从2D转换矩阵中提取旋转比例值 [英] extract rotation, scale values from 2d transformation matrix

问题描述

如何从2d变换矩阵中提取旋转，缩放和平移值?我的意思是进行2D转换

how can i extract rotation, scale and translation values from 2d transformation matrix? i mean a have a 2d transformation

matrix = [1, 0, 0, 1, 0, 0]

matrix.rotate(45 / 180 * PI)
matrix.scale(3, 4)
matrix.translate(50, 100)
matrix.rotate(30 / 180 * PI)
matrix.scale(-2, 4)

现在我的矩阵具有[a，b，c，d，tx，ty]值

now my matrix have values [a, b, c, d, tx, ty]

让我们忘记上面的过程，并假设我们只有值a，b，c，d，tx，ty

lets forget about the processes above and imagine that we have only the values a, b, c, d, tx, ty

如何通过a，b，c，d，tx，ty查找总旋转和比例值

how can i find total rotation and scale values via a, b, c, d, tx, ty

对不起，我的英语水平

感谢您的前进

编辑

我认为这应该是某个地方的答案...

I think it should be an answer somewhere...

我刚刚在Flash Builder(AS3)中这样尝试过

i just tried in Flash Builder (AS3) like this

   var m:Matrix = new Matrix;
   m.rotate(.25 * Math.PI);
   m.scale(4, 5);
   m.translate(100, 50);
   m.rotate(.33 * Math.PI);
   m.scale(-3, 2.5);

   var shape:Shape = new Shape;
   shape.transform.matrix = m;

   trace(shape.x, shape.y, shape.scaleX, shape.scaleY, shape.rotation);

，输出为:

x = -23.6 
y = 278.8 
scaleX = 11.627334873920528 
scaleY = -13.54222263865791 
rotation = 65.56274134518259 (in degrees)

推荐答案

并非a，b，c，d，tx，ty的所有值都会产生有效的旋转序列.我假设上述值是2D中3x3均匀旋转矩阵的一部分

Not all values of a,b,c,d,tx,ty will yield a valid rotation sequence. I assume the above values are part of a 3x3 homogeneous rotation matrix in 2D

    | a  b  tx |
A = | c  d  ty |
    | 0  0  1  |

将坐标[x, y, 1]转换为:

[x', y', 1] = A * |x|
                  |y|
                  |z|

• 因此将转换设置为[dx, dy]=[tx, ty]
• 比例尺为sx = sqrt(a² + c²)和sy = sqrt(b² + d²)
• 旋转角度为t = atan(c/d)或t = atan(-b/a)，它们也应该相同.
• Thus set the traslation into [dx, dy]=[tx, ty]
• The scale is sx = sqrt(a² + c²) and sy = sqrt(b² + d²)
• The rotation angle is t = atan(c/d) or t = atan(-b/a) as also they should be the same.

否则，您将没有有效的旋转矩阵.

Otherwise you don't have a valid rotation matrix.

上述转换扩展为:

x' = tx + sx (x Cos θ - y Sin θ)
y' = ty + sy (x Sin θ + y Cos θ)

顺序为旋转，然后是比例尺，然后是平移.

when the order is rotation, followed by scale and then translation.

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