Fig 1.
The proposed multiplication represents a sphere (reiϕejθ) as the multiplication of the unit circle (eiϕ) in the XY plane with the unit semicircle (ejθ) in the XZ plane and scaling by radius r, as shown in (24) and (27). Here we have considered fixed radius r, variable angles −π ≤ ϕ < π, and −π/2 ≤ θ ≤ π/2 to obtain sphere.
Fig 2.
The π-periodic functions cos π(θ) (top) and sin π(θ) (bottom) plotted for [−3π/2, 3π/2].
The function cos π(θ) is continuous and sin π(θ) is discontinuous at odd multiples of π/2, and both functions are differentiable on . Moreover, within the open interval (−π/2, π/2) of the principal range, both are continuous and differentiable.
Fig 3.
A point P in the considered spherical co-ordinate system, where radius (r), azimuth angle (ϕ rad), and elevation angle (θ rad) measured from XY plane are shown.
Fig 4.
Euler identities (a) exp(iϕ) for ϕ ∈ [0, 2π) with r = 1 and θ = 0, where a vector that represents a point traces full circle in the direction of arrow as ϕ increases, (b) exp(jθ) for θ ∈ [−π/2, π/2] with r = 1 and ϕ = π for X < 0 (blue semicircle), and ϕ = 0 for X > 0 (red semicircle), where a point moves in the direction of arrow in the right semicircle as θ increases with ϕ = 0, and similarly a point moves in the direction of arrow in left semicircle as θ increases with ϕ = π, (c) exp(jθ) for θ ∈ [−π/2, π/2] with r = 1 and ϕ = 3π/2 (blue semicircle) and ϕ = π/2 (red semicircle). The path traced by exp(jθ) follows a semicircular trajectory, as indicated by the blue and red semicircle in (b) and (c) for the appropriate values of ϕ.
Fig 5.
Plot illustrating the azimuth angle ϕ versus (ϕ mod 2π) in the range −π to π, and the elevation angle θ versus (θ mod π) in the range −π/2 to π/2.
The azimuth angle ϕ is considered 2π-periodic, and the elevation angle θ is considered π-periodic.
Fig 6.
A point in the considered spherical co-ordinate system where radius r = 1, azimuth angle ϕ = π/4 and elevation angle θ = π/6 are shown, and point −P with r = 1, ϕ = 5π/4 and elevation angle θ = 7π/6 = −π/6 are also shown.
Fig 7.
The Bloch sphere representation, where the azimuth angle ϕ ∈ [0, 2π) is measured from the positive X axis and the elevation angle θ ∈ [−π/2, π/2] is measured from the XY plane.
The positive Z axis corresponds to the state |0〉, the negative Z axis corresponds to the state |1〉, and any point on the Bloch sphere can be uniquely represented by a pair of angles (ϕ, θ).
Fig 8.
A point cloud image consist of a set of points in 3D coordinate system with geometric coordinates stored in N × 3 matrix (i) Top-row left image (5184×3) and its squared image on right. (ii) Bottom-row left image (20402 × 3) represents two spheres of radius r = 1 (outer sphere) and r = 1/2 (inner sphere) and its squared image on right where outer sphere radius remain same, and inner sphere radius changed to 1/4.
Fig 9.
A set of point cloud images: From the leftmost image (5184 × 3) of the top-row other images are obtained by clanging the phase angle θ ∈ [0, π] in step of π/24.
The right-most image of the bottom-row is the same as leftmost image of the top-row.
Fig 10.
A set of point cloud images of solid cube (132651 × 3): (a) From the leftmost image in the top row, other images are obtained by varying the phase angle ϕ within the range ϕ ∈ [0, 2π], with intervals set at 2π/24. The rightmost image of the bottom row is identical to the leftmost image of the top row. (b) The leftmost image of the top-row is varied by changing the phase angle θ within the range θ ∈ [0, π] in increments of π/24 to produce other images. The right-most image of the bottom-row is identical to the left-most image of the top-row.
Fig 11.
Spherical coordinates system representation by (3) where continuous increase in azimuth angle ϕ with period 2π depicts spin of Earth around Z-axis (north and south pole), and continuous increase in elevation angle θ with period π depicts flow of magnetic flux along longitudes with source at south pole (θ = −π/2) and sink at north pole (θ = π/2).
Fig 12.
A point P in the considered spherical co-ordinate system, where radius (r), azimuth angle (ϕ rad), and polar angle (θ rad) measured from Z-axis are shown.