如何围绕质心旋转/缩放/平移

Question

rotating
x' = x*Cos(angle) - y*Sin(angle)
y' = x*Sin(angle) + y*Cos(angle)
Scaling
x' = x*sx
y' = y*sy
translate
x' = x+tx
y' = y+ty

but all formulas will doing about origin point. If i want to do about Centroid point. (I have (Cx,Cy) ). What the formulas will be.

Sorry, about english, I will practice more.

Thanks.

Answer 1

Translate so that the centroid is the origin, rotate, translate back. That is, if you have coordinates for a point (x,y)=(Cx,Cy)+(xr,yr) where (x,y) is the point you want to rotate around the centroid, (Cx,Cy) is the coordinates of the centroid, and (xr,yr) is the position of the point relative to the centroid. Then you can rotate (xr,yr) and add it to Cx,Cy) .

Answer 2

Translate the object so that the centroid coincides with the origin, then perform whatever transformation, then translate it back.

Depending what you're using to implement geometry, you might be able to linearly compose these operations before performing the heavy lifting. Or your library might provide versions of the operations with an invariant point as an argument, for which you can specify the centroid.

But there's nothing special about transforming about the centroid as opposed to any other point.

Answer 3

That's what the translation is for. Translating moves the origin to a new point (more properly, it establishes a new coordinate system whose origin coincides with the point specified in the original coordinate system). In a typical graphics implementation using affine matrix transformations, this new origin will be the center for any rotations performed after it.

You can see this from the equations, if you compose them. Say we want to place a figure rotating around point (200, 200).

// translate to new origin
x' = x + 200
y' = y + 200
// rotate by 90 degrees
x'' = x'*cos(90) - y'*sin(90) 
    = x*cos(90) + 200*cos(90) - y*sin(90) - 200*sin(90)
    = x*cos(90) - y*sin(90) + 200
y'' = x'*sin(90) + y'*cos(90)
    = x*sin(90) + 200*sin(90) + y*cos(90) + 200*cos(90)
    = x*sin(90) + y*cos(90) + 200