R3 – Rotation – Hyperspherical Coordinates

This post covers the formula for hyperspherical coordinates in R3. Examples of rotation using the formula are given.

Hyperspherical Coordinates

Wikipedia defines the hyperspherical coordinate system as:

‘a coordinate system in an n-dimensional Euclidean space which is analogous to the spherical coordinate system defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate r, and n -1 angular coordinates φ₁, φ₂, … φ_n−1, where the angles φ₁, φ₂, … φ_n−2 range over [0,π] radians (or over [0,180] degrees) and φ_n−1 ranges over [0,2π) radians (or over [0,360) degrees). If x_i are the Cartesian coordinates, then we may compute x₁, … x_n from r, φ₁, … φ_n−1 with:

•             x₁ = rcos(φ₁)
•             x₂ = rsin(φ₁)*cos(φ₂)
•             x₃ = rsin(φ₁)*sin(φ₂)*cos(φ₃)

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•             x_n-1 = rsin(φ₁).  . . sin(φ_n-2)*cos(φ_n-1)
•             x_n = rsin(φ₁).  . . sin(φ_n-2)*sin(φ_n-1)’

Wave Numbers uses a similar definition. In R3 the hyperspherical coordinates are:

• x₁ = rcos(φ₁)*^{^}
• x₂ = rsin(φ₁)*cos(φ2)*i^{^}
• x₃ = rsin(φ₁)*sin(φ₂)*j^{^}

In R3, φ₁ represents the shortest angle around the sphere from the x-axis to the point. This angle is measured as ^{^} and is ≤ π^{^}. For example, φ₁ for the point (0, i^{^}, 0) is a π^{^}/2  rotation from the x-axis at (^{^}, 0, 0)  around the z-axis to (0, i^{^}, 0). The point and the origin form a line. So, examine the 2 angles between this line and the x-axis to find the shortest angle.

In R3, φ₂ represents the angle around the sphere from the y-axis to the point measured in a counterclockwise direction and is ≤ 2π^{^}. For example, φ₂ for the point (0, 0, j^v) is a 3π^{^}/2 counterclockwise rotation from the y-axis at (0, i^{^}, 0) to (0, 0, j^v) around the x-axis. The point and the origin form a line. So, measure the counterclockwise angle between the y-axis and this line to get φ_2.

Examples

Axes Points

Calculate the R3 hyperspherical coordinates of the following points on the x, y and z-axes using the radial coordinate – r, the angular coordinate φ₁ over [0, π^{^}] radians and the angular coordinate φ₂ over [0, 2π^{^}] radians.

• Take the case of the point at (1^{^}, 0, 0).
  • r = 1, φ1 = 0 as on the x-axis and φ2= π^{^}/2 around z-axis
  • x1 = r*cos(0)*1^{^} = 1*1*1^{^} = 1^{^}
  • x2 = r*sin(0)*cos(π^{^}/2)*i^{^} = 1*0*0*i^{^} = 0
  • x3 = r*sin(0)*sin(π^{^}/2)*j^{^} = 1*0*1*j^{^} = 0

• Take the case of the point at (0, i^{^}, 0)
  • r = 1, φ1 = π^{^}/2 around z-axis and φ2= 0 as on the y-axis
    • Note that φ1 = π^{^}/2 even though it is a clockwise rotation from (0, i^{^}, 0) to (1^{^}, 0, 0). The angle to the x-axis is measured as ^{^} independent of the direction of rotation.
  • x1 = r*cos(π^{^}/2)*1^{^} = 1*0*1^{^} = 0
  • x2 = r*sin(π^{^}/2)*cos(0)*i^{^} = 1*1*1*i^{^} = i^{^}
  • x3= r*sin(π^{^}/2)*sin(0)*j^{^} = 1*1*0*j^{^} = 0

• Take the case of the point at (0, 0, j^{^})
  • r = 1, φ1 = π^{^}/2 around y-axis and φ2= π^{^}/2 counterclockwise around x-axis from the y-axis to the point
  • x1 = r*cos(π^{^}/2)*1^{^} = 1*0*1^{^} = 0
  • x2 = r*sin(π^{^}/2)*cos(π^{^}/2)*i^{^} = 1*1*0*i^{^} = 0
  • x3 = r*sin(π^{^}/2)*sin(π^{^}/2)*j^{^} = 1*1*1*j^{^} = j^{^}

• Take the case of the point at (0, 0, j^v)
  • r = 1, φ1 = π^{^}/2 around the y-axis and φ2= 3π^{^}/2 counterclockwise around x-axis from the y-axis to the point
  • x1 = r*cos(π^{^}/2)*1^{^} = 1*0*1^{^} = 0
  • x2 = r*sin(π^{^}/2)*cos(3π^{^}/2)*i^{^} = 1*1*0*i^{^} = 0
  • x3 = r*sin(π^{^}/2)*sin(3π^{^}/2)*j^{^} = 1*1*^–1*j^{^} = j^v

Some Other Points

Calculate the R3 hyperspherical coordinates of some points not on the main axes.

• Take the case of a point with a radius of 1.73. The point and the origin form a line. The angle φ1 = 54.7^{^o} and is the angle from this line to the x-axis. The angle φ2 = 45^{^o} and is the angle from the y-axis to this line.
  • x1 = 1.73*cos(54.7^{^o})*1^{^} = 1.73*0.578*1^{^} = 1^{^}
  • x2 = 1.73*sin(54.7^{^o})*cos(45^{^o})*i^{^} = 1.73*0.816*.707*i^{^} = i^{^}
  • x3 = 1.73*sin(54.7^{^o})*sin(45^{^o})*j^{^} = 1.73*0.816*.707*j^{^} = j^{^}
  • Therefore the R3 hyperspherical coordinates of the point are (1^{^}, i^{^}, j^{^})

• Take the case of a point with a radius of 1.73. The point and the origin form a line. The angle φ1 = 125.26^{^o}  and is the angle from this line to the x-axis. The angle φ2 = 225^{^o} and is the angle from the y-axis to this line.
  • x1 = 1.73*cos(125.26^{^o})*1^{^} = 1.73*0.^–577*1^{^} = 1^v
  • x2 = 1.73*sin(125.26^{^o})*cos(225^{^o})*i^{^} = 1.73*0.817*.^–707*i^{^} = i^v
  • x3 = 1.73*sin(125.26^{^o})*sin(225^{^o})*j^{^} = 1.73*0.817*.^–707*j^{^} = j^v
  • Therefore the R3 hyperspherical coordinates of the point are (1^v, i^v, j^v)

• Take the case of a point with a radius of 2. The point and the origin form a line. The angle φ1 = 75.5^{^o}  is the angle from this line to the x-axis. The angle φ2 = 243.33^{^o} and is the angle from the y-axis to this line.
  • x1 = 2*cos(75.5^{^o})*1^{^} = 2*.25*1^{^} = 0.5^{^}
  • x2 = 2*sin(75.5^{^o})*cos(243.33^{^o})*i^{^} 
    • = 2*0.968*^–0.449*i^{^} =0.87i^v
  • x3 = 2*sin(75.5^{^o})*sin(243.33^{^o})*j^{^} 
    • = 2*0.968*^–0.894*j^{^} =1.73j^v
  • Therefore the R3 hyperspherical coordinates of the point are(0.5^{^}, 0.87i^v, 1.73j^v)

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