给定参数的球面补丁三维最小包围球计算问询

Alright, let's break down how to find the minimal bounding sphere for your spherical patch—this is a fun problem that combines spherical geometry with basic optimization!

1. First, List All Critical Points of the Patch #

Your spherical patch is defined by angular ranges, so its extreme points (the ones that will determine the bounding sphere) are its four corners, plus any poles the patch includes. Convert these from spherical coordinates (φ = polar angle from +z axis, θ = azimuthal angle in the xy-plane) to Cartesian coordinates using:
x = R*sinφ*cosθ, y = R*sinφ*sinθ, z = R*cosφ

• Corner 1: (R*sinφ*cosθ, R*sinφ*sinθ, R*cosφ)
• Corner 2: (R*sinφ*cos(θ+θ_len), R*sinφ*sin(θ+θ_len), R*cosφ)
• Corner 3: (R*sin(φ+φ_len)*cosθ, R*sin(φ+φ_len)*sinθ, R*cos(φ+φ_len))
• Corner 4: (R*sin(φ+φ_len)*cos(θ+θ_len), R*sin(φ+φ_len)*sin(θ+θ_len), R*cos(φ+φ_len))

Add poles if the patch covers them:

• If φ ≤ 0 ≤ φ+φ_len: add the north pole (0, 0, R)
• If φ ≤ π ≤ φ+φ_len: add the south pole (0, 0, -R)

2. Calculate the Patch's Central Direction #

Find the midpoint of the angular ranges to get the "center" of the patch (this is the direction from the origin that's most aligned with the patch):

• φ_mid = φ + φ_len/2
• θ_mid = θ + θ_len/2
• Convert to Cartesian coordinates: C₀ = (R*sinφ_mid*cosθ_mid, R*sinφ_mid*sinθ_mid, R*cosφ_mid)

3. Find the Farthest Point from the Patch's Center #

For each critical point P, calculate the cosine of the angle α between P and C₀ using the dot product. Since all points lie on the original sphere (radius R), this simplifies to:
cosα = (P.x*C₀.x + P.y*C₀.y + P.z*C₀.z) / R²

The smallest value of cosα across all critical points corresponds to the largest angular distance α_max between the patch's center direction and any point in the patch. This is the key value we need.

4. Compute the Minimal Bounding Sphere #

• Center (x, y, z): The bounding sphere's center lies along the line from the origin to C₀. Its distance from the origin is R*cosα_max, so:

  x = (C₀.x / R) * R*cosα_max = C₀.x * cosα_max
  y = C₀.y * cosα_max
  z = C₀.z * cosα_max
  

  (Dividing C₀ by R gives the unit vector in its direction, then multiplying by R*cosα_max gives the center position.)
• Radius r: The radius is the maximum distance from this center to any point in the patch, which simplifies to:
  r = R * sqrt(1 - cos²α_max)

5. Edge Case Checks #

• If the patch covers the entire sphere: α_max = π, so the center is (0,0,0) and r=R (trivially correct).
• If the patch is a tiny "pixel": α_max is very small, so the center is almost at C₀, and r≈R*α_max (matches the local planar patch approximation).
• If the patch includes a pole: double-check that the pole was included in your critical points—it might be the farthest point from C₀.

内容的提问来源于stack exchange，提问作者user3612643