N次元のいかつい極座標変換もメソッドチェーンでしゅっと実装できるお ref: https://qiita.com/piyo7/items/73023be87fef02b32132

file0.txt

\begin{align}
x &= r \cos(\phi) \\
y &= r \sin(\phi) \\
\end{align}

file1.txt

\begin{align}
x &= r \cos(\phi_1) \\
y &= r \sin(\phi_1) \cos(\phi_2) \\
z &= r \sin(\phi_1) \sin(\phi_2) \\
\end{align}

file2.txt

\begin{align}
x_1 &= r \cos(\phi_1) \\
x_2 &= r \sin(\phi_1) \cos(\phi_2) \\
x_3 &= r \sin(\phi_1) \sin(\phi_2) \cos(\phi_3) \\
&\vdots \\
x_{n-1} &= r \sin(\phi_1) \dots \sin(\phi_{n-2}) \cos(\phi_{n-1}) \\
x_n &= r \sin(\phi_1) \dots \sin(\phi_{n-2}) \sin(\phi_{n-1}) \\
\end{align}

file3.scala

phis.map(math.sin).scan(1.0)(_ * _)

file4.scala

phis.map(math.cos) :+ 1.0

file5.scala

def toEuclidean(r: Double, phis: Seq[Double]): Seq[Double] = {
  phis.map(math.sin).scan(1.0)(_ * _).
    zip(phis.map(math.cos) :+ 1.0).
    map { case (a, b) => r * a * b }
}

file6.txt

\begin{align}
r &= \sqrt{x_n^2 + x_{n-1}^2 + \dots + x_1^2} \\
\phi_1 &= \arccos \frac{x_1}{\sqrt{x_n^2 + x_{n-1}^2 + \dots + x_1^2}} \\
\phi_2 &= \arccos \frac{x_2}{\sqrt{x_n^2 + x_{n-1}^2 + \dots + x_2^2}} \\
&\vdots \\
\phi_{n-2} &= \arccos \frac{x_{n-2}}{\sqrt{x_n^2 + x_{n-1}^2 + x_{n-2}^2}} \\
\phi_{n-1} &=
\begin{cases}
\arccos \frac{x_{n-1}}{\sqrt{x_n^2 + x_{n-1}^2}} &x_n \geq 0\\
- \arccos \frac{x_{n-1}}{\sqrt{x_n^2 + x_{n-1}^2}} &x_n < 0 \\
\end{cases}
\end{align}

file7.scala

def toSpherical(xs: Seq[Double]): (Double, Seq[Double]) = {
  val rs = xs.scanRight(0.0) { case (x, sum) => x * x + sum }.map(math.sqrt)
  val phis = xs.init.zip(rs).zipWithIndex.
    map {
      case ((_, 0.0), _) =>
        0.0
      case ((a, b), i) if i == xs.size - 2 && xs.last < 0.0 =>
        -math.acos(a / b)
      case ((a, b), _) =>
        math.acos(a / b)
    }
  (rs.head, phis)
}