tiny_graphics.js

/**
 * @file A file that shows how to organize a complete graphics program.
 * It wraps common WebGL commands and math.
 * The file tiny_graphics_widgets.js additionally wraps web page interactions.  By Garett.
 *
 * This file will consist of a number of class definitions that we will
 * export.  To organize the exports, we will both declare each class in
 * local scope (const) as well as store them in this JS object:
 *
 * Organization of this file:  Math definitions, then graphics definitions.
 *
 * Vector and Matrix algebra are not built into JavaScript at first.  We will add it now.
 *
 * You will be able to declare a 3D vector [x,y,z] supporting various common vector operations
 * with syntax:  vec(x,y), vec3( x,y,z ) or vec4( x,y,z, zero or one ).  For general sized vectors, use
 * class Vector and declare them with standard Array-supported operations like .of().
 *
 * For matrices, you will use class Mat4 to generate the 4 by 4 matrices that are common
 * in graphics, or for general sized matrices you can use class Matrix.
 *
 * To get vector algebra that performs well in JavaScript, we based class Vector on consecutive
 * buffers (using type Float32Array).  Implementations should specialize for common vector
 * sizes 3 and 4 since JavaScript engines can better optimize functions when they can predict
 * argument count.  Implementations should also avoid allocating new array objects since these
 * will all have to be garbage collected.
 *
 * Examples:
 *  ** For size 3 **
 *     equals: "vec3( 1,0,0 ).equals( vec3( 1,0,0 ) )" returns true.
 *       plus: "vec3( 1,0,0 ).plus  ( vec3( 1,0,0 ) )" returns the Vector [ 2,0,0 ].
 *      minus: "vec3( 1,0,0 ).minus ( vec3( 1,0,0 ) )" returns the Vector [ 0,0,0 ].
 * mult-pairs: "vec3( 1,2,3 ).mult_pairs( vec3( 3,2,0 ) )" returns the Vector [ 3,4,0 ].
 *      scale: "vec3( 1,2,3 ).scale( 2 )" overwrites the Vector with [ 2,4,6 ].
 *      times: "vec3( 1,2,3 ).times( 2 )" returns the Vector [ 2,4,6 ].
 * randomized: Returns this Vector plus a random vector of a given maximum length.
 *        mix: "vec3( 0,2,4 ).mix( vec3( 10,10,10 ), .5 )" returns the Vector [ 5,6,7 ].
 *       norm: "vec3( 1,2,3 ).norm()" returns the square root of 15.
 * normalized: "vec3( 4,4,4 ).normalized()" returns the Vector [ sqrt(3), sqrt(3), sqrt(3) ]
 *  normalize: "vec3( 4,4,4 ).normalize()" overwrites the Vector with [ sqrt(3), sqrt(3), sqrt(3) ].
 *        dot: "vec3( 1,2,3 ).dot( vec3( 1,2,3 ) )" returns 15.
 *       cast: "vec3.cast( [-1,-1,0], [1,-1,0], [-1,1,0] )" converts a list of Array literals into a list of vec3's.
 *        to4: "vec3( 1,2,3 ).to4( true or false )" returns the homogeneous vec4 [ 1,2,3, 1 or 0 ].
 *      cross: "vec3( 1,0,0 ).cross( vec3( 0,1,0 ) )" returns the Vector [ 0,0,1 ].  Use only on 3x1 Vecs.
 *  to_string: "vec3( 1,2,3 ).to_string()" returns "[vec3 1, 2, 3]"
 *  ** For size 4, same except: **
 *        to3: "vec4( 4,3,2,1 ).to3()" returns the vec3 [ 4,3,2 ].  Use to truncate vec4 to vec3.
 *  ** To assign by value **
 *       copy: "let new_vector = old_vector.copy()" assigns by value so you get a different vector object.
 *  ** For any size **
 * to declare: Vector.of( 1,2,3,4,5,6,7,8,9,10 ) returns a Vector filled with those ten entries.
 *  ** For multiplication by matrices **
 *             "any_mat4.times( vec4( 1,2,3,0 ) )" premultiplies the homogeneous Vector [1,2,3]
 *              by the 4x4 matrix and returns the new vec4.  Requires a vec4 as input.
 */

export const tiny = {};

/**
 * **Vector** stores vectors of floating point numbers.  Puts vector math into JavaScript.
 * @Note  Vectors should be created with of() due to wierdness with the TypedArray spec.
 * @Tip Assign Vectors with .copy() to avoid referring two variables to the same Vector object.
 * @type {tiny.Vector}
 */
const Vector = tiny.Vector =
  class Vector extends Float32Array {
    static create(...arr) {
      return new Vector(arr);
    }

    // cast(): For compact syntax when declaring lists.
    static cast(...args) {
      return args.map(x => Vector.from(x))
    }

    copy() {
      return new Vector(this)
    }

    equals(b) {
      return this.every((x, i) => x == b[i])
    }

    plus(b) {
      return this.map((x, i) => x + b[i])
    }

    minus(b) {
      return this.map((x, i) => x - b[i])
    }

    times_pairwise(b) {
      return this.map((x, i) => x * b[i])
    }

    scale_by(s) {
      this.forEach((x, i, a) => a[i] *= s)
    }

    times(s) {
      return this.map(x => s * x)
    }

    randomized(s) {
      return this.map(x => x + s * (Math.random() - .5))
    }

    mix(b, s) {
      return this.map((x, i) => (1 - s) * x + s * b[i])
    }

    norm() {
      return Math.sqrt(this.dot(this))
    }

    normalized() {
      return this.times(1 / this.norm())
    }

    normalize() {
      this.scale_by(1 / this.norm())
    }

    dot(b) {
      // Optimize for Vectors of size 2
      if (this.length == 2)
        return this[0] * b[0] + this[1] * b[1];
      return this.reduce((acc, x, i) => {
        return acc + x * b[i];
      }, 0);
    }

    // to3() / to4() / cross():  For standardizing the API with Vector3/Vector4,
    // so the performance hit of changing between these types can be measured.
    to3() {
      return vec3(this[0], this[1], this[2]);
    }

    to4(is_a_point) {
      return vec4(this[0], this[1], this[2], +is_a_point);
    }

    cross(b) {
      return vec3(this[1] * b[2] - this[2] * b[1],
        this[2] * b[0] - this[0] * b[2],
        this[0] * b[1] - this[1] * b[0]);
    }

    to_string() {
      return "[vector " + this.join(", ") + "]"
    }
  }


const Vector3 = tiny.Vector3 =
  class Vector3 extends Float32Array {
    // **Vector3** is a specialization of Vector only for size 3, for performance reasons.
    static create(x, y, z) {
      const v = new Vector3(3);
      v[0] = x;
      v[1] = y;
      v[2] = z;
      return v;
    }

    static cast(...args) {
      // cast(): Converts a bunch of arrays into a bunch of vec3's.
      return args.map(x => Vector3.from(x));
    }

    static unsafe(x, y, z) {
      // unsafe(): returns vec3s only meant to be consumed immediately. Aliases into
      // shared memory, to be overwritten upon next unsafe3 call.  Faster.
      const shared_memory = vec3(0, 0, 0);
      Vector3.unsafe = (x, y, z) => {
        shared_memory[0] = x;
        shared_memory[1] = y;
        shared_memory[2] = z;
        return shared_memory;
      }
      return Vector3.unsafe(x, y, z);
    }

    copy() {
      return Vector3.from(this)
    }

    // In-fix operations: Use these for more readable math expressions.
    equals(b) {
      return this[0] == b[0] && this[1] == b[1] && this[2] == b[2]
    }

    plus(b) {
      return vec3(this[0] + b[0], this[1] + b[1], this[2] + b[2])
    }

    minus(b) {
      return vec3(this[0] - b[0], this[1] - b[1], this[2] - b[2])
    }

    times(s) {
      return vec3(this[0] * s, this[1] * s, this[2] * s)
    }

    times_pairwise(b) {
      return vec3(this[0] * b[0], this[1] * b[1], this[2] * b[2])
    }

    // Pre-fix operations: Use these for better performance (to avoid new allocation).
    add_by(b) {
      this[0] += b[0];
      this[1] += b[1];
      this[2] += b[2]
    }

    subtract_by(b) {
      this[0] -= b[0];
      this[1] -= b[1];
      this[2] -= b[2]
    }

    scale_by(s) {
      this[0] *= s;
      this[1] *= s;
      this[2] *= s
    }

    scale_pairwise_by(b) {
      this[0] *= b[0];
      this[1] *= b[1];
      this[2] *= b[2]
    }

    // Other operations:
    randomized(s) {
      return vec3(this[0] + s * (Math.random() - .5),
        this[1] + s * (Math.random() - .5),
        this[2] + s * (Math.random() - .5));
    }

    mix(b, s) {
      return vec3((1 - s) * this[0] + s * b[0],
        (1 - s) * this[1] + s * b[1],
        (1 - s) * this[2] + s * b[2]);
    }

    norm() {
      return Math.sqrt(this[0] * this[0] + this[1] * this[1] + this[2] * this[2])
    }

    normalized() {
      const d = 1 / this.norm();
      return vec3(this[0] * d, this[1] * d, this[2] * d);
    }

    normalize() {
      const d = 1 / this.norm();
      this[0] *= d;
      this[1] *= d;
      this[2] *= d;
    }

    dot(b) {
      return this[0] * b[0] + this[1] * b[1] + this[2] * b[2]
    }

    cross(b) {
      return vec3(this[1] * b[2] - this[2] * b[1],
        this[2] * b[0] - this[0] * b[2],
        this[0] * b[1] - this[1] * b[0])
    }

    to4(is_a_point)
    // to4():  Convert to a homogeneous vector of 4 values.
    {
      return vec4(this[0], this[1], this[2], +is_a_point)
    }

    to_string() {
      return "[vec3 " + this.join(", ") + "]"
    }
  }

const Vector4 = tiny.Vector4 =
  class Vector4 extends Float32Array {
    // **Vector4** is a specialization of Vector only for size 4, for performance reasons.
    // The fourth coordinate value is homogenized (0 for a vector, 1 for a point).
    static create(x, y, z, w) {
      const v = new Vector4(4);
      v[0] = x;
      v[1] = y;
      v[2] = z;
      v[3] = w;
      return v;
    }

    static unsafe(x, y, z, w) {
      // **unsafe** Returns vec3s to be used immediately only. Aliases into
      // shared memory to be overwritten on next unsafe3 call.  Faster.
      const shared_memory = vec4(0, 0, 0, 0);
      Vec4.unsafe = (x, y, z, w) => {
        shared_memory[0] = x;
        shared_memory[1] = y;
        shared_memory[2] = z;
        shared_memory[3] = w;
      }
    }

    copy() {
      return Vector4.from(this)
    }

    // In-fix operations: Use these for more readable math expressions.
    equals(b) {
      return this[0] == b[0] && this[1] == b[1] && this[2] == b[2] && this[3] == b[3]
    }

    plus(b) {
      return vec4(this[0] + b[0], this[1] + b[1], this[2] + b[2], this[3] + b[3])
    }

    minus(b) {
      return vec4(this[0] - b[0], this[1] - b[1], this[2] - b[2], this[3] - b[3])
    }

    times(s) {
      return vec4(this[0] * s, this[1] * s, this[2] * s, this[3] * s)
    }

    times_pairwise(b) {
      return vec4(this[0] * b[0], this[1] * b[1], this[2] * b[2], this[3] * b[3])
    }

    // Pre-fix operations: Use these for better performance (to avoid new allocation).
    add_by(b) {
      this[0] += b[0];
      this[1] += b[1];
      this[2] += b[2];
      this[3] += b[3]
    }

    subtract_by(b) {
      this[0] -= b[0];
      this[1] -= b[1];
      this[2] -= b[2];
      this[3] -= b[3]
    }

    scale_by(s) {
      this[0] *= s;
      this[1] *= s;
      this[2] *= s;
      this[3] *= s
    }

    scale_pairwise_by(b) {
      this[0] *= b[0];
      this[1] *= b[1];
      this[2] *= b[2];
      this[3] *= b[3]
    }

    // Other operations:
    randomized(s) {
      return vec4(this[0] + s * (Math.random() - .5),
        this[1] + s * (Math.random() - .5),
        this[2] + s * (Math.random() - .5),
        this[3] + s * (Math.random() - .5));
    }

    mix(b, s) {
      return vec4((1 - s) * this[0] + s * b[0],
        (1 - s) * this[1] + s * b[1],
        (1 - s) * this[2] + s * b[2],
        (1 - s) * this[3] + s * b[3]);
    }

    norm() {
      // The norms should behave like for Vector3 because of the homogenous format.
      return Math.sqrt(this[0] * this[0] + this[1] * this[1] + this[2] * this[2])
    }

    normalized() {
      const d = 1 / this.norm();
      return vec4(this[0] * d, this[1] * d, this[2] * d, this[3]);
      // (leaves the 4th coord alone)
    }

    normalize() {
      const d = 1 / this.norm();
      this[0] *= d;
      this[1] *= d;
      this[2] *= d;
      // (leaves the 4th coord alone)
    }

    dot(b) {
      return this[0] * b[0] + this[1] * b[1] + this[2] * b[2] + this[3] * b[3]
    }

    to3() {
      return vec3(this[0], this[1], this[2])
    }

    to_string() {
      return "[vec4 " + this.join(", ") + "]"
    }
  }

const vec = tiny.vec = Vector.create;
const vec3 = tiny.vec3 = Vector3.create;
const vec4 = tiny.vec4 = Vector4.create;
const unsafe3 = tiny.unsafe3 = Vector3.unsafe;
const unsafe4 = tiny.unsafe4 = Vector4.unsafe;

// **Color** is an alias for class Vector4.  Colors should be made as special 4x1
// vectors expressed as ( red, green, blue, opacity ) each ranging from 0 to 1.
const Color = tiny.Color =
  class Color extends Vector4 {
    // Create color from RGBA floats
    static create_from_float(r, g, b, a) {
      const v = new Vector4(4);
      v[0] = r;
      v[1] = g;
      v[2] = b;
      v[3] = a;
      return v;
    }

    // Create color from hexadecimal numbers, e.g., #FFFFFF
    static create_from_hex(hex, alpha = 1.) {
      const result = /^#?([a-f\d]{2})([a-f\d]{2})([a-f\d]{2})$/i.exec(hex);
      const v = new Vector4(4);
      if (result) {
        v[0] = parseInt(result[1], 16) / 255.;
        v[1] = parseInt(result[2], 16) / 255.;
        v[2] = parseInt(result[3], 16) / 255.;
        v[3] = alpha;
      }
      return v;
    }
  }

const color = tiny.color = Color.create_from_float;
const hex_color = tiny.hex_color = Color.create_from_hex;

const Matrix = tiny.Matrix =
  class Matrix extends Array {
    // **Matrix** holds M by N matrices of floats.  Enables matrix and vector math.
    // Example usage:
    //  "Matrix( rows )" returns a Matrix with those rows, where rows is an array of float arrays.
    //  "M.set_identity( m, n )" assigns the m by n identity to Matrix M.
    //  "M.sub_block( start, end )" where start and end are each a [ row, column ] pair returns a sub-rectangle cut out from M.
    //  "M.copy()" creates a deep copy of M and returns it so you can modify it without affecting the original.
    //  "M.equals(b)", "M.plus(b)", and "M.minus(b)" are operations betwen two matrices.
    //  "M.transposed()" returns a new matrix where all rows of M became columns and vice versa.
    //  "M.times(b)" (where the post-multiplied b can be a scalar, a Vector4, or another Matrix) returns a
    //               new Matrix or Vector4 holding the product.
    //  "M.pre_multiply(b)"  overwrites the Matrix M with the product of b * M where b must be another Matrix.
    //  "M.post_multiply(b)" overwrites the Matrix M with the product of M * b where b can be a Matrix or scalar.
    //  "Matrix.flatten_2D_to_1D( M )" flattens input (a Matrix or any array of Vectors or float arrays)
    //                                 into a row-major 1D array of raw floats.
    //  "M.to_string()" where M contains the 4x4 identity returns "[[1, 0, 0, 0] [0, 1, 0, 0] [0, 0, 1, 0] [0, 0, 0, 1]]".

    constructor(...args) {
      super(0);
      this.push(...args)
    }

    static flatten_2D_to_1D(M) {
      let index = 0, floats = new Float32Array(M.length && M.length * M[0].length);
      for (let i = 0; i < M.length; i++) for (let j = 0; j < M[i].length; j++) floats[index++] = M[i][j];
      return floats;
    }

    set(M) {
      this.length = 0;
      this.push(...M);
    }

    set_identity(m, n) {
      this.length = 0;
      for (let i = 0; i < m; i++) {
        this.push(Array(n).fill(0));
        if (i < n) this[i][i] = 1;
      }
    }

    sub_block(start, end) {
      return Matrix.from(this.slice(start[0], end[0]).map(r => r.slice(start[1], end[1])));
    }

    copy() {
      return this.map(r => [...r])
    }

    equals(b) {
      return this.every((r, i) => r.every((x, j) => x == b[i][j]))
    }

    plus(b) {
      return this.map((r, i) => r.map((x, j) => x + b[i][j]))
    }

    minus(b) {
      return this.map((r, i) => r.map((x, j) => x - b[i][j]))
    }

    transposed() {
      return this.map((r, i) => r.map((x, j) => this[j][i]))
    }

    times(b, optional_preallocated_result) {
      const len = b.length;
      if (typeof len === "undefined") return this.map(r => r.map(x => b * x));
      // Matrix * scalar case.
      const len2 = b[0].length;
      if (typeof len2 === "undefined") {
        let result = optional_preallocated_result || new Vector4(this.length);
        // Matrix * Vector4 case.
        for (let r = 0; r < len; r++) result[r] = b.dot(this[r]);
        return result;
      }
      let result = optional_preallocated_result || Matrix.from(new Array(this.length));
      for (let r = 0; r < this.length; r++) {
        // Matrix * Matrix case.
        if (!optional_preallocated_result)
          result[r] = new Array(len2);
        for (let c = 0, sum = 0; c < len2; c++) {
          result[r][c] = 0;
          for (let r2 = 0; r2 < len; r2++)
            result[r][c] += this[r][r2] * b[r2][c];
        }
      }
      return result;
    }

    pre_multiply(b) {
      const new_value = b.times(this);
      this.length = 0;
      this.push(...new_value);
      return this;
    }

    post_multiply(b) {
      const new_value = this.times(b);
      this.length = 0;
      this.push(...new_value);
      return this;
    }

    to_string() {
      return "[" + this.map((r, i) => "[" + r.join(", ") + "]").join(" ") + "]"
    }
  }


const Mat4 = tiny.Mat4 =
  class Mat4 extends Matrix {
    // **Mat4** generates special 4x4 matrices that are useful for graphics.
    // All the methods below return a certain 4x4 matrix.
    static identity() {
      return Matrix.of([1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]);
    };

    static rotation(angle, x, y, z) {
      // rotation(): Requires a scalar (angle) and a three-component axis vector.
      const normalize = (x, y, z) => {
        const n = Math.sqrt(x * x + y * y + z * z);
        return [x / n, y / n, z / n]
      }
      let [i, j, k] = normalize(x, y, z),
        [c, s] = [Math.cos(angle), Math.sin(angle)],
        omc = 1.0 - c;
      return Matrix.of([i * i * omc + c, i * j * omc - k * s, i * k * omc + j * s, 0],
        [i * j * omc + k * s, j * j * omc + c, j * k * omc - i * s, 0],
        [i * k * omc - j * s, j * k * omc + i * s, k * k * omc + c, 0],
        [0, 0, 0, 1]);
    }

    static scale(x, y, z) {
      // scale(): Builds and returns a scale matrix using x,y,z.
      return Matrix.of([x, 0, 0, 0],
        [0, y, 0, 0],
        [0, 0, z, 0],
        [0, 0, 0, 1]);
    }

    static translation(x, y, z) {
      // translation(): Builds and returns a translation matrix using x,y,z.
      return Matrix.of([1, 0, 0, x],
        [0, 1, 0, y],
        [0, 0, 1, z],
        [0, 0, 0, 1]);
    }

    static look_at(eye, at, up) {
      // look_at():  Produce a traditional graphics camera "lookat" matrix.
      // Each input must be a 3x1 Vector.
      // Note:  look_at() assumes the result will be used for a camera and stores its
      // result in inverse space.
      // If you want to use look_at to point a non-camera towards something, you can
      // do so, but to generate the correct basis you must re-invert its result.

      // Compute vectors along the requested coordinate axes. "y" is the "updated" and orthogonalized local y axis.
      let z = at.minus(eye).normalized(),
        x = z.cross(up).normalized(),
        y = x.cross(z).normalized();

      // Check for NaN, indicating a degenerate cross product, which
      // happens if eye == at, or if at minus eye is parallel to up.
      if (!x.every(i => i == i))
        throw "Two parallel vectors were given";
      z.scale_by(-1);                               // Enforce right-handed coordinate system.
      return Mat4.translation(-x.dot(eye), -y.dot(eye), -z.dot(eye))
        .times(Matrix.of(x.to4(0), y.to4(0), z.to4(0), vec4(0, 0, 0, 1)));
    }

    static orthographic(left, right, bottom, top, near, far) {
      // orthographic(): Box-shaped view volume for projection.
      return Mat4.scale(vec3(1 / (right - left), 1 / (top - bottom), 1 / (far - near)))
        .times(Mat4.translation(vec3(-left - right, -top - bottom, -near - far)))
        .times(Mat4.scale(vec3(2, 2, -2)));
    }

    static perspective(fov_y, aspect, near, far) {
      // perspective(): Frustum-shaped view volume for projection.
      const f = 1 / Math.tan(fov_y / 2), d = far - near;
      return Matrix.of([f / aspect, 0, 0, 0],
        [0, f, 0, 0],
        [0, 0, -(near + far) / d, -2 * near * far / d],
        [0, 0, -1, 0]);
    }

    static inverse(m) {
      // inverse(): A 4x4 inverse.  Computing it is slow because of
      // the amount of steps; call fewer times when possible.
      const result = Mat4.identity(), m00 = m[0][0], m01 = m[0][1], m02 = m[0][2], m03 = m[0][3],
        m10 = m[1][0], m11 = m[1][1], m12 = m[1][2], m13 = m[1][3],
        m20 = m[2][0], m21 = m[2][1], m22 = m[2][2], m23 = m[2][3],
        m30 = m[3][0], m31 = m[3][1], m32 = m[3][2], m33 = m[3][3];
      result[0][0] = m12 * m23 * m31 - m13 * m22 * m31 + m13 * m21 * m32 - m11 * m23 * m32 - m12 * m21 * m33 + m11 * m22 * m33;
      result[0][1] = m03 * m22 * m31 - m02 * m23 * m31 - m03 * m21 * m32 + m01 * m23 * m32 + m02 * m21 * m33 - m01 * m22 * m33;
      result[0][2] = m02 * m13 * m31 - m03 * m12 * m31 + m03 * m11 * m32 - m01 * m13 * m32 - m02 * m11 * m33 + m01 * m12 * m33;
      result[0][3] = m03 * m12 * m21 - m02 * m13 * m21 - m03 * m11 * m22 + m01 * m13 * m22 + m02 * m11 * m23 - m01 * m12 * m23;
      result[1][0] = m13 * m22 * m30 - m12 * m23 * m30 - m13 * m20 * m32 + m10 * m23 * m32 + m12 * m20 * m33 - m10 * m22 * m33;
      result[1][1] = m02 * m23 * m30 - m03 * m22 * m30 + m03 * m20 * m32 - m00 * m23 * m32 - m02 * m20 * m33 + m00 * m22 * m33;
      result[1][2] = m03 * m12 * m30 - m02 * m13 * m30 - m03 * m10 * m32 + m00 * m13 * m32 + m02 * m10 * m33 - m00 * m12 * m33;
      result[1][3] = m02 * m13 * m20 - m03 * m12 * m20 + m03 * m10 * m22 - m00 * m13 * m22 - m02 * m10 * m23 + m00 * m12 * m23;
      result[2][0] = m11 * m23 * m30 - m13 * m21 * m30 + m13 * m20 * m31 - m10 * m23 * m31 - m11 * m20 * m33 + m10 * m21 * m33;
      result[2][1] = m03 * m21 * m30 - m01 * m23 * m30 - m03 * m20 * m31 + m00 * m23 * m31 + m01 * m20 * m33 - m00 * m21 * m33;
      result[2][2] = m01 * m13 * m30 - m03 * m11 * m30 + m03 * m10 * m31 - m00 * m13 * m31 - m01 * m10 * m33 + m00 * m11 * m33;
      result[2][3] = m03 * m11 * m20 - m01 * m13 * m20 - m03 * m10 * m21 + m00 * m13 * m21 + m01 * m10 * m23 - m00 * m11 * m23;
      result[3][0] = m12 * m21 * m30 - m11 * m22 * m30 - m12 * m20 * m31 + m10 * m22 * m31 + m11 * m20 * m32 - m10 * m21 * m32;
      result[3][1] = m01 * m22 * m30 - m02 * m21 * m30 + m02 * m20 * m31 - m00 * m22 * m31 - m01 * m20 * m32 + m00 * m21 * m32;
      result[3][2] = m02 * m11 * m30 - m01 * m12 * m30 - m02 * m10 * m31 + m00 * m12 * m31 + m01 * m10 * m32 - m00 * m11 * m32;
      result[3][3] = m01 * m12 * m20 - m02 * m11 * m20 + m02 * m10 * m21 - m00 * m12 * m21 - m01 * m10 * m22 + m00 * m11 * m22;
      // Divide by determinant and return.
      return result.times(1 / (m00 * result[0][0] + m10 * result[0][1] + m20 * result[0][2] + m30 * result[0][3]));
    }
  }


const Keyboard_Manager = tiny.Keyboard_Manager =
  class Keyboard_Manager {
    // **Keyboard_Manager** maintains a running list of which keys are depressed.  You can map combinations of
    // shortcut keys to trigger callbacks you provide by calling add().  See add()'s arguments.  The shortcut
    // list is indexed by convenient strings showing each bound shortcut combination.  The constructor
    // optionally takes "target", which is the desired DOM element for keys to be pressed inside of, and
    // "callback_behavior", which will be called for every key action and allows extra behavior on each event
    // -- giving an opportunity to customize their bubbling, preventDefault, and more.  It defaults to no
    // additional behavior besides the callback itself on each assigned key action.
    constructor(target = document, callback_behavior = (callback, event) => callback(event)) {
      this.saved_controls = {};
      this.actively_pressed_keys = new Set();
      this.callback_behavior = callback_behavior;
      target.addEventListener("keydown", this.key_down_handler.bind(this));
      target.addEventListener("keyup", this.key_up_handler.bind(this));
      window.addEventListener("focus", () => this.actively_pressed_keys.clear());
      // Deal with stuck keys during focus change.
    }

    key_down_handler(event) {
      if (["INPUT", "TEXTAREA"].includes(event.target.tagName))
        return;
      // Don't interfere with typing.
      this.actively_pressed_keys.add(event.key);
      // Track the pressed key.
      for (let saved of Object.values(this.saved_controls)) {
        // Re-check all the keydown handlers.
        if (saved.shortcut_combination.every(s => this.actively_pressed_keys.has(s))
          && event.ctrlKey == saved.shortcut_combination.includes("Control")
          && event.shiftKey == saved.shortcut_combination.includes("Shift")
          && event.altKey == saved.shortcut_combination.includes("Alt")
          && event.metaKey == saved.shortcut_combination.includes("Meta"))
          // Modifiers must exactly match.
          this.callback_behavior(saved.callback, event);
        // The keys match, so fire the callback.
      }
    }

    key_up_handler(event) {
      const lower_symbols = "qwertyuiopasdfghjklzxcvbnm1234567890-=[]\\;',./",
        upper_symbols = "QWERTYUIOPASDFGHJKLZXCVBNM!@#$%^&*()_+{}|:\"<>?";

      const lifted_key_symbols = [event.key, upper_symbols[lower_symbols.indexOf(event.key)],
        lower_symbols[upper_symbols.indexOf(event.key)]];
      // Call keyup for any shortcuts
      for (let saved of Object.values(this.saved_controls))                          // that depended on the released
        if (lifted_key_symbols.some(s => saved.shortcut_combination.includes(s)))  // key or its shift-key counterparts.
          this.callback_behavior(saved.keyup_callback, event);                  // The keys match, so fire the callback.
      lifted_key_symbols.forEach(k => this.actively_pressed_keys.delete(k));
    }

    add(shortcut_combination, callback = () => {
    }, keyup_callback = () => {
    }) {                                 // add(): Creates a keyboard operation.  The argument shortcut_combination wants an
      // array of strings that follow standard KeyboardEvent key names. Both the keyup
      // and keydown callbacks for any key combo are optional.
      this.saved_controls[shortcut_combination.join('+')] = {shortcut_combination, callback, keyup_callback};
    }
  }


const Graphics_Card_Object = tiny.Graphics_Card_Object =
  class Graphics_Card_Object {
    // ** Graphics_Card_Object** Extending this base class allows an object to
    // copy itself onto a WebGL context on demand, whenever it is first used for
    // a GPU draw command on a context it hasn't seen before.
    constructor() {
      this.gpu_instances = new Map()
    }     // Track which GPU contexts this object has copied itself onto.
    copy_onto_graphics_card(context, intial_gpu_representation) {
      // copy_onto_graphics_card():  Our object might need to register to multiple
      // GPU contexts in the case of multiple drawing areas.  If this is a new GPU
      // context for this object, copy the object to the GPU.  Otherwise, this
      // object already has been copied over, so get a pointer to the existing
      // instance.  The instance consists of whatever GPU pointers are associated
      // with this object, as returned by the WebGL calls that copied it to the
      // GPU.  GPU-bound objects should override this function, which builds an
      // initial instance, so as to populate it with finished pointers.
      const existing_instance = this.gpu_instances.get(context);

      // Warn the user if they are avoidably making too many GPU objects.  Beginner
      // WebGL programs typically only need to call copy_onto_graphics_card once
      // per object; doing it more is expensive, so warn them with an "idiot
      // alarm". Don't trigger the idiot alarm if the user is correctly re-using
      // an existing GPU context and merely overwriting parts of itself.
      if (!existing_instance) {
        Graphics_Card_Object.idiot_alarm |= 0;     // Start a program-wide counter.
        if (Graphics_Card_Object.idiot_alarm++ > 200)
          throw `Error: You are sending a lot of object definitions to the GPU, probably by mistake!  
                    Many of them are likely duplicates, which you don't want since sending each one is very slow.  
                    To avoid this, from your display() function avoid ever declaring a Shape Shader or Texture (or 
                    subclass of these) with "new", thus causing the definition to be re-created and re-transmitted every
                    frame. Instead, call these in your scene's constructor and keep the result as a class member, 
                    or otherwise make sure it only happens once.  In the off chance that you have a somehow deformable 
                    shape that MUST change every frame, then at least use the special arguments of 
                    copy_onto_graphics_card to limit which buffers get overwritten every frame to only 
                    the necessary ones.`;
      }
      // Check if this object already exists on that GPU context.
      return existing_instance ||             // If necessary, start a new object associated with the context.
        this.gpu_instances.set(context, intial_gpu_representation).get(context);
    }

    activate(context, ...args) {                            // activate():  To use, super call it to retrieve a container of GPU
      // pointers associated with this object.  If none existed one will be created.
      // Then do any WebGL calls you need that require GPU pointers.
      return this.gpu_instances.get(context) || this.copy_onto_graphics_card(context, ...args)
    }
  }


const Vertex_Buffer = tiny.Vertex_Buffer =
  class Vertex_Buffer extends Graphics_Card_Object {
    // **Vertex_Buffer** organizes data related to one 3D shape and copies it into GPU memory.  That data
    // is broken down per vertex in the shape.  To use, make a subclass of it that overrides the
    // constructor and fills in the "arrays" property.  Within "arrays", you can make several fields that
    // you can look up in a vertex; for each field, a whole array will be made here of that data type and
    // it will be indexed per vertex.  Along with those lists is an additional array "indices" describing
    // how vertices are connected to each other into shape primitives.  Primitives could includes
    // triangles, expressed as triples of vertex indices.
    constructor(...array_names) {
      // This superclass constructor expects a list of names of arrays that you plan for.
      super();
      [this.arrays, this.indices] = [{}, []];
      // Initialize a blank array member of the Shape with each of the names provided:
      for (let name of array_names) this.arrays[name] = [];
    }

    copy_onto_graphics_card(context, selection_of_arrays = Object.keys(this.arrays), write_to_indices = true) {
      // copy_onto_graphics_card():  Called automatically as needed to load this vertex array set onto
      // one of your GPU contexts for its first time.  Send the completed vertex and index lists to
      // their own buffers within any of your existing graphics card contexts.  Optional arguments
      // allow calling this again to overwrite the GPU buffers related to this shape's arrays, or
      // subsets of them as needed (if only some fields of your shape have changed).

      // Define what this object should store in each new WebGL Context:
      const initial_gpu_representation = {webGL_buffer_pointers: {}};
      // Our object might need to register to multiple GPU contexts in the case of
      // multiple drawing areas.  If this is a new GPU context for this object,
      // copy the object to the GPU.  Otherwise, this object already has been
      // copied over, so get a pointer to the existing instance.
      const did_exist = this.gpu_instances.get(context);
      const gpu_instance = super.copy_onto_graphics_card(context, initial_gpu_representation);

      const gl = context;

      const write = did_exist ? (target, data) => gl.bufferSubData(target, 0, data)
        : (target, data) => gl.bufferData(target, data, gl.STATIC_DRAW);

      for (let name of selection_of_arrays) {
        if (!did_exist)
          gpu_instance.webGL_buffer_pointers[name] = gl.createBuffer();
        gl.bindBuffer(gl.ARRAY_BUFFER, gpu_instance.webGL_buffer_pointers[name]);
        write(gl.ARRAY_BUFFER, Matrix.flatten_2D_to_1D(this.arrays[name]));
      }
      if (this.indices.length && write_to_indices) {
        if (!did_exist)
          gpu_instance.index_buffer = gl.createBuffer();
        gl.bindBuffer(gl.ELEMENT_ARRAY_BUFFER, gpu_instance.index_buffer);
        write(gl.ELEMENT_ARRAY_BUFFER, new Uint32Array(this.indices));
      }
      return gpu_instance;
    }

    execute_shaders(gl, gpu_instance, type)     // execute_shaders(): Draws this shape's entire vertex buffer.
    {       // Draw shapes using indices if they exist.  Otherwise, assume the vertices are arranged as triples.
      if (this.indices.length) {
        gl.bindBuffer(gl.ELEMENT_ARRAY_BUFFER, gpu_instance.index_buffer);
        gl.drawElements(gl[type], this.indices.length, gl.UNSIGNED_INT, 0)
      } else gl.drawArrays(gl[type], 0, Object.values(this.arrays)[0].length);
    }

    draw(webgl_manager, program_state, model_transform, material, type = "TRIANGLES") {
      // draw():  To appear onscreen, a shape of any variety goes through this function,
      // which executes the shader programs.  The shaders draw the right shape due to
      // pre-selecting the correct buffer region in the GPU that holds that shape's data.
      const gpu_instance = this.activate(webgl_manager.context);
      material.shader.activate(webgl_manager.context, gpu_instance.webGL_buffer_pointers, program_state, model_transform, material);
      // Run the shaders to draw every triangle now:
      this.execute_shaders(webgl_manager.context, gpu_instance, type);
    }
  }


const Shape = tiny.Shape =
  class Shape extends Vertex_Buffer {
    // **Shape** extends Vertex_Buffer to give it an awareness that it holds data about 3D space.  This class
    // is used the same way as Vertex_Buffer, by subclassing it and writing a constructor that fills in the
    // "arrays" property with some custom arrays.

    // Shape extends Vertex_Buffer's functionality for copying shapes into buffers the graphics card's memory,
    // adding the basic assumption that each vertex will have a 3D position and a 3D normal vector as available
    // fields to look up.  This means there will be at least two arrays for the user to fill in:  "positions"
    // enumerating all the vertices' locations, and "normals" enumerating all vertices' normal vectors pointing
    // away from the surface.  Both are of type Vector3.

    // By including  these, Shape adds to class Vertex_Buffer the ability to compound shapes in together into a
    // single performance-friendly Vertex_Buffer, placing this shape into a larger one at a custom transforms by
    // adjusting positions and normals with a call to insert_transformed_copy_into().  Compared to Vertex_Buffer
    // we also gain the ability via flat-shading to compute normals from scratch for a shape that has none, and
    // the ability to eliminate inter-triangle sharing of vertices for any data we want to abruptly vary as we
    // cross over a triangle edge (such as texture images).

    // Like in class Vertex_Buffer we have an array "indices" to fill in as well, a list of index triples
    // defining which three vertices belong to each triangle.  Call new on a Shape and fill its arrays (probably
    // in an overridden constructor).

    // IMPORTANT: To use this class you must define all fields for every single vertex by filling in the arrays
    // of each field, so this includes positions, normals, any more fields a specific Shape subclass decides to
    // include per vertex, such as texture coordinates.  Be warned that leaving any empty elements in the lists
    // will result in an out of bounds GPU warning (and nothing drawn) whenever the "indices" list contains
    // references to that position in the lists.
    static insert_transformed_copy_into(recipient, args, points_transform = Mat4.identity()) {
      // insert_transformed_copy_into():  For building compound shapes.  A copy of this shape is made
      // and inserted into any recipient shape you pass in.  Compound shapes help reduce draw calls
      // and speed up performane.

      // Here if you try to bypass making a temporary shape and instead directly insert new data into
      // the recipient, you'll run into trouble when the recursion tree stops at different depths.
      const temp_shape = new this(...args);
      recipient.indices.push(...temp_shape.indices.map(i => i + recipient.arrays.position.length));
      // Copy each array from temp_shape into the recipient shape:
      for (let a in temp_shape.arrays) {
        // Apply points_transform to all points added during this call:
        if (a == "position" || a == "tangents")
          recipient.arrays[a].push(...temp_shape.arrays[a].map(p => points_transform.times(p.to4(1)).to3()));
        // Do the same for normals, but use the inverse transpose matrix as math requires:
        else if (a == "normal")
          recipient.arrays[a].push(...temp_shape.arrays[a].map(n =>
            Mat4.inverse(points_transform.transposed()).times(n.to4(1)).to3()));
        // All other arrays get copied in unmodified:
        else recipient.arrays[a].push(...temp_shape.arrays[a]);
      }
    }

    make_flat_shaded_version() {
      // make_flat_shaded_version(): Auto-generate a new class that re-uses any
      // Shape's points, but with new normals generated from flat shading.
      return class extends this.constructor {
        constructor(...args) {
          super(...args);
          this.duplicate_the_shared_vertices();
          this.flat_shade();
        }
      }
    }

    duplicate_the_shared_vertices() {
      // (Internal helper function)
      //  Prepare an indexed shape for flat shading if it is not ready -- that is, if there are any
      // edges where the same vertices are indexed by both the adjacent triangles, and those two
      // triangles are not co-planar.  The two would therefore fight over assigning different normal
      // vectors to the shared vertices.
      const arrays = {};
      for (let arr in this.arrays) arrays[arr] = [];
      for (let index of this.indices)
        for (let arr in this.arrays)
          arrays[arr].push(this.arrays[arr][index]);
      // Make re-arranged versions of each data field, with
      Object.assign(this.arrays, arrays);
      // copied values every time an index was formerly re-used.
      this.indices = this.indices.map((x, i) => i);
      // Without shared vertices, we can use sequential numbering.
    }

    flat_shade() {
      // (Internal helper function)
      // Automatically assign the correct normals to each triangular element to achieve flat shading.
      // Affect all recently added triangles (those past "offset" in the list).  Assumes that no
      // vertices are shared across seams.   First, iterate through the index or position triples:
      for (let counter = 0; counter < (this.indices ? this.indices.length : this.arrays.position.length); counter += 3) {
        const indices = this.indices.length ? [this.indices[counter], this.indices[counter + 1], this.indices[counter + 2]]
          : [counter, counter + 1, counter + 2];
        const [p1, p2, p3] = indices.map(i => this.arrays.position[i]);
        // Cross the two edge vectors of this triangle together to get its normal:
        const n1 = p1.minus(p2).cross(p3.minus(p1)).normalized();
        // Flip the normal if adding it to the triangle brings it closer to the origin:
        if (n1.times(.1).plus(p1).norm() < p1.norm()) n1.scale_by(-1);
        // Propagate this normal to the 3 vertices:
        for (let i of indices) this.arrays.normal[i] = Vector3.from(n1);
      }
    }

    normalize_positions(keep_aspect_ratios = true) {
      let p_arr = this.arrays.position;
      const average_position = p_arr.reduce((acc, p) => acc.plus(p.times(1 / p_arr.length)),
        vec3(0, 0, 0));
      p_arr = p_arr.map(p => p.minus(average_position));           // Center the point cloud on the origin.
      const average_lengths = p_arr.reduce((acc, p) =>
        acc.plus(p.map(x => Math.abs(x)).times(1 / p_arr.length)), vec3(0, 0, 0));
      if (keep_aspect_ratios) {
        // Divide each axis by its average distance from the origin.
        this.arrays.position = p_arr.map(p => p.map((x, i) => x / average_lengths[i]));
      } else
        this.arrays.position = p_arr.map(p => p.times(1 / average_lengths.norm()));
    }
  }


const Light = tiny.Light =
  class Light {
    // **Light** stores the properties of one light in a scene.  Contains a coordinate and a
    // color (each are 4x1 Vectors) as well as one size scalar.
    // The coordinate is homogeneous, and so is either a point or a vector.  Use w=0 for a
    // vector (directional) light, and w=1 for a point light / spotlight.
    // For spotlights, a light also needs a "size" factor for how quickly the brightness
    // should attenuate (reduce) as distance from the spotlight increases.
    constructor(position, color, size) {
      Object.assign(this, {position, color, attenuation: 1 / size});
    }
  }


const Graphics_Addresses = tiny.Graphics_Addresses =
  class Graphics_Addresses {
    // **Graphics_Addresses** is used internally in Shaders for organizing communication with the GPU.
    // Once we've compiled the Shader, we can query some things about the compiled program, such as
    // the memory addresses it will use for uniform variables, and the types and indices of its per-
    // vertex attributes.  We'll need those for building vertex buffers.
    constructor(program, gl) {
      const num_uniforms = gl.getProgramParameter(program, gl.ACTIVE_UNIFORMS);
      for (let i = 0; i < num_uniforms; ++i) {
        // Retrieve the GPU addresses of each uniform variable in the shader
        // based on their names, and store these pointers for later.
        let u = gl.getActiveUniform(program, i).name.split('[')[0];
        this[u] = gl.getUniformLocation(program, u);
      }

      this.shader_attributes = {};
      // Assume per-vertex attributes will each be a set of 1 to 4 floats:
      const type_to_size_mapping = {0x1406: 1, 0x8B50: 2, 0x8B51: 3, 0x8B52: 4};
      const numAttribs = gl.getProgramParameter(program, gl.ACTIVE_ATTRIBUTES);
      for (let i = 0; i < numAttribs; i++) {
        // https://github.com/greggman/twgl.js/blob/master/dist/twgl-full.js for another example:
        const attribInfo = gl.getActiveAttrib(program, i);
        // Pointers to all shader attribute variables:
        this.shader_attributes[attribInfo.name] = {
          index: gl.getAttribLocation(program, attribInfo.name),
          size: type_to_size_mapping[attribInfo.type],
          enabled: true, type: gl.FLOAT,
          normalized: false, stride: 0, pointer: 0
        };
      }
    }
  }


const Container = tiny.Container =
  class Container {
    // **Container** allows a way to create patch JavaScript objects within a single line.  Some properties get
    // replaced with substitutes that you provide, without having to write out a new object from scratch.
    // To override, simply pass in "replacement", a JS Object of keys/values you want to override, to generate
    // a new object.  For shorthand you can leave off the key and only provide a value (pass in directly as
    // "replacement") and a guess will be used for which member you want overridden based on type.
    override(replacement)
    // override(): Generate a copy by value, replacing certain properties.
    {
      return this.helper(replacement, Object.create(this.constructor.prototype))
    }

    replace(replacement) {
      // replace(): Like override, but modifies the original object.
      return this.helper(replacement, this)
    }

    helper(replacement, target) {
      // (Internal helper function)
      Object.assign(target, this);
      // If a JS object was given, use its entries to override:
      if (replacement.constructor === Object)
        return Object.assign(target, replacement);
      // Otherwise we'll try to guess the key to override by type:
      const matching_keys_by_type = Object.entries(this).filter(([key, value]) => replacement instanceof value.constructor);
      if (!matching_keys_by_type[0]) throw "Container: Can't figure out which value you're trying to replace; nothing matched by type.";
      return Object.assign(target, {[matching_keys_by_type[0][0]]: replacement});
    }
  }


const Material = tiny.Material =
  class Material extends Container {
    // **Material** contains messages for a shader program.  These configure the shader
    // for the particular color and style of one shape being drawn.  A material consists
    // of a pointer to the particular Shader it uses (to select that Shader for the draw
    // command), as well as a collection of any options wanted by the shader.
    constructor(shader, options) {
      super();
      Object.assign(this, {shader}, options);
    }
  }


const Shader = tiny.Shader =
  class Shader extends Graphics_Card_Object {
    // **Shader** loads a GLSL shader program onto your graphics card, starting from a JavaScript string.
    // To use it, make subclasses of Shader that define these strings of GLSL code.  The base class will
    // command the GPU to recieve, compile, and run these programs.  In WebGL 1, the shader runs once per
    // every shape that is drawn onscreen.

    // Extend the class and fill in the abstract functions, some of which define GLSL strings, and others
    // (update_GPU) which define the extra custom JavaScript code needed to populate your particular shader
    // program with all the data values it is expecting, such as matrices.  The shader pulls these values
    // from two places in your JavaScript:  A Material object, for values pertaining to the current shape
    // only, and a Program_State object, for values pertaining to your entire Scene or program.
    copy_onto_graphics_card(context) {
      // copy_onto_graphics_card():  Called automatically as needed to load the
      // shader program onto one of your GPU contexts for its first time.

      // Define what this object should store in each new WebGL Context:
      const initial_gpu_representation = {
        program: undefined, gpu_addresses: undefined,
        vertShdr: undefined, fragShdr: undefined
      };
      // Our object might need to register to multiple GPU contexts in the case of
      // multiple drawing areas.  If this is a new GPU context for this object,
      // copy the object to the GPU.  Otherwise, this object already has been
      // copied over, so get a pointer to the existing instance.
      const gpu_instance = super.copy_onto_graphics_card(context, initial_gpu_representation);

      const gl = context;
      const program = gpu_instance.program || context.createProgram();
      const vertShdr = gpu_instance.vertShdr || gl.createShader(gl.VERTEX_SHADER);
      const fragShdr = gpu_instance.fragShdr || gl.createShader(gl.FRAGMENT_SHADER);

      if (gpu_instance.vertShdr) gl.detachShader(program, vertShdr);
      if (gpu_instance.fragShdr) gl.detachShader(program, fragShdr);

      gl.shaderSource(vertShdr, this.vertex_glsl_code());
      gl.compileShader(vertShdr);
      if (!gl.getShaderParameter(vertShdr, gl.COMPILE_STATUS))
        throw "Vertex shader compile error: " + gl.getShaderInfoLog(vertShdr);

      gl.shaderSource(fragShdr, this.fragment_glsl_code());
      gl.compileShader(fragShdr);
      if (!gl.getShaderParameter(fragShdr, gl.COMPILE_STATUS))
        throw "Fragment shader compile error: " + gl.getShaderInfoLog(fragShdr);

      gl.attachShader(program, vertShdr);
      gl.attachShader(program, fragShdr);
      gl.linkProgram(program);
      if (!gl.getProgramParameter(program, gl.LINK_STATUS))
        throw "Shader linker error: " + gl.getProgramInfoLog(this.program);

      Object.assign(gpu_instance, {
        program,
        vertShdr,
        fragShdr,
        gpu_addresses: new Graphics_Addresses(program, gl)
      });
      return gpu_instance;
    }

    activate(context, buffer_pointers, program_state, model_transform, material) {
      // activate(): Selects this Shader in GPU memory so the next shape draws using it.
      const gpu_instance = super.activate(context);

      context.useProgram(gpu_instance.program);

      // --- Send over all the values needed by this particular shader to the GPU: ---
      this.update_GPU(context, gpu_instance.gpu_addresses, program_state, model_transform, material);

      // --- Turn on all the correct attributes and make sure they're pointing to the correct ranges in GPU memory. ---
      for (let [attr_name, attribute] of Object.entries(gpu_instance.gpu_addresses.shader_attributes)) {
        if (!attribute.enabled) {
          if (attribute.index >= 0) context.disableVertexAttribArray(attribute.index);
          continue;
        }
        context.enableVertexAttribArray(attribute.index);
        context.bindBuffer(context.ARRAY_BUFFER, buffer_pointers[attr_name]);
        // Activate the correct buffer.
        context.vertexAttribPointer(attribute.index, attribute.size, attribute.type,
          attribute.normalized, attribute.stride, attribute.pointer);
        // Populate each attribute
        // from the active buffer.
      }
    }

    // Your custom Shader has to override the following functions:
    vertex_glsl_code() {
    }

    fragment_glsl_code() {
    }

    update_GPU() {
    }

    // *** How those four functions work (and how GPU shader programs work in general):

    // vertex_glsl_code() and fragment_glsl_code() should each return strings that contain
    // code for a custom vertex shader and fragment shader, respectively.

    // The "Vertex Shader" is code that is sent to the graphics card at runtime, where on each
    // run it gets compiled and linked there.  Thereafter, all of your calls to draw shapes will
    // launch the vertex shader program, which runs every line of its code upon every vertex
    // stored in your buffer simultaneously (each instruction executes on every array index at
    // once).  Any GLSL "attribute" variables will appear to refer to some data field of just
    // one vertex, but really they affect all the stored vertices at once in parallel.

    // The purpose of this vertex shader program is to calculate the final resting place of
    // vertices in screen coordinates.  Each vertex starts out in local object coordinates
    // and then undergoes a matrix transform to land somewhere onscreen, or else misses the
    // drawing area and is clipped (cancelled).  One this has program has executed on your whole
    // set of vertices, groups of them (three if using triangles) are connected together into
    // primitives, and the set of pixels your primitive overlaps onscreen is determined.  This
    // launches an instance of the "Fragment Shader", starting the next phase of GPU drawing.

    // The "Fragment Shader" is more code that gets sent to the graphics card at runtime.  The
    // fragment shader runs after the vertex shader on a set of pixels (again, executing in
    // parallel on all pixels at once that were overlapped by a primitive).  This of course can
    // only happen once the final onscreen position of a primitive is known, which the vertex
    // shader found.

    // The fragment shader fills in (shades) every pixel (fragment) overlapping where the triangle
    // landed.  It retrieves different values (such as vectors) that are stored at three extreme
    // points of the triangle, and then interpolates the values weighted by the pixel's proximity
    // to each extreme point, using them in formulas to determine color.  GLSL variables of type
    // "varying" appear to have come from a single vertex, but are actually coming from all three,
    // and are computed for every pixel in parallel by interpolated between the different values of
    // the variable stored at the three vertices in this fashion.

    // The fragment colors may or may not become final pixel colors; there could already be other
    // triangles' fragments occupying the same pixels.  The Z-Buffer test is applied to see if the
    // new triangle is closer to the camera, and even if so, blending settings may interpolate some
    // of the old color into the result.  Finally, an image is displayed onscreen.

    // You must define an update_GPU() function that includes the extra custom JavaScript code
    // needed to populate your particular shader program with all the data values it is expecting.
  }


const Texture = tiny.Texture =
  class Texture extends Graphics_Card_Object {
    // **Texture** wraps a pointer to a new texture image where
    // it is stored in GPU memory, along with a new HTML image object.
    // This class initially copies the image to the GPU buffers,
    // optionally generating mip maps of it and storing them there too.
    constructor(filename, min_filter = "LINEAR_MIPMAP_LINEAR") {
      super();
      Object.assign(this, {filename, min_filter});
      // Create a new HTML Image object:
      this.image = new Image();
      this.image.onload = () => this.ready = true;
      this.image.crossOrigin = "Anonymous";           // Avoid a browser warning.
      this.image.src = filename;
    }

    copy_onto_graphics_card(context, need_initial_settings = true) {
      // copy_onto_graphics_card():  Called automatically as needed to load the
      // texture image onto one of your GPU contexts for its first time.

      // Define what this object should store in each new WebGL Context:
      const initial_gpu_representation = {texture_buffer_pointer: undefined};
      // Our object might need to register to multiple GPU contexts in the case of
      // multiple drawing areas.  If this is a new GPU context for this object,
      // copy the object to the GPU.  Otherwise, this object already has been
      // copied over, so get a pointer to the existing instance.
      const gpu_instance = super.copy_onto_graphics_card(context, initial_gpu_representation);

      if (!gpu_instance.texture_buffer_pointer) gpu_instance.texture_buffer_pointer = context.createTexture();

      const gl = context;
      gl.bindTexture(gl.TEXTURE_2D, gpu_instance.texture_buffer_pointer);

      if (need_initial_settings) {
        gl.pixelStorei(gl.UNPACK_FLIP_Y_WEBGL, true);
        gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MAG_FILTER, gl.LINEAR);
        // Always use bi-linear sampling when zoomed out.
        gl.texParameteri(gl.TEXTURE_2D, gl.TEXTURE_MIN_FILTER, gl[this.min_filter]);
        // Let the user to set the sampling method
        // when zoomed in.
      }

      gl.texImage2D(gl.TEXTURE_2D, 0, gl.RGBA, gl.RGBA, gl.UNSIGNED_BYTE, this.image);
      if (this.min_filter == "LINEAR_MIPMAP_LINEAR")
        gl.generateMipmap(gl.TEXTURE_2D);
      // If the user picked tri-linear sampling (the default) then generate
      // the necessary "mips" of the texture and store them on the GPU with it.
      return gpu_instance;
    }

    activate(context, texture_unit = 0) {
      // activate(): Selects this Texture in GPU memory so the next shape draws using it.
      // Optionally select a texture unit in case you're using a shader with many samplers.
      // Terminate draw requests until the image file is actually loaded over the network:
      if (!this.ready)
        return;
      const gpu_instance = super.activate(context);
      context.activeTexture(context["TEXTURE" + texture_unit]);
      context.bindTexture(context.TEXTURE_2D, gpu_instance.texture_buffer_pointer);
    }
  }


const Program_State = tiny.Program_State =
  class Program_State extends Container {
    // **Program_State** stores any values that affect how your whole scene is drawn,
    // such as its current lights and the camera position.  Class Shader uses whatever
    // values are wrapped here as inputs to your custom shader program.  Your Shader
    // subclass must override its method "update_GPU()" to define how to send your
    // Program_State's particular values over to your custom shader program.
    constructor(camera_transform = Mat4.identity(), projection_transform = Mat4.identity()) {
      super();
      this.set_camera(camera_transform);
      const defaults = {projection_transform, animate: true, animation_time: 0, animation_delta_time: 0};
      Object.assign(this, defaults);
    }

    set_camera(matrix) {
      // set_camera():  Applies a new (inverted) camera matrix to the Program_State.
      // It's often useful to cache both the camera matrix and its inverse.  Both are needed
      // often and matrix inversion is too slow to recompute needlessly.
      // Note that setting a camera matrix traditionally means storing the inverted version,
      // so that's the one this function expects to receive; it automatically sets the other.
      Object.assign(this, {camera_transform: Mat4.inverse(matrix), camera_inverse: matrix})
    }
  }


const Webgl_Manager = tiny.Webgl_Manager =
  class Webgl_Manager {
    // **Webgl_Manager** manages a whole graphics program for one on-page canvas, including its
    // textures, shapes, shaders, and scenes.  It requests a WebGL context and stores Scenes.
    constructor(canvas, background_color, dimensions) {
      const members = {
        instances: new Map(),
        scenes: [],
        prev_time: 0,
        canvas,
        scratchpad: {},
        program_state: new Program_State()
      };
      Object.assign(this, members);
      // Get the GPU ready, creating a new WebGL context for this canvas:
      for (let name of ["webgl", "experimental-webgl", "webkit-3d", "moz-webgl"]) {
        this.context = this.canvas.getContext(name);
        if (this.context) break;
      }
      if (!this.context) throw "Canvas failed to make a WebGL context.";
      const gl = this.context;

      this.set_size(dimensions);

      gl.clearColor.apply(gl, background_color);           // Tell the GPU which color to clear the canvas with each frame.
      gl.getExtension("OES_element_index_uint");           // Load an extension to allow shapes with more than 65535 vertices.
      gl.enable(gl.DEPTH_TEST);                            // Enable Z-Buffering test.
      // Specify an interpolation method for blending "transparent" triangles over the existing pixels:
      gl.enable(gl.BLEND);
      gl.blendFunc(gl.SRC_ALPHA, gl.ONE_MINUS_SRC_ALPHA);
      // Store a single red pixel, as a placeholder image to prevent a console warning:
      gl.bindTexture(gl.TEXTURE_2D, gl.createTexture());
      gl.texImage2D(gl.TEXTURE_2D, 0, gl.RGBA, 1, 1, 0, gl.RGBA, gl.UNSIGNED_BYTE, new Uint8Array([255, 0, 0, 255]));

      // Find the correct browser's version of requestAnimationFrame() needed for queue-ing up re-display events:
      window.requestAnimFrame = (w =>
        w.requestAnimationFrame || w.webkitRequestAnimationFrame
        || w.mozRequestAnimationFrame || w.oRequestAnimationFrame || w.msRequestAnimationFrame
        || function (callback, element) {
          w.setTimeout(callback, 1000 / 60);
        })(window);
    }

    set_size(dimensions = [1080, 600]) {
      // set_size():  Allows you to re-size the canvas anytime.  To work, it must change the
      // size in CSS, wait for style to re-flow, and then change the size again within canvas
      // attributes.  Both are needed because the attributes on a canvas ave a special effect
      // on buffers, separate from their style.
      const [width, height] = dimensions;
      this.canvas.style["width"] = width + "px";
      this.canvas.style["height"] = height + "px";
      Object.assign(this, {width, height});
      Object.assign(this.canvas, {width, height});
      // Build the canvas's matrix for converting -1 to 1 ranged coords (NCDS) into its own pixel coords:
      this.context.viewport(0, 0, width, height);
    }

    render(time = 0) {
      // render(): Draw a single frame of animation, using all loaded Scene objects.  Measure
      // how much real time has transpired in order to animate shapes' movements accordingly.
      this.program_state.animation_delta_time = time - this.prev_time;
      if (this.program_state.animate) this.program_state.animation_time += this.program_state.animation_delta_time;
      this.prev_time = time;

      const gl = this.context;
      gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT);
      // Clear the canvas's pixels and z-buffer.

      const open_list = [...this.scenes];
      while (open_list.length) {
        // Traverse all Scenes and their children, recursively.
        open_list.push(...open_list[0].children);
        // Call display() to draw each registered animation:
        open_list.shift().display(this, this.program_state);
      }
      // Now that this frame is drawn, request that render() happen
      // again as soon as all other web page events are processed:
      this.event = window.requestAnimFrame(this.render.bind(this));
    }
  }


const Scene = tiny.Scene =
  class Scene {
    // **Scene** is the base class for any scene part or code snippet that you can add to a
    // canvas.  Make your own subclass(es) of this and override their methods "display()"
    // and "make_control_panel()" to make them draw to a canvas, or generate custom control
    // buttons and readouts, respectively.  Scenes exist in a hierarchy; their child Scenes
    // can either contribute more drawn shapes or provide some additional tool to the end
    // user via drawing additional control panel buttons or live text readouts.
    constructor() {
      this.children = [];
      // Set up how we'll handle key presses for the scene's control panel:
      const callback_behavior = (callback, event) => {
        callback(event);
        event.preventDefault();    // Fire the callback and cancel any default browser shortcut that is an exact match.
        event.stopPropagation();   // Don't bubble the event to parent nodes; let child elements be targetted in isolation.
      }
      this.key_controls = new Keyboard_Manager(document, callback_behavior);
    }

    new_line(parent = this.control_panel) {
      // new_line():  Formats a scene's control panel with a new line break.
      parent.appendChild(document.createElement("br"))
    }

    live_string(callback, parent = this.control_panel) {
      // live_string(): Create an element somewhere in the control panel that
      // does reporting of the scene's values in real time.  The event loop
      // will constantly update all HTML elements made this way.
      parent.appendChild(Object.assign(document.createElement("div"), {
        className: "live_string",
        onload: callback
      }));
    }

    key_triggered_button(description, shortcut_combination, callback, color = '#6E6460',
                         release_event, recipient = this, parent = this.control_panel) {
      // key_triggered_button():  Trigger any scene behavior by assigning
      // a key shortcut and a labelled HTML button to fire any callback
      // function/method of a Scene.  Optional release callback as well.
      const button = parent.appendChild(document.createElement("button"));
      button.default_color = button.style.backgroundColor = color;
      const press = () => {
          Object.assign(button.style, {
            'background-color': color,
            'z-index': "1", 'transform': "scale(1.5)"
          });
          callback.call(recipient);
        },
        release = () => {
          Object.assign(button.style, {
            'background-color': button.default_color,
            'z-index': "0", 'transform': "scale(1)"
          });
          if (!release_event) return;
          release_event.call(recipient);
        };
      const key_name = shortcut_combination.join('+').split(" ").join("Space");
      if (key_name) {
        button.textContent = "(" + key_name + ") " + description;
      } else {
        button.textContent = description;
      }
      button.addEventListener("mousedown", press);
      button.addEventListener("mouseup", release);
      button.addEventListener("touchstart", press, {passive: true});
      button.addEventListener("touchend", release, {passive: true});
      if (!shortcut_combination) return;
      this.key_controls.add(shortcut_combination, press, release);
    }

    // To use class Scene, override at least one of the below functions,
    // which will be automatically called by other classes:
    display(context, program_state) {
    }                            // display(): Called by Webgl_Manager for drawing.
    make_control_panel() {
    }                            // make_control_panel(): Called by Controls_Widget for generating interactive UI.
    show_explanation(document_section) {
    }                            // show_explanation(): Called by Text_Widget for generating documentation.
  }