Scalaで四元数を扱うためのオブジェクト(Quaternionオブジェクト)及びクラス(Quaternionクラス)の実装例。

Quaternion.scala

package info.pandanote.test

import scala.reflect.ClassTag
import scala.math._

// A class to represent and to handle a quaternion.
// 四元数を扱うためのクラスを書いてみました。
case class Quaternion(real: Double, _i: Double, _j: Double, _k: Double) {
  override def toString(): String = real + (if (_i == 0.0) "" else ((if (_i>0.0)"+"else"-")+(if (_i>0.0)_i else -_i) + "i")) + (if (_j == 0.0) "" else ((if (_j>0.0)"+"else"-")+(if (_j>0.0)_j else -_j) + "j")) + (if (_k == 0.0) "" else ((if (_k>0.0)"+"else"-")+(if (_k>0.0)_k else -_k) + "k"))
  
  def re() = real
  def i() = _i
  def j() = _j
  def k() = _k
  
  def +(that: Quaternion) = Quaternion(this.real+that.real,this._i+that._i,this._j+that._j,this._k+that._k)
  def +(that: Int) = Quaternion(this.real+that,this._i,this._j,this._k)
  def +(that: Long) = Quaternion(this.real+that,this._i,this._j,this._k)
  def +(that: Float) = Quaternion(this.real+that,this._i,this._j,this._k)
  def +(that: Double) = Quaternion(this.real+that,this._i,this._j,this._k)
  def -(that: Quaternion) = Quaternion(this.real-that.real,this._i-that._i,this._j-that._j,this._k-that._k)
  def -(that: Int) = Quaternion(this.real-that,this._i,this._j,this._k)
  def -(that: Long) = Quaternion(this.real-that,this._i,this._j,this._k)
  def -(that: Float) = Quaternion(this.real-that,this._i,this._j,this._k)
  def -(that: Double) = Quaternion(this.real-that,this._i,this._j,this._k)
  def *(that: Quaternion) 
    = Quaternion(this.real*that.real-this._i*that._i-this._j*that._j-this._k*that._k,
                  this.real*that._i+this._i*that.real+this._j*that._k-this._k*that._j,
                  this.real*that._j-this._i*that._k+this._j*that.real+this._k*that._i,
                  this.real*that._k+this._i*that._j-this._j*that._i+this._k*that.real)
  def *(that: Int) = Quaternion(this.real*that,this._i*that,this._j*that,this._k*that)
  def *(that: Long) = Quaternion(this.real*that,this._i*that,this._j*that,this._k*that)
  def *(that: Float) = Quaternion(this.real*that,this._i*that,this._j*that,this._k*that)
  def *(that: Double) = Quaternion(this.real*that,this._i*that,this._j*that,this._k*that)
  
  def unary_- = Quaternion(-this.real,-this._i,-this._j,-this._k)
  def abs = math.sqrt(this.real*this.real+this._i*this._i+this._j*this._j+this._k*this._k)
  def conjugate = Quaternion(this.real,-this._i,-this._j,-this._k)

  // 除算: 四元数を除数とする除算は、逆数を右からかける演算として定義します。
  def /(that: Int) = Quaternion(this.real/that,this._i/that,this._j/that,this._k/that)
  def /(that: Long) = Quaternion(this.real/that,this._i/that,this._j/that,this._k/that)
  def /(that: Float) = Quaternion(this.real/that,this._i/that,this._j/that,this._k/that)
  def /(that: Double) = Quaternion(this.real/that,this._i/that,this._j/that,this._k/that)
  def /(that: Quaternion) = {
    val xnorm = that.real*that.real+that._i*that._i+that._j*that._j+that._k*that._k
    val q = this*that.conjugate
    Quaternion(q.real/xnorm,q._i/xnorm,q._j/xnorm,q._k/xnorm)
  }
  
  // 四元数の剰余計算というものが定義されているのかどうかわかりませんが、書いてみました。
  // 複素数と同様の考え方で定義しています。
  def %(that: Quaternion) = {
    val div = this./(that)
    this - Quaternion(Math.round(div.real),Math.round(div.i),Math.round(div.j),Math.round(div.k))*that
  }
  def %(that: Int): Quaternion = this.%(Quaternion(that, 0, 0, 0))
  def %(that: Long): Quaternion = %(Quaternion(that, 0, 0, 0))
  def %(that: Float): Quaternion = %(Quaternion(that, 0, 0, 0))
  def %(that: Double): Quaternion = %(Quaternion(that, 0, 0, 0))

  def exp = {
    val expreal = math.exp(real)
    val vlen = math.sqrt(_i*_i+_j*_j+_k*_k)
    Quaternion(math.cos(vlen),math.sin(vlen)*_i/vlen,math.sin(vlen)*_j/vlen,math.sin(vlen)*_k/vlen)*expreal
  }
  def log = {
    val logimag = math.acos(real/abs)/math.sqrt(_i*_i+_j*_j+_k*_k)
    Quaternion(math.log(abs),_i*logimag,_j*logimag,_k*logimag)
  }
  def pow(that: Int): Quaternion = pow(Quaternion(that,0,0,0))
  def pow(that: Double): Quaternion = pow(Quaternion(that,0,0,0))
  def pow(that: Quaternion): Quaternion = {
    if (that == Quaternion.zero) Quaternion.one
    else if (this == Quaternion.zero) {
      // 実数成分は正である場合に限り0を返す。
      if (that.real < 0.0 || that._i != 0.0 || that._j != 0.0 || that._k != 0.0) Quaternion.nan
      else Quaternion.zero
    } else {
      Quaternion.exp(log*that)
    }
  }
  
  // こんなのも定義してみます。
  def **(that: Quaternion): Quaternion = pow(that)
  def **(that: Double): Quaternion = pow(that)
  def **(that: Int): Quaternion = pow(that)
  
  override def equals(that: Any) = that match {
    case that: Quaternion => this.real == that.real && this._i == that._i && this._j == that._j && this._k == that._k
    case real: Double => this.real == real && this._i == 0 && this._j == 0 && this._k == 0
    case real: Int => this.real == real && this._i == 0 && this._j == 0 && this._k == 0
    case real: Short => this.real == real && this._i == 0 && this._j == 0 && this._k == 0
    case real: Long => this.real == real && this._i == 0 && this._j == 0 && this._k == 0
    case real: Float => this.real == real && this._i == 0 && this._j == 0 && this._k == 0
    case _ => false
  }
  
  // 試しにこんな演算子を定義してみます。
  def <>(that: Quaternion) =
      (this.real != that.real || this._i != that._i || this._j != that._j || this._k != that._k)

  override def hashCode() = real.## ^ _i.## ^ _j.## ^ _k.##
}

object Quaternion {
  outer =>
    val one = new Quaternion(1,0,0,0)
    val zero = new Quaternion(0,0,0,0)
    val nan = new Quaternion(Double.NaN,Double.NaN,Double.NaN,Double.NaN)
    // Javaでいうところのstaticなメソッドみたいなものです。
    def exp(that: Quaternion): Quaternion = that.exp
    def log(that: Quaternion): Quaternion = that.log

    // 実数に四元数を作用させる演算が定義されていなかったので、追加しました。
    // 試行錯誤の末、以下の一行になりました…
    implicit def fromDouble(d: Double) = new Quaternion(d,0,0,0)
}