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3 changed files with 47 additions and 185 deletions
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2
index.html
View file

@ -10,14 +10,12 @@ body {
padding: 0; padding: 0;
margin: 0; margin: 0;
background-color: #1b1b1b; background-color: #1b1b1b;
overflow-y: hidden;
} }
</style> </style>
</head> </head>
<body> <body>
<main> <main>
</main> </main>
<script src="thomson-problem.js"></script>
<script src="sketch.js"></script> <script src="sketch.js"></script>
</body> </body>
</html> </html>

186
sketch.js
View file

@ -1,33 +1,16 @@
let camera;
let red;
let charges = []; let charges = [];
let faces = [];
let sphere_mode = 'circles';
let sphere_radius = 200; let sphere_radius = 200;
const SURFACE_NONE = 0;
const SURFACE_CIRCLES = 1;
const SURFACE_EARTH = 2;
let surface = SURFACE_CIRCLES;
let physics = false; let physics = false;
let skeleton = false;
let polyhedron = false;
function preload() { function preload() {
earth_image = loadImage("atlas1.jpg"); earth = loadImage("atlas1.jpg");
} }
function setup() { function setup() {
createCanvas(windowWidth, windowHeight, WEBGL); createCanvas(600, 600, WEBGL);
camera = createCamera();
red = color(0xbf, 0x00, 0x00);
}
function windowResized() {
resizeCanvas(windowWidth, windowHeight);
} }
function draw() { function draw() {
@ -35,117 +18,51 @@ function draw() {
background(50); background(50);
make_lights(); make_lights();
if (physics) move_charges(charges); if (physics) {
move_charges();
draw_charges(sphere_radius);
if (skeleton) draw_skeleton(sphere_radius);
if (polyhedron) {
if (physics || faces.length === 0) find_faces();
draw_faces(sphere_radius);
} }
draw_charges(sphere_radius);
draw_sphere(sphere_radius, 25); draw_sphere(sphere_radius, 25);
} }
function face_dist_sq([v1, v2, v3]) { function move_charges() {
const center = p5.Vector.add(v1, v2).add(v3).mult(1 / 3); for (charge of charges) {
return createVector(camera.eyeX, camera.eyeY, camera.eyeZ).sub(center).magSq(); charge.acceleration.setMag(0);
}; }
for (let i = 0; i < charges.length; i += 1) {
function find_faces() { for (let j = 0; j < i; j += 1) {
faces = []; const displacement = p5.Vector.sub(
for (let i = 2; i < charges.length; i += 1) { charges[i].position,
for (let j = 1; j < i; j += 1) { charges[j].position,
for (let k = 0; k < j; k += 1) { );
// Check if p1 p2 p3 form a face of the convex polyhedron const acceleration_mag = 1 / displacement.mag() * 0.005;
// enclosing all vertices ... let ai = displacement.copy().normalize().mult(acceleration_mag);
const p1 = charges[i].position; let aj = p5.Vector.mult(ai, -1);
const p2 = charges[j].position; project_onto_plane(ai, charges[i].position);
const p3 = charges[k].position; project_onto_plane(aj, charges[j].position);
const normal = p5.Vector.sub(p2, p1).cross(p5.Vector.sub(p3, p1)); charges[i].acceleration.add(ai);
// ... by checking if the other vertices are on the same charges[j].acceleration.add(aj);
// side of the plane generated by p1 p2 p3
let plane_separates_vertices = false;
let euler_formula = false;
for (let r = 1; r < charges.length; r += 1) {
for (let s = 0; s < r; s += 1) {
if (
r === i || r === j || r === k ||
s === i || s === j || s === k
) continue;
const q1 = charges[r].position;
const q2 = charges[s].position;
// Let l(t) := q1 + (q2 - q1) * t.
// L := { l(t) : 0 <= t <= 1 } is the line segment
// between q1 and q2. L intersects the plane
// generated by p1 p2 p3 iff l(t) intersects with
// the plane for some 0 <= t <= 1.
//
// If L intersects the plane, q1 and q2 are on
// opposite sides of the plane generated by p1 p2 p3,
// so p1 p2 p3 can't be a face. If we want the
// polyhedron to be convex. Which we do.
//
// A point k is on the plane generated by p1 p2 p3 iff
// dot(k - p1, normal) = 0. Let n := normal.
//
// dot(l(t) - p1, n) = 0
// iff dot(q1 + (q2 - q1) * t - p1, n) = 0
// iff dot(q1 - p1, n) + dot(q2 - q2, n) * t = 0
// iff t = dot(p1 - q1, n) / dot(q2 - q2, n)
const t = (
p5.Vector.dot(p5.Vector.sub(p1, q1), normal) /
p5.Vector.dot(p5.Vector.sub(q2, q1), normal)
);
plane_separates_vertices ||= t >= 0 && t <= 1;
if (plane_separates_vertices) break;
euler_formula ||= charges.length * 2 - faces.length == 4;
if (euler_formula) break;
}
if (plane_separates_vertices || euler_formula) break;
}
if (euler_formula) return;
if (!plane_separates_vertices) {
faces.push([p1, p2, p3]);
}
}
} }
} }
for (let i = 0; i < charges.length; i += 1) {
let charge = charges[i];
charge.velocity = charge.velocity.add(charge.acceleration);
//project_onto_plane(charge.velocity, charge.position);
charge.position = charge.position.add(charge.velocity);
charge.position.normalize();
}
} }
function draw_faces(radius) { function project_onto_unit_vector(v, unit_vector) {
// fix OpenGL stacking alpha behaviour let size = p5.Vector.dot(v, unit_vector);
faces.sort((a, b) => face_dist_sq(b) - face_dist_sq(a)); return p5.Vector.mult(unit_vector, size);
push();
strokeWeight(2);
stroke(0x00);
for ([p1, p2, p3] of faces) {
fill(0xbf, 0x7f);
beginShape(TRIANGLES);
vertex(p1.x * radius, p1.y * radius, p1.z * radius);
vertex(p2.x * radius, p2.y * radius, p2.z * radius);
vertex(p3.x * radius, p3.y * radius, p3.z * radius);
endShape();
}
pop();
} }
function draw_skeleton(radius) { /// Project `v` onto the plane normal to `unit_vector`
push(); /// Mutates `v`
noStroke(); function project_onto_plane(v, unit_vector) {
fill(0xff); let v_proj = project_onto_unit_vector(v, unit_vector);
sphere(4); v.sub(v_proj)
stroke(0xbf);
for (let charge of charges) {
line(
0,
0,
0,
charge.position.x * radius,
charge.position.y * radius,
charge.position.z * radius,
);
}
pop();
} }
function make_charges(n) { function make_charges(n) {
@ -167,7 +84,6 @@ function make_charges(n) {
position: position, position: position,
velocity: createVector(), velocity: createVector(),
acceleration: createVector(), acceleration: createVector(),
color: red,
}); });
} }
} }
@ -175,8 +91,8 @@ function make_charges(n) {
function draw_charges(radius) { function draw_charges(radius) {
push(); push();
noStroke(); noStroke();
ambientMaterial(0xbf, 0x00, 0x00);
for (let charge of charges.values()) { for (let charge of charges.values()) {
ambientMaterial(charge.color);
let position = charge.position.copy(); let position = charge.position.copy();
position.mult(radius); position.mult(radius);
push(); push();
@ -191,14 +107,11 @@ function draw_sphere(radius, n_axis_circles) {
stroke(0x3f); stroke(0x3f);
noFill(); noFill();
if (surface === SURFACE_NONE) n_axis_circles = 0;
else if (surface === SURFACE_EARTH) n_axis_circles = 2;
push(); push();
rotateX(TAU / 4); rotateX(TAU / 4);
draw_circles( draw_circles(
radius, radius,
n_axis_circles, sphere_mode === 'earth' ? 2 : n_axis_circles,
color(0x00, 0x9f, 0xff), color(0x00, 0x9f, 0xff),
color(0xff, 0x9f, 0x00), color(0xff, 0x9f, 0x00),
); );
@ -207,18 +120,18 @@ function draw_sphere(radius, n_axis_circles) {
rotateY(TAU / 4); rotateY(TAU / 4);
draw_circles( draw_circles(
radius, radius,
n_axis_circles, sphere_mode === 'earth' ? 2 : n_axis_circles,
color(0xff, 0x00, 0xff), color(0xff, 0x00, 0xff),
color(0x00, 0xff, 0x00), color(0x00, 0xff, 0x00),
); );
pop(); pop();
if (surface === SURFACE_EARTH) { if (sphere_mode === 'earth') {
push();
noStroke(); noStroke();
noFill(); noFill();
tint(0xff, 0x9f); tint(0xff, 0x9f);
texture(earth_image); texture(earth);
push();
rotateY(TAU / 4); rotateY(TAU / 4);
sphere(radius); sphere(radius);
pop(); pop();
@ -254,16 +167,11 @@ function make_lights() {
} }
function keyPressed() { function keyPressed() {
if (key == ' ') { if (key == 'd') {
sphere_mode = sphere_mode === 'earth' ? 'circles' : 'earth';
} else if (key == ' ') {
physics = !physics; physics = !physics;
} else if (key == 'd') {
surface = (surface + 1) % 3;
} else if (key == 'f') {
skeleton = !skeleton;
} else if (key == 'g') {
polyhedron = !polyhedron;
} else if (key >= '0' && key <= '9') { } else if (key >= '0' && key <= '9') {
make_charges(int(key)); make_charges(int(key));
faces = [];
} }
} }

44
thomson-problem.js
View file

@ -1,44 +0,0 @@
function move_charges(charges) {
for (charge of charges) {
charge.acceleration.setMag(0);
}
for (let i = 0; i < charges.length; i += 1) {
for (let j = 0; j < i; j += 1) {
const displacement = p5.Vector.sub(
charges[i].position,
charges[j].position,
);
let acceleration_mag = 1 / displacement.mag() * 0.001;
let ai;
if (acceleration_mag === Infinity) {
ai = createVector(random(-1, 1), random(-1, 1), random(-1, 1));
} else {
ai = displacement.copy();
}
ai = ai.normalize().mult(acceleration_mag);
let aj = p5.Vector.mult(ai, -1);
project_onto_plane(ai, charges[i].position);
project_onto_plane(aj, charges[j].position);
charges[i].acceleration.add(ai);
charges[j].acceleration.add(aj);
}
}
for (let i = 0; i < charges.length; i += 1) {
let charge = charges[i];
charge.velocity = charge.velocity.add(charge.acceleration);
charge.position = charge.position.add(charge.velocity);
charge.position.normalize();
}
}
function project_onto_unit_vector(v, unit_vector) {
let size = p5.Vector.dot(v, unit_vector);
return p5.Vector.mult(unit_vector, size);
}
/// Project `v` onto the plane normal to `unit_vector`
/// Mutates `v`
function project_onto_plane(v, unit_vector) {
let v_proj = project_onto_unit_vector(v, unit_vector);
v.sub(v_proj)
}