height = m2.length, m2width = m2[0].length
if (m1width !== m2width || m1height !== m2height) {
throw 'Incompatible matrices'
}
let result = Matrix.new(m1height, m1width)
let y, x
for (y=0; y<m1height; y++) {
for (x=0; x<m1width; x++) {
result[y][x] = m1[y][x] + m2[y][x]
}
}
return result
}
Matrix.subtract = function(m1, m2) {
const m1height = m1.length, m1width = m1[0].length
const m2height = m2.length, m2width = m2[0].length
if (m1width !== m2width || m1height !== m2height) {
throw 'Incompatible matrices'
}
let result = Matrix.new(m1height, m1width)
let y, x
for (y=0; y<m1height; y++) {
for (x=0; x<m1width; x++) {
result[y][x] = m1[y][x] - m2[y][x]
}
}
return result
}
Matrix.dot = function(m1, m2) {
const m1height = m1.length, m1width = m1[0].length
const m2height = m2.length, m2width = m2[0].length
if (m1width !== m2height) {
throw 'Incompatible matrices'
}
let result = Matrix.new(m1height, m2width)
let size = m1width
let y, x, n, sum
for (y=0; y<m1height; y++) {
for (x=0; x<m2width; x++) {
sum = 0
for (n=0; n<size; n++) {
sum += m1[y][n] * m2[n][x]
}
result[y][x] = sum
}
}
return result
}
Matrix.scale = function(m, factor) {
const mheight = m.length, mwidth = m[0].length
let result = Matrix.new(mheight, mwidth)
let y, x
for (y=0; y<mheight; y++) {
for (x=0; x<mwidth; x++) {
result[y][x] = m[y][x] * factor
}
}
return result
}
Matrix.transpose = function(m) {
const height = m.length, width = m[0].length
let result = Matrix.new(width, height)
let y, x
for (y=0; y<width; y++) {
for (x=0; x<height; x++) {
result[y][x] = m[x][y]
}
}
return result
}
Matrix.from = function(input) {
const height = input.length
const width = input[0].length
let matrix = new Array(height)
let y, x
for (y=0; y<height; y++) {
matrix[y] = Float32Array.from(input[y])
}
// Return the result
return matrix
}
Matrix.new = function(height, width) {
let matrix = new Array(height)
let y
for (y=0; y<height; y++) {
matrix[y] = new Float32Array(width)
}
return matrix
}
Matrix.identity = function(dimension) {
let matrix = new Array(dimension)
let y
for (y=0; y<dimension; y++) {
matrix[y] = new Float32Array(dimension)
matrix[y][y] = 1
}
return matrix
}
//
//
// VERTEX.JS
//
//
/*
Note:
Generating all the vertices is similar to counting up in binary
By incrementing the binary number by 1 each time until 2^{dimensions},
we are sure to hit every possible combinations of 1's and 0's in {dimensions} places
Example:
0 --> 00 --> -1, -1, -1
1 --> 01 --> -1, -1, 1
2 --> 10 --> -1, 1, -1
3 --> 11 --> -1, 1, 1
*/
function getVertices(dimension) {
const n = Math.pow(2, dimension)
let vertices = new Array(n)
let vertex
for (let i=0; i<Math.pow(2, dimension); i++) {
// Generate binary number
vertex = i.toString(2)
// Add leading 0's
/*
Note:
In base 2, the number of digits we need to encode the number is log2(n)
Here, we have log2(2^{dimensions}) = {dimensions}
*/
while (vertex.length < dimension) {
vertex = `0${vertex}`
}
// Generate the vertex by splitting then substituting 0 --> -1, 1 --> 1
// I could use array.forEach, but that's not supported in all browsers...
vertex = vertex.split('')
for (let j=0; j<vertex.length; j++) {
vertex[j] = parseInt(vertex[j]) === 0 ? -1 : 1
}
// Add the vertex to the list of vertices
vertices[i] = vertex
}
// Convert to matrix
return Matrix.from(vertices)
}
//
//
// PROJECTION.JS
//
//
function getProjectionMatrix(dimension, perspective) {
/*
Return a simple projection matrix that depends on p (perspective)
[p, 0, ..., 0, 0]
[0, p, ..., 0, 0]
[ ... ]
[0, 0, ..., p, 0]
[0, 0, ..., 0, 0]
But we are excluding the last line because we want (x, y, z) --> (x, y) and not (x, y, 0)
If the dimension is 2, we don't need to project it to dimension 1 --> identity matix
If the dimension is 1, we need to project into 2 dimensions (add a dimension) --> [[1], [0]] matrix
*/
/*
I won't need 1D projections because in draw.js projections only happen
for dimensions strictly higher than 2
But for reference purposes, here it is
if (dimensions === 1) {
matrix = Matrix.from([[1], [0]])
}
*/
let matrix
if (dimension === 2) {
matrix = Matrix.identity(2)
} else {
matrix = Matrix.new(dimension-1, dimension)
for (let i=0; i<dimension-1; i++) {
matrix[i][i] = perspective
}
}
return matrix
}
//
//
// ROTATION.JS
//
//
/*
Note:
Rotation is the rotation of a plane, defined by 2 orthogonal vectors
So when I say XY rotation, it's the plane defined by the X and Y axes
rotating around the other axes (Z for 3D, ZW for 4D)
https://www.fourthdimensionapp.com/4dmanip/ has a great explanation on rotation
*/
function getRotationMatrix(axis1, axis2, dimension, angle) {
/*
Note:
Looking at rotations matrices for 2D, 3D and 4D, we can extrapolate a general formula for them
Each axis has a vertical matrix index corresponding to it
- X --> 0 (1D) ; X_0 --> 0 (1D)
- Y --> 1 (2D) ; X_1 --> 1 (2D)
- Z --> 2 (3D) ; X_2 --> 2 (3D)
- W --> 3 (4D) ; X_3 --> 3 (4D)
- ...
- X_{n} --> {n} ({n+1}D)
Algorithm:
- Generate an identity matrix with dimension {dimension}
- Replace the following coordinates with the following values
- [axis1, axis1] --> cos(a)
- [axis1, axis2] --> -sin(a)
- [axis2, axis1] --> sin(a)
- [axis2, axis2] --> cos(a)
*/
let matrix = Matrix.identity(dimension)
matrix[axis1][axis1] = Math.cos(angle)
matrix[axis1][axis2] = -Math.sin(angle)
matrix[axis2][axis1] = Math.sin(angle)
matrix[axis2][axis2] = Math.cos(angle)
return matrix
}
function getAxisName(index) {
/*
Note:
Take the index of the axis and get the associated letter
Unicode caps alphabet start at 65, 'X' char is at index 65+23 = 88
'W' (axis index 3) is at 65+25-3 = 90-3 = 87
So we have X, Y, Z as 0, 1, 2 ; then we do the alphabet in reverse from 'W'
I'm not really caring about dimensions larger than 25 because
- Such a dimension is very laggy
- String.fromCharCode() doesn't throw an error for negative/large values
- They will just have weird axis names
*/
if (index < 3) {
return String.fromCharCode(index + 88)
}
return String.fromCharCode(90 - index)
}
function getListOfRotations(dimension) {
/*
Note:
Return a list of all the planes you can rotate for a given dimension
0D --> []
1D --> []
2D --> [0 1] --> ['XY']
3D --> [0 1, 0 2, 1 2] --> ['XY', 'XZ', 'YZ']
4D --> [0 1, 0 2, 0 3, 1 2, 1 3, 2 3] --> ['XY', 'XZ', 'XW', 'YZ', 'YW', 'ZW']
Pattern:
- The first digit goes from 0 to dimension-2
- The last digit goes from the first digit + 1 to dimension-1
Combine these two in a loop and BAM!
*/
let list = []
let first, last
for (first=0; first<dimension-1; first++) {
for (last=first+1; last<dimension; last++) {
list.push([first, last])
}
}
return list
}
/*
Fun fact:
The number of rotations per dimension is given by (n-1)*n/2
Observation:
- 0D has no rotation possible
(because to rotate you need at least 2 orthogonal vectors (2 axes) that define a plane)
- 1D gets the number of rotations for 0D (so 0) and still no rotation possible,
so a total of 0 rotations
- 2D gets the number of rotations for 1D (0) + 1 (2 axes = 1 plane)
- 3D gets 1 + 2 = 3
- 4D gets 3 + 3 = 6
- 5D gets 6 + 4 = 10
- 6D gets 10 + 5 = 15
- Hopefully you see some kind of pattern here
Generalization:
- Every higher dimension {n} gets
- The rotation planes from the previous dimension {n-1} (obviously)
- The number of axes from the previous dimension (so {n-1} axes)
that each can be combined with the new axis to create {n} rotation planes
- That translates to a well-know progression:
- 0, 0, 1, 3, 6, 10, 15, ...
- u(n) = u(n-1) + n - 1
- These are the triangular numbers, of general formula
- n(n+1)/2
- 2 choose n+1 (binomial coefficients)
- We only shift everything over by 1 to correspond with the dimension number
- More on them: https://en.wikipedia.org/wiki/Triangular_number
Proof (copied from http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/runsums/triNbProof.html):
T(n) + T(n) = 1 + 2 + 3 + ... + (n-1) + n
+ n + (n-1) + ... + 3 + 2 + 1
= (1+n) + (2+[n-1]) + (3+[n-2]) + ... + ([n-1]+2) + (n+1)
= (n+1) + (n+1) + ... + (n+1) + (n+1)
= n*(n+1)
T(n) = (T(n) + T(n)) / 2 = n*(n+1)/2
*/
//
//
// DRAW.JS
//
//
function draw() {
// Clear the screen
ctx.fillStyle = '#fff'
ctx.fillRect(-WIDTH/2, -HEIGHT/2, WIDTH, HEIGHT)
// Matrices
let point, rotated, projected
// Scalars and indices
let perspective, rotation
let points = new Array(FIGURE.length)
// Rotating, projecting the vertices, and storing them
for (let i=0; i<FIGURE.length; i++) {
point = Matrix.from([FIGURE[i]])
point = Matrix.transpose(point)
/*
Note: Here I'm combining the rotation matrices by dotting them
Starting out with the identity matrix, this has the effect of leaving the point intact
if there are no rotations
*/
rotated = Matrix.identity(DIMENSION)
for (let j=0; j<ROTATIONS.length; j++) {
rotation = ROTATIONS[j]
rotated = Matrix.dot(rotated, getRotationMatrix(rotation[0], rotation[1], DIMENSION, ANGLE))
}
rotated = Matrix.dot(rotated, point)
/*
Note: for every dimension strictly higher than 2
we are projecting {rotated} from dimension n to n-1, and repeat until we hit 2D
*/
projected = Matrix.from(rotated)
for (let j=DIMENSION; j>2; j--) {
if (ISOMETRIC) {
perspective = 1
} else {
perspective = getPerspectiveScalar(projected[j-1][0])
}
projected = Matrix.dot(getProjectionMatrix(j, perspective), projected)
}
projected = Matrix.scale(projected, SCALING)
points[i] = [projected[0][0], projected[1][0]]
}
drawVertices(points)
connectVertices(points)
ANGLE += SPEED
NEXTFRAME = requestAnimationFrame(draw)
}
function drawVertices(points) {
// Pretty self-explanatory
for (let i=0; i<points.length; i++) {
ctx.beginPath()
ctx.arc(points[i][0], points[i][1], 3, 0, 2*Math.PI)
ctx.fillStyle = '#000'
ctx.fill()
}
}
function connectVertices(points) {
/*
Here is the pattern to connect the vertices correctly
This works for every dimension starting from 1D (2 vertices, connect them together)
- every 2 points (i%2 === 0): for the next 1 point: connect i and i+1
- every 4 points (i%4 === 0): for the next 2 points: connect i and i+2
- every 8 points (i%8 === 0): for the next 4 points: connect i and i+4
- every 16 points (i%16 === 0): for the next 8 points: connect i and i+8
- ...
- every 2^{dimensions}: for the next 2^{dimensions+!}: connect i and i+2^{dimensions-1}
I'm calling the every ... points the step, and the number points each step the size
*/
ctx.beginPath()
for (let dimension=1; dimension<DIMENSION+1; dimension++) {
let step = Math.pow(2, dimension)
let size = Math.pow(2, dimension-1)
for (let i=0; i<points.length; i+=step) {
for (let j=0; j<size; j++) {
connect(i+j, i+j+size)
}
}
}
function connect(i1, i2) {
ctx.moveTo(points[i1][0], points[i1][1])
ctx.lineTo(points[i2][0], points[i2][1])
}
ctx.strokeStyle = '#000'
ctx.stroke()
}
function getPerspectiveScalar(scalar) {
return 1/(DISTANCE + scalar)
}
//
//
// INDEX.JS
//
//
console.clear()
const canvas = document.getElementById('canvas')
const ctx = canvas.getContext('2d')
// User-set options
let DISTANCE = 3
let DIMENSION = 4
let SCALING = 500
let SPEED = 0.01
let ISOMETRIC = false
let ROTATIONS = [[0, 3], [1, 2]]
// Settings and constants
const HEIGHT = 400, WIDTH = 400
let FIGURE
let NEXTFRAME
let ANGLE
// I needed a list of all the rotation planes (2 axes) indices
// to avoid generating it every time the user changed the rotation
let ALLROTATIONS
// Figure stats, the code doesn't need it, but it's nice to have
const NAMES = ['Square', 'Cube', 'Tesseract', 'Penteract', 'Hexeract', 'Hepteract', 'Octeract', 'Enneract', 'Dekeract']
let VERTICES, EDGES
// The init function assumes the
function init() {
// Stop animation if there is one
cancelAnimationFrame(NEXTFRAME)
// Set canvas dimensions
canvas.width = WIDTH
canvas.height = HEIGHT
// Center the canvas
ctx.setTransform(1, 0, 0, 1, WIDTH/2, HEIGHT/2);
// Init global variable
FIGURE = getVertices(DIMENSION)
ALLROTATIONS = getListOfRotations(DIMENSION)
ANGLE = 0
// Update the info text
VERTICES = Math.pow(2, DIMENSION)
EDGES = DIMENSION * Math.pow(2, DIMENSION-1)
document.getElementById('name').innerHTML = NAMES[DIMENSION-2]
document.getElementById('info').innerHTML = `${VERTICES} vertices, ${EDGES} edges`
// Dräw, bröther
draw()
}
// Start everything
init()
//
//
// CONROLS.JS
//
//
/*
Note: Every form is for ONE option
the form ID should match the initalizer CONTROLS.init.{name}
and the oninput callback CONTROLS.callback.{name}
*/
function CONTROLS() {}
CONTROLS.DOM = {
// All the container form elements
'forms': {
'rotations': document.getElementById('rotations'),
'dimension': document.getElementById('dimension'),
'scaling': document.getElementById('scaling'),
'isometric': document.getElementById('isometric'),
'speed': document.getElementById('speed'),
'distance': document.getElementById('distance'),
},
// The child values, for reference
'children': {
}
}
//
// Initalization
//
CONTROLS.init = function() {
/*
Note: Looping through all the individual initializers to call them all
Also add the callbacks and disable form submission
*/
let forms = Object.keys(this.init)
for (let i=0; i<forms.length; i++) {
this.init[forms[i]].call(this)
this.DOM.forms[forms[i]].oninput = this.callback[forms[i]].bind(this)
// onchange is for the checkboxes on mobile touch devices
this.DOM.forms[forms[i]].onchange = this.callback[forms[i]].bind(this)
this.DOM.forms[forms[i]].onsubmit = function() { return false }
}
}
CONTROLS.init.dimension = function() {
/*
Note: Create a number input for the dimension number
*/
let input = document.createElement('input')
input.type = 'number'
input.value = DIMENSION
this.DOM.children.dimension = input
this.DOM.forms.dimension.appendChild(input)
}
CONTROLS.init.scaling = function() {
/*
Note: Create a number input for the scaling
*/
let input = document.createElement('input')
input.type = 'number'
input.value = SCALING
input.step = 20
this.DOM.children.scaling = input
this.DOM.forms.scaling.appendChild(input)
}
CONTROLS.init.speed = function() {
/*
Note: Create a number input for the scaling
*/
let input = document.createElement('input')
input.type = 'number'
input.value = SPEED
input.step = 0.001
this.DOM.children.speed = input
this.DOM.forms.speed.appendChild(input)
}
CONTROLS.init.distance = function() {
/*
Note: Create a number input for the scaling
*/
let input = document.createElement('input')
input.type = 'number'
input.value = DISTANCE
input.step = 0.1
this.DOM.children.distance = input
this.DOM.forms.distance.appendChild(input)
}
CONTROLS.init.isometric = function() {
/*
Note: Create a checkbox input for the isometric projection
*/
let checkbox = document.createElement('input')
checkbox.type = 'checkbox'
checkbox.id = 'isometric-checkbox'
if (ISOMETRIC) {
checkbox.checked = true
}
let label = document.createElement('label')
label.innerHTML = '' // Adding content with CSS, boi
label.htmlFor = 'isometric-checkbox'
this.DOM.children.isometric = checkbox
this.DOM.forms.isometric.appendChild(checkbox)
this.DOM.forms.isometric.appendChild(label)
}
CONTROLS.init.rotations = function() {
/*
Note: Here I'm creating the inputs for the rotation matrices
The checkboxes are in a form which gets triggered when any input changes
I'm creating checkboxes + label then appending it to the form innerHTML
*/
let checkbox, label, text
let activeRotations = JSON.stringify(ROTATIONS)
// Reset the form
this.DOM.forms.rotations.innerHTML = ''
// Reset the children
this.DOM.children.rotations = []
for (let i=0; i<ALLROTATIONS.length; i++) {
text = getAxisName(ALLROTATIONS[i][0]) + getAxisName(ALLROTATIONS[i][1])
checkbox = document.createElement('input')
checkbox.type = 'checkbox'
checkbox.id = text
// Check if the rotation of the checkbox is active = present in ROTATIONS
if (activeRotations.indexOf(JSON.stringify(ALLROTATIONS[i])) !== -1) {
checkbox.checked = true
}
label = document.createElement('label')
label.htmlFor = text
label.innerHTML = text
this.DOM.children.rotations[i] = checkbox
this.DOM.forms.rotations.appendChild(checkbox)
this.DOM.forms.rotations.appendChild(label)
}
}
//
// Callbacks
//
CONTROLS.callback = {}
CONTROLS.callback.dimension = function() {
let value = parseInt(this.DOM.children.dimension.value)
// If the value is not a number (NaN), quit
if (isNaN(value)) {
return false
}
// Take the value and clamp it between [2, 10]
// Dimension 10 has 1024 vertices, which is a lot
value = Math.min(10, Math.max(2, value)) // <-- change the 10 value to your max dimension!
// Ka-bam, reset everything
DIMENSION = value
ROTATIONS = []
init()
// Update the rotation options for the new dimension
CONTROLS.init.rotations.call(this)
}
CONTROLS.callback.scaling = function() {
let value = parseFloat(this.DOM.children.scaling.value)
// If the value is not a number (NaN), quit
if (isNaN(value)) {
return false
}
SCALING = value
}
CONTROLS.callback.speed = function() {
let value = parseFloat(this.DOM.children.speed.value)
// If the value is not a number (NaN), quit
if (isNaN(value)) {
return false
}
SPEED = value
}
CONTROLS.callback.distance = function() {
let value = parseFloat(this.DOM.children.distance.value)
// If the value is not a number (NaN), quit
if (isNaN(value)) {
return false
}
DISTANCE = value
}
CONTROLS.callback.isometric = function() {
ISOMETRIC = this.DOM.children.isometric.checked
}
CONTROLS.callback.rotations = function() {
/*
Note: Here I'm looping through all the rotation checkboxes
If they are checked, add its rotation name (= elmt.value) to the global ROTATIONS variable
*/
ROTATIONS = []
let checkbox
for (let i=0; i<this.DOM.children.rotations.length; i++) {
checkbox = this.DOM.children.rotations[i]
if (checkbox.checked) {
ROTATIONS.push(ALLROTATIONS[i])
}
}
}
//
// Start everything
//
CONTROLS.init()
Also see: Tab Triggers