旋转矩阵给定角度和点在X，Y，Z [英] Rotation Matrix given angle and point in X,Y,Z

问题描述

我做的图像处理，我想旋转都基于一个角度，起源，和X，Y在XYZ空间像素和z坐标。

I am doing image manipulation and I want to rotate all of the pixels in xyz space based on an angle, the origin, and an x,y, and z coordinate.

我只需要设置正确的矩阵（4×4），然后我会好从那里。角度是在度，而不是弧度和将x，y，z的所有将要从-1到1（浮子）

I just need to setup the proper matrix (4x4) and then I will be good from there. The Angle is in degrees, not radians and the x,y,z are all going to be from -1 to 1 (floats)

编辑：

好吧，这里是code，我掀起了做有关的起源和X定义了一个给定行的旋转，Y，Z coorinate。

Ok, here is the code that I whipped up to do the rotation about a given line defined by the origin and an X, Y, Z coorinate.

        float ang = angD * (float)(Math.PI / 180);  // from degrees to radians, if needed
        //U = n*n(t) + cos(a)*(I-n*n(t)) + sin(a)*N(x).

        var u = MatrixDouble.Identity(4);  // 4x4 Identity Matrix
        u = u.Multiply(Math.Cos(ang));

        var n = new MatrixDouble(1, 4, new List<double> { x, y, z, 0 });
        var nt = n.Transpose();

        // This next part is the N(x) matrix.  The data is inputted in Column
        // first order and fills in the 4x4 matrix with the given 16 Doubles
        var nx = new MatrixDouble(4, 4, new List<double> { 0, z, -y, 0, -z, 0, x, 0, y, -x, 0, 0, 0, 0, 0, 1 });

        nx = nx.Multiply(Math.Sin(ang));

        var ret = nt.Multiply(n);
        ret[3, 3] = 1;

        u = u.Subtract(ret);

        u = ret.Add(u.Add(nx));

这是一个有点复杂，我用一个自定义的矩阵库，但没有在那里应该不会太难任何功能矩阵LIB实现。

It's a little complicated and I'm using a custom Matrix library, but nothing up there should be too hard to implement with any functioning Matrix lib.

呼，很多数学的！

推荐答案

完整的旋转矩阵推导，并在给定的<一个href=\"https://sites.google.com/site/glennmurray/Home/rotation-matrices-and-formulas\">https://sites.google.com/site/glennmurray/Home/rotation-matrices-and-formulas.

The complete rotation matrices are derived and given at https://sites.google.com/site/glennmurray/Home/rotation-matrices-and-formulas.

从纸：

5.2关于原点旋转简化矩阵

5.2 The simplified matrix for rotations about the origin

请注意这个假设（U，V，W）是旋转轴，美^ 2 + V ^ 2 + W ^ 2 = 1。

Note this assumes that (u, v, w) is a direction vector for the axis of rotation and that u^2 + v^2 + w^2 = 1.

如果您有要旋转点（X，Y，Z），那么我们可以得到的七个变量能够产生旋转点的功能：

If you have a point (x, y, z) that you want to rotate, then we can obtain a function of of seven variables that yields the rotated point:

的˚F的（X，Y，Z，U，V，W，THETA）=

f(x, y, z, u, v, w, theta) =

文中还包括了约一个任意轴的旋转矩阵和公式（不一定通过原点），爪哇$ C $提供Apache许可证下的c和一个Web应用程序，说明旋转的链接。

The paper also includes matrices and formulas for rotations about an arbitrary axis (not necessarily through the origin), Java code available under the Apache license, and a link to a web app that illustrates rotations.

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