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<!DOCTYPE html>
<!--
mobius-transforms - Exploring Möbius transformations and implementing the book Indra's Pearls
Copyright (C) 2022 Tim J. Hutton
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <https://www.gnu.org/licenses/>.
-->
<html>
<head>
<meta charset="UTF-8">
<script src="code/math.js"></script>
<script>
function toScreen( p ) {
return add( origin_screen, complex_conjugate( mul( p, scale ) ) );
}
function fromScreen( p ) {
return mul( complex_conjugate( sub( p, origin_screen ) ), 1.0 / scale );
}
function inViewingFrustrum( p ) {
return true;
}
function getMousePos(canvas, evt) {
var rect = canvas.getBoundingClientRect();
return p2( evt.clientX - rect.left, evt.clientY - rect.top );
}
function onMouseMove( evt ) {
var pos = getMousePos( canvas, evt );
onMove( pos );
}
function onTouchMove( evt ) {
if( point_is_being_dragged )
evt.preventDefault();
onMove( fixTouch( evt.changedTouches[0] ) );
}
function onMove( pos ) {
if( point_is_being_dragged ) {
control_points[ i_point ] = fromScreen( add( pos, drag_offset ) );
if(hq_mode) {
document.getElementById('hq_mode_button').value = "Rendering...";
}
setTimeout(() => { // (delay to allow the button text to update)
clearIFS();
runIFS();
draw();
if(hq_mode) {
document.getElementById('hq_mode_button').value = "High quality mode: ON";
}
}, 1);
}
else {
// find the closest point, if within a minimum radius
var iClosest = -1;
var closest_dist2 = touch_radius * touch_radius;
var num_control_points = control_points.length;
for( var i = 0; i < num_control_points; i++ ) {
var d2 = dist2( pos, toScreen( control_points[ i ] ) );
if( d2 < closest_dist2 ) {
closest_dist2 = d2;
iClosest = i;
}
}
if( iClosest >= 0 ) {
point_is_highlighted = true;
i_point = iClosest;
draw();
}
else if( point_is_highlighted ) {
point_is_highlighted = false;
draw();
}
}
}
function onMouseDown( evt ) {
var pos = getMousePos( canvas, evt );
onDown( pos );
}
function onTouchStart( evt ) {
onDown( fixTouch( evt.changedTouches[0] ) );
}
function onDown( pos ) {
// find the closest point, if within a minimum radius
var iClosest = -1;
var closest_dist2 = touch_radius * touch_radius;
var num_control_points = control_points.length;
for( var i = 0; i < num_control_points; i++ ) {
var d2 = dist2( pos, toScreen( control_points[ i ] ) );
if( d2 < closest_dist2 ) {
closest_dist2 = d2;
iClosest = i;
}
}
if( iClosest >= 0 ) {
point_is_being_dragged = true;
point_is_highlighted = false;
i_point = iClosest;
drag_offset = sub( toScreen( control_points[ i_point ] ), pos );
draw();
}
else if( point_is_being_dragged ) {
point_is_being_dragged = false;
draw();
}
else if( point_is_highlighted ) {
point_is_highlighted = false;
draw();
}
}
function onMouseUp( evt ) {
if( point_is_highlighted || point_is_being_dragged ) {
point_is_highlighted = point_is_being_dragged = false;
draw();
}
is_mouse_down = false;
}
// thanks to: http://stackoverflow.com/a/12714084/126823
function fixTouch (touch) {
var winPageX = window.pageXOffset,
winPageY = window.pageYOffset,
x = touch.clientX,
y = touch.clientY;
if (touch.pageY === 0 && Math.floor(y) > Math.floor(touch.pageY) ||
touch.pageX === 0 && Math.floor(x) > Math.floor(touch.pageX)) {
// iOS4 clientX/clientY have the value that should have been
// in pageX/pageY. While pageX/page/ have the value 0
x = x - winPageX;
y = y - winPageY;
} else if (y < (touch.pageY - winPageY) || x < (touch.pageX - winPageX) ) {
// Some Android browsers have totally bogus values for clientX/Y
// when scrolling/zooming a page. Detectable since clientX/clientY
// should never be smaller than pageX/pageY minus page scroll
x = touch.pageX - winPageX;
y = touch.pageY - winPageY;
}
var rect = canvas.getBoundingClientRect();
return p2( x - rect.left, y - rect.top );
}
function init() {
canvas = document.getElementById('canvas');
ctx = canvas.getContext('2d');
canvas_rect = { x:0, y:0, width:canvas.width, height:canvas.height };
canvas.addEventListener( 'mousemove', onMouseMove, false );
canvas.addEventListener( 'touchmove', onTouchMove, false );
canvas.addEventListener( 'mousedown', onMouseDown, false );
canvas.addEventListener( 'touchstart', onTouchStart, false );
canvas.addEventListener( 'mouseup', onMouseUp, false );
canvas.addEventListener( 'mouseout', onMouseUp, false );
canvas.addEventListener( 'touchend', onMouseUp, false );
canvas.addEventListener( 'touchcancel', onMouseUp, false );
num_transforms = 3;
control_points = [];
velocities = [];
for(var i = 0; i < num_transforms * 4; i++) {
control_points.push( p2( Math.random()*4.0 - 2.0, Math.random()*4.0 - 2.0 ) );
var dir = Math.random() * 2.0 * Math.PI;
velocities.push( p2( Math.cos( dir ) * 0.005, Math.sin( dir ) * 0.005 ) );
}
// some presets:
// skull
//control_points = [{x:0.4866733034189119,y:-0.3457824044206985,z:0},{x:1.313262575615668,y:-1.397026140339055,z:0},{x:0.6516350951229584,y:1.638719823441038,z:0},{x:-1.8569883564943233,y:0.08818231772017304,z:0},{x:0.5795578225793658,y:1.3549839078686818,z:0},{x:-1.8961028244298554,y:1.6087998022720393,z:0},{x:-1.6882600255129114,y:0.3069160294563704,z:0},{x:0.0036256316221829365,y:-2.0331095041409233,z:0},{x:0.31508558391625563,y:1.0498951917336012,z:0},{x:-0.7344985142475648,y:-0.9506556944928284,z:0},{x:-1.288507550763319,y:-0.5351060846302649,z:0},{x:-1.5200243034970713,y:0.21087848965428035,z:0}];
// stars
//control_points = [{x:1.496509486320491,y:-0.5973510559562649,z:0},{x:1.363031770057287,y:0.5030253265047588,z:0},{x:-0.4006162540899725,y:-0.31598846100639305,z:0},{x:0.3877012648959322,y:-1.1534511520185347,z:0},{x:0.37948263847029284,y:1.0729183770897013,z:0},{x:-1.6818327845132162,y:-1.5667178744100405,z:0},{x:-0.6804687668035543,y:-1.8026611211435384,z:0},{x:0.18355448066088614,y:1.1028015889392742,z:0},{x:-0.5482198806114792,y:1.0682003658572363,z:0},{x:0.1303466381970876,y:-0.924249479448636,z:0},{x:0.06475305759483793,y:-0.6112945687883007,z:0},{x:1.130285084727879,y:-1.554137395983805,z:0}];
// dragon:
//control_points = [{x:-1.5049655316774313,y:-1.0046110176648915,z:0},{x:0.6950357700194231,y:-0.432217929539951,z:0},{x:1.0743308469633628,y:0.6074866445391349,z:0},{x:-1.4504110751965176,y:1.084841058616748,z:0},{x:0.8879006445000018,y:0.1631992819881769,z:0},{x:1.8187741523098753,y:1.4747701825935757,z:0},{x:0.9488978374634134,y:0.7531535508928877,z:0},{x:0.2046657793807769,y:-2.1349204141857965,z:0},{x:0.6810830544856391,y:0.41883146175194397,z:0},{x:-1.247827026913857,y:1.3800558371943485,z:0},{x:1.2236590009001986,y:0.5638551016962444,z:0},{x:-1.9784761192044398,y:-0.8425079666566594,z:0}];
// gasket:
//control_points = [{x:-0.1018252798481565,y:-1.6993659680125772,z:0},{x:1.379590288668221,y:1.1523609868431468,z:0},{x:0.4689151107386992,y:0.4121648021857115,z:0},{x:1.7244514564858764,y:-1.4367706357328038,z:0},{x:-0.7373058434565494,y:-0.08334095212148668,z:0},{x:-1.6491235833287208,y:0.9879685699677004,z:0},{x:-0.09822345862515205,y:-0.7683141638693157,z:0},{x:-0.9538636622332818,y:-0.4259876509291862,z:0},{x:-1.022065959402516,y:0.2559407944576475,z:0},{x:1.6979795385246752,y:-0.501974961351221,z:0},{x:1.410013313430138,y:-1.2657227272429628,z:0},{x:1.0331107145329936,y:-0.25404098701767536,z:0}];
origin_screen = p2( canvas.width/2, canvas.height/2 );
scale = Math.min( canvas.width, canvas.height ) / 4.5;
point_is_highlighted = false;
point_is_being_dragged = false;
i_point = -1;
grab_offset = p2( 0, 0 );
touch_radius = 50;
draw_control_points = true;
ctx.clearRect(0, 0, canvas.width, canvas.height);
image = ctx.getImageData(0, 0, canvas.width, canvas.height);
pixels = image.data;
hq_mode = false;
num_ifs_iterations = 100000;
inc = 100;
clearIFS();
runIFS();
draw();
move_points = false;
animate();
}
function drawPoint( p, r, t, t_offset ) {
ctx.beginPath();
ctx.arc( p.x, p.y, r, 0, 2.0 * Math.PI );
ctx.fillText( t, p.x + 5 + t_offset.x, p.y - 5 + t_offset.y );
ctx.fill();
}
function draw() {
// draw ifs
ctx.putImageData(image, 0, 0);
var extent = 6;
// draw the axes
if( draw_control_points ) {
ctx.strokeStyle = "rgba(0,0,0,0.2)";
ctx.beginPath();
drawLine( p2( -extent, 0 ), p2( extent, 0 ) );
drawLine( p2( 0, -extent ), p2( 0, extent ) );
// unit marks
for( var i = -extent; i <= extent; i++ ) {
drawLine( p2( i, -0.05 ), p2( i, 0.05 ) );
drawLine( p2( -0.05, i ), p2( 0.05, i ) );
}
ctx.stroke();
}
// draw the control points
if( draw_control_points ) {
ctx.font="22px Arial";
var rest_color = "rgb(0,0,0)";
var highlighted_color = "rgb(0,255,255)";
var dragging_color = "rgb(0,255,0)";
var num_moveable_control_points = control_points.length;
for( var i = 0; i < control_points.length; i++ ) {
if( point_is_highlighted && i_point == i )
ctx.fillStyle = ctx.strokeStyle = highlighted_color;
else if( point_is_being_dragged && i_point == i )
ctx.fillStyle = ctx.strokeStyle = dragging_color;
else
ctx.fillStyle = ctx.strokeStyle = rest_color;
var radius = ( i < num_moveable_control_points ) ? 4 : 2;
drawPoint( toScreen( control_points[ i ] ), radius, String.fromCharCode(65 + i), p2(0,0) );
}
}
// convert to code we can paste into the Observable
control_points_code = '<p>For a really high-quality render you can paste the following definition into the GPU implementation here: <a href="https://observablehq.com/@timhutton/mobius-ifs">https://observablehq.com/@timhutton/mobius-ifs</a></p><blockquote><pre><tt>IFS custom = IFS(\n';
for( let i = 0; i < 3; i++) {
control_points_code += " Mobius( ";
for( let j = 0; j < 4; j++ ) {
let cp = control_points[i*3+j];
control_points_code += "vec2( " +cp.x.toString()+", "+cp.y.toString()+" )"
if( j < 3 )
control_points_code += ", ";
else
control_points_code += " ";
}
control_points_code += ")";
if( i < 2 )
control_points_code += ", \n";
else
control_points_code += " \n";
}
control_points_code += ");\n</tt></pre></blockquote>";
document.getElementById("control_points").innerHTML = control_points_code;
}
function drawLine( a, b ) {
var sa = toScreen( a );
var sb = toScreen( b );
ctx.moveTo( sa.x, sa.y );
ctx.lineTo( sb.x, sb.y );
}
function onDrawControlPointsButton(element) {
draw_control_points = !draw_control_points;
element.value = draw_control_points ? "Draw control points: ON" : "Draw control points: OFF";
draw();
}
function onHighQualityModeButton(element) {
hq_mode = !hq_mode;
element.value = hq_mode ? "High quality mode: ON" : "High quality mode: OFF";
if( hq_mode ) {
num_ifs_iterations = 10000000;
inc = 1;
}
else {
num_ifs_iterations = 100000;
inc = 100;
}
if(hq_mode) {
element.value = "Rendering...";
}
setTimeout(() => { // (delay to allow the button text to update)
clearIFS();
runIFS();
draw();
if(hq_mode) {
element.value = "High quality mode: ON";
}
}, 1);
}
function onAnimateModeButton(element) {
move_points = !move_points;
element.value = move_points ? "Animate control points: ON" : "Animate control points: OFF";
draw();
if( move_points) {
animate();
}
}
function animate( currentFrameTime ) {
if( move_points ) {
for( var i = 0; i < control_points.length; i++) {
control_points[i] = add( control_points[i], velocities[i] );
if( control_points[i].x < -2 ) {
control_points[i].x = -4.0 - control_points[i].x;
velocities[i].x *= -1.0;
}
else if( control_points[i].x > 2 ) {
control_points[i].x = 4.0 - control_points[i].x;
velocities[i].x *= -1.0;
}
if( control_points[i].y < -2 ) {
control_points[i].y = -4.0 - control_points[i].y;
velocities[i].y *= -1.0;
}
else if( control_points[i].y > 2 ) {
control_points[i].y = 4.0 - control_points[i].y;
velocities[i].y *= -1.0;
}
}
clearIFS();
runIFS();
draw();
requestAnimationFrame( animate );
}
}
function clearIFS() {
pixels.fill( 255 );
}
function runIFS() {
// do IFS
var transforms = [];
for( var i = 0; i < num_transforms; i++) {
transforms.push( [control_points[i * 4 + 0], control_points[i * 4 + 1], control_points[i * 4 + 2], control_points[i * 4 + 3]] );
}
var p = p2(0.0, 0.0);
for( var i = 0; i < num_ifs_iterations; i++) {
p = mobius_on_point( transforms[Math.floor(Math.random() * transforms.length)], p );
var sp = toScreen( p );
if( sp.x >= 0 && sp.y >=0 && sp.x < image.width && sp.y < image.height ) {
var pixel_offset = ( Math.floor(sp.y) * image.width + Math.floor(sp.x) ) * 4;
// sort of log-mapping: pixels get darker at a slower rate the darker they get
// avoids saturating too much
var gray = pixels[pixel_offset + 0];
if( Math.random() < gray / 255.0 ) {
pixels[pixel_offset + 0] = pixels[pixel_offset + 1] = pixels[pixel_offset + 2] = Math.max( 0, gray - inc );
}
}
}
}
window.onload = init;
</script>
<noscript>
<p>For full functionality of this site it is necessary to enable JavaScript.
Here are the <a href="http://www.enable-javascript.com/" target="_blank">
instructions how to enable JavaScript in your web browser</a>.
</p></noscript>
<style>
input[type='button'] {
font-size: 20px;
font-family: Helvetica Neue,Helvetica,Arial,sans-serif;
}
</style>
</head>
<body>
<div style="float:left;touch-action:none"><p><canvas id="canvas" width="700" height="600">(Canvas drawing not supported by your browser.)</canvas></p></div>
<div style="float:left;padding-left: 20px">
<p><input type="button" value="Draw control points: ON" onclick="onDrawControlPointsButton(this)" /></p>
<p><input type="button" id="hq_mode_button" value="High quality mode: OFF" onclick="onHighQualityModeButton(this)" /></p>
<p><input type="button" value="Animate control points: OFF" onclick="onAnimateModeButton(this)" /></p>
</div>
<div style="clear:both">
<p>The image shows the points generated by an <a href="https://en.wikipedia.org/wiki/Iterated_function_system">IFS</a> using three <a href="index.html">Möbius transformations</a>, controlled by points A - L. Drag them around to change the transforms, or reload the page to randomize them.</p>
<p>With <b>High quality mode: ON</b> we draw a lot more points.</p>
<div id="control_points"></div>
<hr>
<p>Source code: <a href="https://github.com/timhutton/mobius-transforms">https://github.com/timhutton/mobius-transforms</a></p>
</div>
</html>