Coordinate Transformations

Date: 06/25/2003 at 20:21:24
From: Dan
Subject: Rotate Point Around Logical Origin

What is the equation that can be used to determine the new point when 
you rotate a point some number of degrees around a false origin?

Example:

Point: 10,10
Rotation: 90 Degrees
Origin: 2,6

Date: 06/26/2003 at 14:06:18
From: Doctor Douglas
Subject: Re: Rotate Point Around Logical Origin

Hi Dan,

Thanks for writing to the Math Forum.

First I would change coordinates so that they are centered on the
new origin:

  (u,v) = (10 - 2,10 - 6) = (8,4)

Thus the point is 8 units east and 4 units north of the new origin.

Now we rotate this by using polar coordinates.  First convert the 
(u,v) ordered pair into polar (r,@) coordinates:

  r = sqrt(u^2 + v^2) = sqrt(64 + 16) = sqrt(80)
  @ = arctan(4/8) = 26.5651 deg

At this point we have not done the rotation, just expressed the
coordinates of the given point in different coordinates. The new 
rotated point has

  r' = sqrt(80)
  @' = 26.5651 deg + 90 deg
     = 116.5651 deg

So you can see that the rotation is easy to do in these polar 
coordinates.  Now we go back to (u,v) coordinates:

  u' = r' cos(@') = sqrt(80)*cos(116.5651 deg) = -4.000007 or -4
  v' = r' sin(@') = sqrt(80)*sin(116.5651 deg) = 7.999997 or 8

In terms of the original (x,y) coordinates, the new point is

  (x',y') = (2 + u',6 + v') = (-2,14)

In this procedure we use coordinate transformations to go to polar
coordinates where the rotation is "easy" (in fact nearly trivial)
to perform, then transform back to the original coordinates for the
final answer.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/