如何将平面上的3D点转换为UV坐标? [英] How to convert a 3D point on a plane to UV coordinates?

问题描述

我有一个3d点，由[x0, y0, z0]定义.

I have a 3d point, defined by [x0, y0, z0].

此点属于由[a, b, c, d]定义的平面.

This point belongs to a plane, defined by [a, b, c, d].

normal = [a, b, c]和ax + by + cz + d = 0

如何将3d点转换或映射到一对(u,v)坐标?

How can convert or map the 3d point to a pair of (u,v) coordinates ?

这一定很简单，但我无法弄清楚.

This must be something really simple, but I can't figure it out.

推荐答案

首先，您需要计算u和v向量. u和v应该与您的平面的法线正交，并且彼此正交.没有定义它们的独特方法，但是一种便捷的方法可能是这样的:

First of all, you need to compute your u and v vectors. u and v shall be orthogonal to the normal of your plane, and orthogonal to each other. There is no unique way to define them, but a convenient and fast way may be something like this:

n = [a, b, c] 
u = normalize([b, -a, 0]) // Assuming that a != 0 and b != 0, otherwise use c.
v  = cross(n, u) // If n was normalized, v is already normalized. Otherwise normalize it.

现在，一个简单的点积就可以了:

Now a simple dot product will do:

u_coord = dot(u,[x0 y0 z0])
v_coord = dot(v,[x0 y0 z0])

请注意，这假定u-v坐标的原点是世界原点(0,0,0).

Notice that this assumes that the origin of the u-v coordinates is the world origin (0,0,0).

这将起作用，即使您的矢量[x0 y0 z0]并不完全位于飞机上也是如此.如果是这种情况，它将仅将其投影到飞机上.     

This will work even if your vector [x0 y0 z0] does not lie exactly on the plane. If that is the case, it will just project it to the plane.      

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