圆锥顶部投影场景下已知点坐标求解未知点C的可行性及补充方案问询

Great question! Let's break this down clearly, starting with the original problem and then moving to your adjusted approach.

原问题的局限 #

As Gregor Thomas pointed out, you can't directly derive point C's coordinates from just points A and B in a cone-top projection scenario. Here's why: a cone-top (perspective) projection relies on a fixed vertex (the cone's tip) and a projection plane (like the cone's base). Without explicit constraints on how C relates to A, B, the projection vertex, or the plane, there's no unique mathematical relationship to solve for C—there are infinitely many possible points C that could fit vague positional assumptions.

调整后方案的可行性 #

Your adjusted approach does work, provided we clearly define the "equal-length projection" of B through A. Let's walk through the logic and concrete steps:

1. Define point D clearly #

First, we need a precise definition for "point D is the equal-length projection of B through A." A common and practical interpretation here is that D is a point such that:

• The projection follows the direction of the cone's projection rays (i.e., lines from the cone's vertex O to points in space)
• The length of segment AD equals the length of segment AB (or alternatively, A is the midpoint of BD—adjust based on your exact projection rule, as long as it's mathematically definable)

For example, if we take "equal-length projection" to mean A is the midpoint of BD, D's coordinates are straightforward to calculate using vector math:

xD = 2*xA - xB
yD = 2*yA - yB
zD = 2*zA - zB

2. Calculate projection E of D #

Next, find E, the projection of D onto your target plane (e.g., the cone's base) using the cone-top perspective projection rules. Let's assume the cone's vertex is at O(xO, yO, zO) and the projection plane is defined by a simple equation like z = 0 (adjust to your actual plane):

• The line from O to D can be parameterized as: (xO + t*(xD - xO), yO + t*(yD - yO), zO + t*(zD - zO))
• Solve for t when the line intersects the projection plane (e.g., set z = 0 and solve for t)
• Plug t back into the parameterized line to get E's coordinates (xE, yE, zE)

3. Compute midpoint C of EB #

Once you have E and B, calculating C (the midpoint of EB) is trivial with the midpoint formula:

xC = (xE + xB) / 2
yC = (yE + yB) / 2
zC = (zE + zB) / 2

Key Takeaway #

The critical difference between the original and adjusted problem is that your new condition adds a concrete, calculable constraint (the definition of D). This removes the ambiguity that made the original problem unsolvable, giving you a clear path to compute C's coordinates.

内容的提问来源于stack exchange，提问作者P. Vauclin