Coordinate TransformationsDate: 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/ |
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