球面共面n点的3D无交多边形构建及边求解技术问询

Alright, let's tackle your problem head-on—since all your points lie on both a sphere and a single common plane, we can use that dual property to make this way simpler than it might sound at first.

1. First: Figure Out the Polygon Edges #

The key insight here is that any set of coplanar points on a sphere must all lie on the circle formed by the intersection of their common plane and the sphere. So to find non-intersecting edges, we just need to order these points along that circle, then connect adjacent points (including wrapping back from the last to the first). Here's how to do it step by step:

• Calculate the plane's normal vector: Grab three non-collinear points from your set (if all points are collinear, you can't form a polygon anyway). Let’s call them A, B, C. Compute the cross product (B - A) × (C - A) to get a normal vector for the plane, then normalize it to make it unit length.
• Project points to a 2D coordinate system: We need to flatten the 3D coplanar points into 2D to easily sort them by their position on the circle.
  1. Pick one of your points (say P₀) as the origin for this local 2D space.
  2. Create two orthogonal vectors that lie entirely within the plane: use u = normalize(P₁ - P₀) (where P₁ is another distinct point), then v = normalize(normal_vector × u) (this ensures v is perpendicular to u and stays in the plane).
  3. For every point Pᵢ in your set, compute its 2D coordinates as (dot(Pᵢ - P₀, u), dot(Pᵢ - P₀, v)).
• Sort points by polar angle: Calculate the polar angle (relative to your local 2D origin) for each projected 2D coordinate, then sort the original 3D points based on these angles (either clockwise or counterclockwise—your choice).
• Define edges: The edges of your polygon are the line segments connecting each consecutive pair of sorted points, plus a segment from the last point back to the first. This guarantees no intersecting edges because we're following the natural circular order of the points.

2. Second: Build the Non-Intersecting 3D Polygon #

Once you have your sorted list of vertices, constructing the polygon is straightforward:

• Simply connect each vertex to the next one in the sorted order with a straight line segment (all these segments will lie in the common plane of your points).
• Close the polygon by connecting the final vertex back to the first one.

Quick Notes to Avoid Pitfalls #

• Remove duplicate points first: If any of your input points are identical, they'll break the polygon (creating zero-length edges), so filter those out before starting.
• Handle collinear edge cases: If your initial three points are collinear, just pick another trio—since you’re working with coplanar points that form a valid polygon, there must be at least three non-collinear points in the set.
• Convexity guarantee: Since we’re connecting points in circular order, the resulting polygon will always be convex, which inherently means no intersecting edges.

内容的提问来源于stack exchange，提问作者George Valentin Datcu