Maths - Calculation of Matrix for 2D Rotation about a point

Result

First we state the result, then we will go on to derive and explain it.

xout
yout
1
=
r00 r01 x - r00*x - r01*y
r10 r11 y - r10*x - r11*y
0 0 1
xin
yin
1

where:

Or multiplying out the matrix and vector terms to give in ordinary equations:

xout = r00* xin + r01* yin + x - r00*x - r01*y
yout = r10* xin + r11* yin + y - r10*x - r11*y

Explanation

In order to calculate the rotation about any arbitrary point we need to calculate its new rotation and translation. In other words rotation about a point is an 'proper' isometry transformation' which means that it has a linear and a rotational component.

Assume we have a matrix [R0] which defines a rotation about the origin:

We now want to apply this same rotation but about an arbitrary point P:

As we can see its orientation is the same as if it had been rotated about the origin, but it has been translated to a different point on space by the rotation.

In order to prove this and to calculate the amount of linear translation we need to replace:

With the following 3 simpler transforms which, when done in order, are equivalent:

So if we are using the global frame-of-reference (as explained here) then,

[resulting transform] = [third transform] * [second transform] * [first transform]

[resulting transform] = [+Px,+Py] * [rotation] * [-Px,-Py]

Note for matrix algebra, the order of operations is important, so these translations do not cancel out.

So matrix representing rotation about a given point is:

[R] = [T]-1 * [R0] * [T]

where:

[T]-1 = inverse transform = translation of point to origin

1 0 x
0 1 y
0 0 1

[R0] = rotation about origin (if this is not clear see this discussion)

r00 = cos(θ) r01 = -sin(θ) 0
r10 = sin(θ) r11 = cos(θ) 0
0 0 1

[T] = translation of origin to point

1 0 -x
0 1 -y
0 0 1

when these matrices are multiplied this will give the following result for rotation about x,y:

1 0 x
0 1 y
0 0 1
*
r00 r01 0
r10 r11 0
0 0 1
*
1 0 -x
0 1 -y
0 0 1

multiply out second pair

1 0 x
0 1 y
0 0 1
*
r00 r01 -r00*x - r01*y
r10 r11 -r10*x - r11*y
0 0 1

multiply remaining pair:

r00 r01 x - r00*x - r01*y
r10 r11 y - r10*x - r11*y
0 0 1

So the rotational components are the same but the rotation moves the position of the centre.


Further Reading

You may be interested in other means to represent orientation and rotational quantities such as:

Or you may be interested in how these quantities are used to simulate physical objects:


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