我想我的项目多边形沿着矢量到平面的三维空间。我想preferably使用一个转换矩阵要做到这一点，但我不知道如何建立这样的一个矩阵。

I want to project my Polygon along a vector to a plane in 3d Space. I would preferably use a single transformation matrix to do this, but I don't know how to build a matrix of this kind.

由于的

目标的 其执行所需的投影-a 4×4变换矩阵，

goal -a 4x4 transformation matrix which performs the required projection,

或

更新

感谢您的回答，它可以作为预期。

Thank you for the answer, it works as intended.

谨慎，以谁发现这个人的一句话：如果投影的正常的平面平行于投影矢量，分母Ð会变成（几乎）0，所以要避免发生奇怪的事情，某种处理的需要这种特殊情况。我通过检查解决它，如果D＆LT; 1E-5，如果是的话，只是翻译我多边形沿着费尽挤出矢量

A word of caution to the people who found this: If the Plane of projection's normal is parallel to the projection vector, the Denominator D will become (almost) 0, so to avoid strange things from happening, some kind of handling for this special case is needed. I solved it by checking if D < 1e-5, and if so, just translate my polygon along hte extrusion vector.

推荐答案

假设多边形的顶点之一（X0，Y0，Z0）和方向向量（DX，DY，DZ）。

Suppose one of the polygon's vertices is (x0, y0, z0), and the direction vector is (dx,dy,dz).

在投影线上的点是：（X，Y，Z）=（X0 +​​ T * DX，Y0 + T * DY，Z0 + T * DZ）。

您想找到这条线与面的交点，所以将其插入平面方程由+ CZ + D = 0 AX + 并解决在t：

You want to find the intersection of this line with the plane, so plug it into the plane equation ax+by+cz+d = 0 and solve for t:

t = (-a*x0 - b*y0 - c*z0 - d) / (a*dx + b*dy + c*dz)

然后，你有你的目标顶点： X = X0 + DX * T 等

由于这是一个仿射变换，它可以由一个4×4矩阵执行。你应该能够通过写三个方程的x，y和z为X0，Y0，Z0的函数和服用的系数来确定矩阵元素

Since this is an affine transformation, it can be performed by a 4x4 matrix. You should be able to determine the matrix elements by writing the three equations for x,y,z as a function of x0,y0,z0 and taking the coefficients.

例如，对于x：

x = x0 - (a*dx*x0 + b*dx*y0 + c*dx*z0 + d*dx) / D
x = (1 - a*dx/D)*x0 - (b*dx/D)*y0 - (c*dx/D)*z0 - d*dx/D

其中， D = A * DX + B * DY + C * DZ 从上面的分母。 y和z的工作类似。

Where D = a*dx + b*dy + c*dz is the denominator from above. y and z work similarly.

结果矩阵：

1-a*dx/D    -b*dx/D    -c*dx/D   -d*dx/D
 -a*dy/D   1-b*dy/D    -c*dy/D   -d*dy/D
 -a*dz/D    -b*dz/D   1-c*dz/D   -d*dz/D
    0          0          0         1

（注：在Direct3D的这个矩阵应该调换，因为它使用的列向量的行向量代替）

(Note: On Direct3D this matrix should be transposed, because it uses row vectors instead of column vectors).