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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 repositionCamera() {
var look_at = p3(0,0,0.25);
var vd = 1.2; // vertical distance
var hd = vd * 2; // horizontal distance
camera = {};
camera.p = p3( hd*Math.cos(theta), hd*Math.sin(theta), vd );
camera.z = normalize( sub( look_at, camera.p ) );
var up = p3(0,0,1);
camera.x = normalize( cross( camera.z, up ) );
camera.y = normalize( cross( camera.x, camera.z ) );
camera.f = canvas.height;
camera.pp = p2(canvas.width/2,canvas.height/2);
}
function enforce_constraints() {
// hacky approach for now: enforce the constraints over and over until the control points stop moving
var cp = control_points.slice();
var iterations = 0;
do {
if( normalize_mobius )
mobius_normalize( cp );
if( unitary )
mobius_make_unitary( cp );
if( nonloxodromic )
mobius_make_nonloxodromic( cp );
if( unit_circle_group ) {
// [ u, v; complex_conjugate(u), complex_conjugate(v) ] where |u|^2 - |v|^2 = 1
// implies |v| = sqrt( |u|^2 - 1 )
// implies |u| >= 1
var u = cp[0];
var u_mag = magnitude( u );
if( u_mag == 0.0 ) {
cp[0] = p2( 1.0, 0.0 );
cp[1] = p2( 0.0, 0.0 );
cp[2] = p2( 0.0, 0.0 );
cp[3] = p2( 1.0, 0.0 );
}
else if( u_mag < 1.0 ) {
u = normalize( u );
cp[0] = u;
cp[1] = p2( 0.0, 0.0 );
cp[2] = p2( 0.0, 0.0 );
cp[3] = complex_conjugate( u );
}
else {
var v = cp[1];
var v_mag_old = magnitude( v );
var v_mag_new = Math.sqrt( len2( u ) - 1.0 );
if( v_mag_old == 0.0 || isNaN( v_mag_old ) ) {
v = p2( v_mag_new, 0.0);
}
else {
v = mul( v, v_mag_new / v_mag_old );
}
cp[0] = u;
cp[1] = v;
cp[2] = complex_conjugate( v );
cp[3] = complex_conjugate( u );
}
}
var d = 0;
for( var i = 0; i < 4; i++ ) {
d += dist2( cp[i], control_points[i] );
control_points[i] = cp[i];
}
iterations++;
} while( iterations < 50 && d > 1E-8 );
if( iterations > 40 ) {
console.log('enforce_constraints iterations:', iterations);
}
}
function toScreen( p ) {
if( show_3D )
return camera_projection( p );
else
return add( origin_screen, complex_conjugate( mul( p, scale ) ) );
}
function fromScreen( p ) {
if( show_3D ) {
// what point on the xy plane, tp, would have camera_projection( tp ) = p?
// we know it lies on the line from the camera position through the virtual camera's screen at an arbitrary distance
var vsp = p3( ( p.x - camera.pp.x ) / camera.f, ( ( canvas.height - p.y) - camera.pp.y ) / camera.f, 1 ); // virtual screen point in camera space
var ray = add( add( mul( camera.x, vsp.x ), mul( camera.y, vsp.y ) ), mul( camera.z, vsp.z ) ); // into world space
var u = -camera.p.z / ray.z;
return add( camera.p, mul( ray, u ) );
}
else
return mul( complex_conjugate( sub( p, origin_screen ) ), 1.0 / scale );
}
function camera_projection( p ) {
var ray = sub( p, camera.p ); // the ray from camera center to point
var cp = p3( dot( camera.x, ray ), dot( camera.y, ray ), dot( camera.z, ray ) ); // into camera space
return p3( cp.x * camera.f / cp.z + camera.pp.x,
canvas.height - ( cp.y * camera.f / cp.z + camera.pp.y ),
cp.z );
}
function inViewingFrustrum( p ) {
if( !show_3D) return true;
var near_plane = 0.5;
var far_plane = 1000;
return p.z > near_plane && p.z < far_plane;
}
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( drawing ) {
if( is_mouse_down ) {
// add the point to the drawn_lines if sufficiently far from the previous point
var oldPath = drawn_lines[ drawn_lines.length - 1 ];
var oldPos = toScreen( oldPath[ oldPath.length - 1 ] );
if( dist( oldPos, pos ) > 3 ) {
oldPath.push( fromScreen( pos ) );
}
draw();
}
}
else if( point_is_being_dragged ) {
control_points[ i_point ] = fromScreen( add( pos, drag_offset ) );
enforce_constraints();
draw();
}
else {
// find the closest point, if within a minimum radius
var iClosest = -1;
var closest_dist2 = touch_radius * touch_radius;
var num_control_points = 4;
if( unitary || unit_circle_group ) num_control_points = 2; // only allow a and b to be dragged
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 ) {
if( drawing ) {
drawn_lines.push([fromScreen(pos)]);
is_mouse_down = true;
}
else {
// find the closest point, if within a minimum radius
var iClosest = -1;
var closest_dist2 = touch_radius * touch_radius;
var num_control_points = 4;
if( unitary || unit_circle_group ) num_control_points = 2; // only allow a and b to be dragged
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 );
control_points = [ p2( 0.9, -2.0 ), p2( 1.1, 0.0 ), p2( 0.9, 0.0 ), p2( 0.5, -2.0 ) ];
origin_screen = p2( canvas.width/2, canvas.height/2 );
scale = Math.min( canvas.width, canvas.height ) / 6;
point_is_highlighted = false;
point_is_being_dragged = false;
i_point = -1;
grab_offset = p2( 0, 0 );
touch_radius = 50;
draw_grid = true;
draw_arrows = false;
normalize_mobius = false;
draw_fixed_points = false;
unitary = false;
nonloxodromic = false;
show_ad_minus_bc = false;
show_a_plus_d = false;
unit_circle_group = false;
theta = -Math.PI / 2;
repositionCamera();
show_3D = false;
riemann_inversion_sphere = { p:p3(0,0,1), r:1 };
drawn_lines = [];
drawing = false;
is_mouse_down = false;
draw();
is_spinning = true;
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();
}
// adapted from http://stackoverflow.com/a/6333775/126823
function drawArrow( from, to ) {
var headlen = 5; // length of head in pixels
var angle = Math.atan2(to.y-from.y,to.x-from.x);
ctx.moveTo(from.x, from.y);
ctx.lineTo(to.x, to.y);
ctx.moveTo(to.x-headlen*Math.cos(angle-Math.PI/6),to.y-headlen*Math.sin(angle-Math.PI/6));
ctx.lineTo(to.x, to.y);
ctx.lineTo(to.x-headlen*Math.cos(angle+Math.PI/6),to.y-headlen*Math.sin(angle+Math.PI/6));
}
function drawSmoothTransformedLine( a, c, mobius_transform, on_sphere, recursion_depth = 1 ) {
var b = average( a, c );
var qa = a,qb = b,qc = c;
if( mobius_transform ) {
qa = mobius_on_point( mobius_transform, qa );
qb = mobius_on_point( mobius_transform, qb );
qc = mobius_on_point( mobius_transform, qc );
}
if( on_sphere ) {
qa = sphereInversion( qa, riemann_inversion_sphere );
qb = sphereInversion( qb, riemann_inversion_sphere );
qc = sphereInversion( qc, riemann_inversion_sphere );
}
var sa = toScreen( qa );
var sb = toScreen( qb );
var sc = toScreen( qc );
// if nearly collinear then just draw straight lines
var close_enough = 3;
if( recursion_depth > 5
|| dist( sa, sc ) < close_enough
|| distanceOfPointFromLine( sa, sc, sb ) < close_enough ) {
if( inViewingFrustrum( sa ) && inViewingFrustrum( sb ) ) {
ctx.moveTo( sa.x, sa.y );
ctx.lineTo( sb.x, sb.y );
}
if( inViewingFrustrum( sb ) && inViewingFrustrum( sc ) ) {
ctx.moveTo( sb.x, sb.y );
ctx.lineTo( sc.x, sc.y );
}
return;
}
// otherwise recurse down
drawSmoothTransformedLine( a, b, mobius_transform, on_sphere, recursion_depth + 1 );
drawSmoothTransformedLine( b, c, mobius_transform, on_sphere, recursion_depth + 1 );
}
function drawSmoothTransformedArrow( p, on_sphere ) {
// interpolate between the identity transform and the current transform
if( on_sphere )
op = sphereInversion( p, riemann_inversion_sphere );
else
op = p;
var sp2 = toScreen( op );
ctx.moveTo( sp2.x, sp2.y );
var num_steps = 30;
var cp = [];
var target_p = control_points.slice();
mobius_normalize( target_p );
for( var i = 0; i < num_steps; i++ ) {
var u = i / num_steps;
mobius_identity( cp );
for( var j = 0; j < 4; j++ )
cp[ j ] = add( mul( target_p[ j ], u ), mul( cp[ j ], 1 - u ) );
var q = mobius_on_point( cp, p );
if( on_sphere )
q = sphereInversion( q, riemann_inversion_sphere );
sq = toScreen( q );
if( inViewingFrustrum( sq ) )
ctx.lineTo( sq.x, sq.y );
else
ctx.moveTo( sq.x, sq.y );
}
// draw the arrow using the last line segment
var q = mobius_on_point( control_points, p );
if( on_sphere )
q = sphereInversion( q, riemann_inversion_sphere );
var sq2 = toScreen( q );
if( inViewingFrustrum( sq ) && inViewingFrustrum( sq2 ) )
drawArrow( sq, sq2 );
}
function draw() {
// clear the area
ctx.strokeStyle = "rgb(0,0,0)";
ctx.lineWidth = 1;
ctx.clearRect(0, 0, canvas.width, canvas.height);
ctx.strokeRect(0, 0, canvas.width, canvas.height);
// draw the axes
var extent = 6;
ctx.strokeStyle = "rgba(0,0,0,0.2)";
ctx.beginPath();
for( var i = -extent; i <= extent - 1; i++ ) {
drawSmoothTransformedLine( p2( i, 0 ), p2( i+1, 0 ), null, false );
drawSmoothTransformedLine( p2( 0, i ), p2( 0, i+1 ), null, false );
}
for( var i = -extent; i <= extent; i++ ) {
drawSmoothTransformedLine( p2( i, -0.05 ), p2( i, 0.05 ), null, false );
drawSmoothTransformedLine( p2( -0.05, i ), p2( 0.05, i ), null, false );
}
ctx.stroke();
if( unit_circle_group ) {
// draw the unit circle
ctx.strokeStyle = "rgba(0,0,0,0.8)";
ctx.beginPath();
var p = p2( 1.0, 0.0 );
var sp = toScreen( p );
ctx.moveTo( sp.x, sp.y );
for( i = 5; i <= 360; i+= 5 ) {
var theta = i * Math.PI / 180.0;
p = p2( Math.cos( theta ), Math.sin( theta ) );
sp = toScreen( p );
ctx.lineTo( sp.x, sp.y );
}
ctx.stroke();
}
// draw arrows showing the transform
if( draw_arrows ) {
if( show_3D ) {
spacing = 0.5;
min_x = -extent + spacing/2; max_x = extent;
min_y = -extent + spacing/2; max_y = extent;
}
else {
spacing = 40;
min_x = canvas_rect.x + spacing/2; max_x = canvas_rect.x + canvas_rect.width;
min_y = canvas_rect.y + spacing/2; max_y = canvas_rect.y + canvas_rect.height;
}
ctx.strokeStyle = "rgba(0,0,128,0.4)";
ctx.beginPath();
for( var x = min_x; x < max_x; x += spacing ) {
for( var y = min_y; y < max_y; y += spacing ) {
if( unit_circle_group ) {
// skip this arrow if outside the unit circle
if( show_3D && len2( p2( x, y ) ) > 1.0 ) { // (x,y is in world space)
continue;
}
if( !show_3D && len2( fromScreen( p2( x, y ) ) ) > 1.0 ) { // (x,y is on the screen)
continue;
}
}
if( show_3D ) {
drawSmoothTransformedArrow( p2(x,y), null );
drawSmoothTransformedArrow( p2(x,y), control_points );
}
else
drawSmoothTransformedArrow( fromScreen( p2(x,y) ), null );
}
}
ctx.stroke();
}
// draw a distorted grid to show the transform
if( draw_grid ) {
if( show_3D ) {
spacing = 0.5;
min_x = -extent; max_x = extent;
min_y = -extent; max_y = extent;
}
else {
spacing = 40;
min_x = canvas_rect.x; max_x = canvas_rect.x + canvas_rect.width;
min_y = canvas_rect.y; max_y = canvas_rect.y + canvas_rect.height;
}
ctx.strokeStyle = "rgba(0,0,0,0.4)";
ctx.beginPath();
for( var x = min_x; x < max_x; x += spacing ) {
for( var y = min_y; y < max_y; y += spacing ) {
if( show_3D ) {
var wp1 = p2( x, y );
var wp2 = p2( x + spacing, y );
drawSmoothTransformedLine( wp1, wp2, control_points, false );
drawSmoothTransformedLine( wp1, wp2, control_points, true );
wp2 = p2( x, y + spacing );
drawSmoothTransformedLine( wp1, wp2, control_points, false );
drawSmoothTransformedLine( wp1, wp2, control_points, true );
}
else {
var wp1 = fromScreen( p2( x, y ) );
var wp2 = fromScreen( p2( x + spacing, y ) );
drawSmoothTransformedLine( wp1, wp2, control_points, false );
wp2 = fromScreen( p2( x, y + spacing ) );
drawSmoothTransformedLine( wp1, wp2, control_points, false );
}
}
}
ctx.stroke();
}
// draw the four 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 = 4;
if( unitary || unit_circle_group ) num_moveable_control_points = 2;
for( var i = 0; i < 4; 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, "abcd"[i], p2(0,0) );
}
if( draw_fixed_points ) {
ctx.fillStyle = ctx.strokeStyle = "rgb(0,0,0)";
var f_plus, f_minus;
[ f_plus, f_minus ] = get_mobius_fixed_points( control_points );
drawPoint( toScreen( f_plus ), 2, "f+", p2(0,0) );
drawPoint( toScreen( f_minus ), 2, "f-", p2(0,0) );
if( show_3D ) {
f_plus = sphereInversion( f_plus, riemann_inversion_sphere );
f_minus = sphereInversion( f_minus, riemann_inversion_sphere );
drawPoint( toScreen( f_plus ), 2, "f+", p2(0,0) );
drawPoint( toScreen( f_minus ), 2, "f-", p2(0,0) );
}
}
if( show_a_plus_d ) {
if( normalize_mobius ) {
// show the classification region
ctx.fillStyle = ctx.strokeStyle = "rgba(0,160,0,0.2)";
ctx.lineWidth = 5;
var sp1 = toScreen( p2( -2, 0 ) );
var sp2 = toScreen( p2( 2, 0 ) );
ctx.beginPath();
ctx.moveTo( sp1.x, sp1.y );
ctx.lineTo( sp2.x, sp2.y );
ctx.stroke();
drawPoint( toScreen( p2( -2, 0 ) ), 5, "", p2(0,0) );
drawPoint( toScreen( p2( 2, 0 ) ), 5, "", p2(0,0) );
}
var a_plus_d = add( control_points[0], control_points[3] );
ctx.fillStyle = ctx.strokeStyle = "rgb(0,0,0)";
drawPoint( toScreen( a_plus_d ), 2, "a+d", p2(0,0) );
}
if( show_ad_minus_bc ) {
var ad_minus_bc = sub( mul_complex( control_points[0], control_points[3] ), mul_complex( control_points[1], control_points[2] ) );
ctx.fillStyle = ctx.strokeStyle = "rgb(0,0,0)";
drawPoint( toScreen( ad_minus_bc ), 2, "ad-bc", p2(0,0) );
}
if( show_3D ) {
ctx.fillStyle = ctx.strokeStyle = "rgb(0,0,0)";
drawPoint( toScreen( riemann_inversion_sphere.p ), 2, "", p2(0,0) );
}
// draw the drawn lines
var inverse_mobius = get_mobius_inverse(control_points);
ctx.strokeStyle = "rgba(0,0,150,1.0)";
drawSketch( null );
// draw the iterated mobius transformed versions
var t = control_points;
var ti = inverse_mobius;
for(var i = 0; i < 50; i++) {
ctx.strokeStyle = "rgba(0,150,0,1.0)";
drawSketch( t );
ctx.strokeStyle = "rgba(255,0,0,1.0)";
drawSketch( ti );
t = get_mobius_composed(t, control_points);
ti = get_mobius_composed(ti, inverse_mobius);
}
}
function drawSketch(mobius_transform) {
ctx.beginPath();
for(var iPath = 0; iPath < drawn_lines.length; iPath++)
{
var path = drawn_lines[iPath];
for(var iPt = 0; iPt < path.length-1; iPt++)
{
drawSmoothTransformedLine( path[iPt], path[iPt + 1], mobius_transform, false );
if(show_3D)
drawSmoothTransformedLine( path[iPt], path[iPt + 1], mobius_transform, true );
}
}
ctx.stroke();
}
function onGridButton(element) {
draw_grid = !draw_grid;
element.value = draw_grid ? "Grid: ON" : "Grid: OFF";
draw();
}
function onArrowsButton(element) {
draw_arrows = !draw_arrows;
element.value = draw_arrows ? "Arrows: ON" : "Arrows: OFF";
draw();
}
function onNormalizeButton(element) {
normalize_mobius = !normalize_mobius;
if( normalize_mobius ) {
mobius_normalize( control_points );
}
element.value = normalize_mobius ? "Normalize: ON" : "Normalize: OFF";
draw();
}
function onFixedPointsButton(element) {
draw_fixed_points = !draw_fixed_points;
element.value = draw_fixed_points ? "Fixed points: ON" : "Fixed points: OFF";
draw();
}
function onShowAPlusDButton(element) {
show_a_plus_d = !show_a_plus_d;
element.value = show_a_plus_d ? "Show a+d: ON" : "Show a+d: OFF";
draw();
}
function onShowADMinusBCButton(element) {
show_ad_minus_bc = !show_ad_minus_bc;
element.value = show_ad_minus_bc ? "Show ad-bc: ON" : "Show ad-bc: OFF";
draw();
}
function onUnitaryButton(element) {
unitary = !unitary;
element.value = unitary ? "Unitary: ON" : "Unitary: OFF";
enforce_constraints();
draw();
}
function onNonLoxodromicButton(element) {
nonloxodromic = !nonloxodromic;
element.value = nonloxodromic ? "Non-loxodromic: ON" : "Non-loxodromic: OFF";
enforce_constraints();
draw();
}
function onUnitCircleGroupButton(element) {
unit_circle_group = !unit_circle_group;
element.value = unit_circle_group ? "Unit circle group: ON" : "Unit circle group: OFF";
enforce_constraints();
draw();
}
function onIdentityButton() {
mobius_identity( control_points );
enforce_constraints();
draw();
}
function onShowIn3DButton(element) {
show_3D = !show_3D;
element.value = show_3D ? "Show in 3D: ON" : "Show in 3D: OFF";
if( show_3D ) {
lastFrameTime = undefined; // reset the spin to avoid jumps
repositionCamera();
animate();
}
draw();
}
function onSpinningButton(element) {
is_spinning = !is_spinning;
element.value = is_spinning ? "Spinning: ON" : "Spinning: OFF";
if( is_spinning ) {
lastFrameTime = undefined; // reset the spin to avoid jumps
animate();
}
}
function onDrawingModeButton(element) {
drawing = !drawing;
element.value = drawing ? "Drawing mode: ON" : "Drawing mode: OFF";
draw();
}
function onClearDrawingButton(element) {
drawn_lines = [];
draw();
}
function animate( currentFrameTime ) {
if( show_3D && is_spinning ) {
// rotate round
if( typeof currentFrameTime !== 'undefined' ) {
if( typeof lastFrameTime === 'undefined' ) {
lastFrameTime = currentFrameTime;
}
var elapsedSinceLastFrame = currentFrameTime - lastFrameTime;
var theta_increment = elapsedSinceLastFrame * 0.0003;
lastFrameTime = currentFrameTime;
theta += theta_increment;
repositionCamera();
}
draw();
requestAnimationFrame( animate );
}
}
window.onload = init;
</script>
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<style>
input[type='button'] {
font-size: 20px;
font-family: Helvetica Neue,Helvetica,Arial,sans-serif;
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</head>
<body>
<p>A <b><a href="https://en.wikipedia.org/wiki/M%C3%B6bius_transformation">Möbius transformation</a></b> is <b>f( z ) = ( az + b ) / ( cz + d )</b> where <b>z</b>, <b>a</b>,
<b>b</b>, <b>c</b> and <b>d</b> are complex numbers. Drag the points to change the transformation.</p>
<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="Arrows: OFF" onclick="onArrowsButton(this)" /></p>
<p><input type="button" value="Grid: ON" onclick="onGridButton(this)" /></p>
<p><input type="button" value="Identity" onclick="onIdentityButton()" /></p>
<p><input type="button" value="Normalize: OFF" onclick="onNormalizeButton(this)" /></p>
<p><input type="button" value="Show ad-bc: OFF" onclick="onShowADMinusBCButton(this)" /></p>
<p><input type="button" value="Fixed points: OFF" onclick="onFixedPointsButton(this)" /></p>
<p><input type="button" value="Unitary: OFF" onclick="onUnitaryButton(this)" /></p>
<p><input type="button" value="Non-loxodromic: OFF" onclick="onNonLoxodromicButton(this)" /></p>
<p><input type="button" value="Unit circle group: OFF" onclick="onUnitCircleGroupButton(this)" /></p>
<p><input type="button" value="Show a+d: OFF" onclick="onShowAPlusDButton(this)" /></p>
<p><input type="button" value="Show in 3D: OFF" onclick="onShowIn3DButton(this)" /></p>
<p><input type="button" value="Spinning: ON" onclick="onSpinningButton(this)" /></p>
<p><input type="button" value="Drawing mode: OFF" onclick="onDrawingModeButton(this)" /></p>
<p><input type="button" value="Clear drawing" onclick="onClearDrawingButton(this)" /></p>
</div>
<div style="clear:both">
<p>To simplify the transformation, set <b>c</b> to zero (move it to the origin). Now <b>a</b> and <b>b</b> between
them give a similarity transformation: <b>a</b> controls the scale and rotation while <b>b</b> controls the translation.</p>
<p>The identity transformation is given when <b>b</b>=<b>c</b>=0 and <b>a</b>=<b>d</b>.</p>
<p>A <i>normalized</i> Möbius transformation is one where <b>ad - bc = 1</b>.</p>
<p>A <i>unitary</i> Möbius transformation has <b>c = -b̅</b> and <b>d = a̅</b> (where <b>z̅</b> is the <a href="https://en.wikipedia.org/wiki/Complex_conjugate">complex conjugate</a> of <b>z</b>).</p>
<p>A normalized Möbius transformation is <i>loxodromic</i> if <b>a+d</b> is not on the real line. Set <b>Non-loxodromic: ON</b> to enforce that <b>a+d</b> is real, and <b>Normalize: ON</b> to ensure that the transformation is normalized. Then if <b>|a+d| < 2</b> the Möbius transformation is <i>elliptic</i>. If <b>|a+b|=2</b> then it is <i>parabolic</i>. Otherwise, with <b>|a+d| > 2</b>, it is <i>hyperbolic</i>.
<p>A Möbius transformation with <b>|a|² - |b|² = 1, c = b̅, d = a̅</b> maps the unit circle to itself. Set <b>Unit circle group: ON</b> to enforce this contraint. When <b>a</b> is real this gives a translation of the hyperbolic plane in the Poincaré disk model.</p>
<p>With <b>Drawing mode: ON</b>, draw on the screen. Multiple copies of the drawing (blue) under iterations of the Möbius transformation (green) and its inverse (red) are shown.
<p>Read the book <a href="http://www.amazon.co.uk/Visual-Complex-Analysis-Tristan-Needham/dp/0198534469">Visual Complex Analysis</a> by Tristan Needham. To see the relationship with the Riemann sphere, see <a href="https://www.youtube.com/watch?v=0z1fIsUNhO4">this video</a>. To explore the possibilities of iterating Möbius transforms, see the book <a href="https://en.wikipedia.org/wiki/Indra%27s_Pearls_(book)">Indra's Pearls</a>, by David Mumford, Caroline Series and David Wright.</p>
<hr>
<p>Source code: <a href="https://github.com/timhutton/mobius-transforms">https://github.com/timhutton/mobius-transforms</a></p>
</div>
</html>