🔰四元數 ℍ

Quaternions: ring of real quaternions ╱🚧 under construction

數學 ⟩ 數系 ⟩ 四元數 ℍ

四元數 $(\mathbb{H},+,\cdot)$ 是在四維向量空間 $\mathbb{R}^4$ 上定義自己的乘法，並擁有以下性質：

• $\Bbb{i}^2=\Bbb{j}^2=\Bbb{k}^2= \Bbb{ijk}=-1$

其中： $\mathbb{1}=(1,0,0,0)$, $\mathbb{i}=(0,1,0,0)$, $\mathbb{j}=(0,0, 1,0)$, $\mathbb{k}=(0, 0, 0,1)$

🔴 主題

• 四元數表示法

• 共軛四元數

• 四元數長度

• 四元數運算

🖥️ 影片

Visualizing Quaternons

⭐️ 重點

通常省略四元數的乘法符號，以避免與向量空間的內積符號混淆。

四元數沒有「乘法交換律」❗️

四元數 $(\mathbb{H},+,\cdot)$

• 是一個 skew field。
• 可視為複數 $(\mathbb{C},+,\cdot)$ 的擴充。

四元數的優點就是：

• 可同時處理純量與向量。( 👉 四元數表示法 )
• 它的乘法可以同時處理內積與外積。( 👉 四元數旋轉 )

👥 相關

• 複數 ℂ

📗 參考

• wiki ⟩ Quaternions ⭐️
• MathWorld ⟩ Quaternion
• Math for 3D Games ⟩ 3.6 Quaternions

• Abstract and Linear Algebra (Burton) ⟩ 3.3 Integral Domain and Field, p. 210

• Quaternions for Computer Graphics (John Vince (John A.))
• 3D Game Engine Programming ⟩ Understanding Quaternions
• Quaternions (Advanced Methods in Computer Graphics) Part 1

💾 程式

• replit ⟩ Quaternions

// 2023.03.04 - 06:50 (+) Quaternion, n.asQuaternion
// 2023.03.01 - 13:30 (/) isEqualTo -> equal, Number.EPSILON -> 1e-10
//                    (-) areEqualNumbers()
//                    (+) .round()
// 2023.01.20 - 16:45 (+) deg
// 2023.01.20 - 11:47 (+) .clamp(), .isEqualTo (*first recorded)
// -----------------------------------------------------

const { Random } = require('./Random.js');
const { sqrt, abs } = Math;
const {log} = console;

// deg
const deg = Math.PI / 180;

// ⭐ Number+ext
// -----------------------------------------------------
//   (constants)
// - deg                // degree = PI/180
// -----------------------------------------------------
//   (quaternion)
// 🔸 n.asQuaternion
//
// -----------------------------------------------------
//   (integer)
// 🔸 n.isEven
// 🔸 n.isOdd
// 🔸 n.binary
// 🔸 n.octal
// 🔸 n.hex
// 🔹 n.toBinary({prefix:, pad: {length:, char:}})
// 🔹 n.toOctal({prefix:, pad: {length:, char:})
// 🔹 n.toHex({prefix:, pad: {length:, char:})
// -----------------------------------------------------
// 🔹 .clamp()
// 🔹 .equal()
// 🔹 .round()
//
Object.defineProperties(Number.prototype, {

    // ---------------
    // quaternion
    // ---------------

    // 🔸 n.asQuaternion
    asQuaternion: {
        get() { return new Quaternion(this) },
    },

    // ---------------
    // integer
    // ---------------

    // 🔸 n.isEven
    isEven: {
        get() { return this % 2 === 0 },
    },

    // 🔸 n.isOdd
    isOdd: {
        get() { return Math.abs(this % 2) === 1 },
    },

    // 🔸 n.binary
    binary: {
        get() { return this.toString(2) },
    },

    // 🔸 n.octal
    octal: {
        get() { return this.toString(8) },
    },

    // 🔸 n.hex
    hex: {
        get() { return this.toString(16).toUpperCase() },
    },

    // 🔹 n.toBinary({prefix:, pad: {length:, char:}})
    toBinary: {
        value: formatFunc('0b', 2),
    },

    // 🔹 n.toOctal({prefix:, pad: {length:, char:}})
    toOctal: {
        value: formatFunc('0o', 8),
    },

    // 🔹 n.toHex({prefix:, pad: {length:, char:}})
    toHex: {
        value: formatFunc('0x', 16),
    },

    // ---------------
    // range
    // ---------------

    // 🔹 .clamp(min, max)
    clamp: {
        value: function(min, max){
            if (min > max) [min, max] = [max, min];
            return Math.min(max, Math.max(min, this));
        },
    },

    // 🔹 .equal()
    equal: {
        value: function(y, {threshold=1e-10}={}){
            return abs(this - y) < threshold; 
        },
    },

    // 🔹 .round()
    round: {
        value: function(places){
            return +this.toFixed(places); 
        },
    },

});

// ⭐ helpers
// -----------------------------------------------------

// general format function
function formatFunc(pre, base) {
    return function format({
        prefix = pre,
        pad = { length: 0, char: '0' },
    } = {}) {
        // ⭐ `pad` might be overridden, better destructure again
        const { length = 0, char = '0' } = pad;
        return prefix + 
            this.toString(base).padStart(length, char).toUpperCase();
    }
}

// 2023.03.04 - 07:19 (+) .scalarPart, .vectorPart
// 2023.03.04 - 06:50 (•) -------- merged into "Number+ext" --------
// 2023.03.03 - 17:08 (+) .dot, .cross
// 2023.03.01 - 13:42 (+) .randomInt, .randomFloat, .toString
// 2023.02.24 - 15:33 (+) static members, .conjugate, .norm, .inverse, ...
// 2023.02.24 - 13:35 (•) first draft (by chatGPT)
// -------------------------------------------------

// ⭐ Quaternion
// --------------------------------------------------
// - Quaternion.zero, .one, .i, .j, .k
// - Quaternion.randomInt()
// - Quaternion.randomFloat()
// --------------------------------------------------
// - .scalarPart, vectorPart
// - .conjugate
// - .normSquare
// - .norm
// - .inverse
// --------------------------------------------------
// - .add()            p + q
// - .subtract()       p - q
// - .multiply()       pq
// - .divide()         p/q = p q^(-1)
// - .scale()          kp
// - .dot()            p • q
// - .cross()          p x q
// --------------------------------------------------
// - .distance()       |p-q|
// - .equal()          p == q
//
class Quaternion {

    // i, j, k
    static get zero() { return new Quaternion(0, 0, 0, 0) }
    static get one() { return new Quaternion(1, 0, 0, 0) }
    static get i() { return new Quaternion(0, 1, 0, 0) }
    static get j() { return new Quaternion(0, 0, 1, 0) }
    static get k() { return new Quaternion(0, 0, 0, 1) }

    // init
    constructor(a=0, b=0, c=0, d=0) {
        this.a = a;
        this.b = b;
        this.c = c;
        this.d = d;
    }

    // ----------------------
    //     static methods
    // ----------------------

    // Quaternion.randomInt()
    static randomInt(min, max) {
        return new Quaternion(
            Random.int(min, max),
            Random.int(min, max),
            Random.int(min, max),
            Random.int(min, max),
        )
    }

    // Quaternion.randomFloat()
    static randomFloat(min, max) {
        return new Quaternion(
            Random.float(min, max),
            Random.float(min, max),
            Random.float(min, max),
            Random.float(min, max),
        )
    }


    // ----------------------
    //     getters
    // ----------------------

    // scalarPart
    get scalarPart() {
        return this.a;
    }

    // vectorPart
    get vectorPart() {
        const { b, c, d } = this;
        return new Quaternion(0, b, c, d);
    }

    // conjugate
    get conjugate() {
        const { a, b, c, d } = this;
        return new Quaternion(a, -b, -c, -d);
    }

    // inverse (reciprocal)
    get inverse() {
        const n = this.normSquare;
        return this.conjugate.scale(1 / n);
    }

    // |p|^2 = p • p
    get normSquare() {
        return this.dot(this);
    }

    // norm
    get norm() {
        const { a, b, c, d } = this;
        return sqrt(this.normSquare);
    }

    // ---------------------------
    //   ⭐ + - * / operations 
    // ---------------------------

    add(q) {
        const { a, b, c, d } = this;
        return new Quaternion(
            a + q.a, b + q.b, c + q.c, d + q.d
        );
    }

    subtract(q) {
        const { a, b, c, d } = this;
        return new Quaternion(
            a - q.a, b - q.b, c - q.c, d - q.d
        );
    }

    multiply(q) {
        const { a, b, c, d } = this;
        return new Quaternion(
            a * q.a - b * q.b - c * q.c - d * q.d,
            a * q.b + b * q.a + c * q.d - d * q.c,
            a * q.c - b * q.d + c * q.a + d * q.b,
            a * q.d + b * q.c - c * q.b + d * q.a,
        );
    }

    // p/q = p q^(-1)
    // ❗ note: p q^(-1) !== q^(-1) p
    divide(q) {
        return this.multiply(q.inverse);
    }

    // kq
    scale(k) {
        const { a, b, c, d } = this;
        return new Quaternion(k * a, k * b, k * c, k * d);
    }

    // p • q
    dot(q) {
        const { a, b, c, d } = q;
        return this.a * a + this.b * b + this.c * c + this.d * d;
    }

    // p x q = (p • q) - bar(p) q
    cross(q) {
        return this.dot(q).asQuaternion.subtract(
            this.conjugate.multiply(q)
        );
    }

    // ---------------------------
    //   ⭐ helpers 
    // ---------------------------

    // distance
    distance(q) {
        return this.subtract(q).norm;
    }

    // equal
    equal(q, {threshold=1e-10}={}) {
        return this.distance(q).equal(0, {threshold});
    }

    // toString
    toString(places=3) {
        const { a, b, c, d } = this;
        return `(${a.round(places)},${b.round(places)},${c.round(places)},${d.round(places)})`;
    }
}


// ⭐ CommonJS export
module.exports = {
    Quaternion,
    deg,
};

// export {deg};    // ES module export