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Rotation Matrix about a Point Other Than the Origin

Date: 12/18/2006 at 19:53:19
From: Jon
Subject: Rotation Matrices about points, not origin

I know that in order to rotate a point around the origin you use the 
rotation matrix

[cos x   -sin x]  
[sin x    cos x]  

I was wondering how you rotate a point around a point other than the 

I thought that you'd have to place a value in front of the sine and 
cosine (3sin x or 5cos x, for example), or add to the sine or cosine 
value (3+sin x or 7+cos x).  I don't know what to add to where, I get 
very inconsistent results.

Date: 12/19/2006 at 06:38:42
From: Doctor Schwa
Subject: Re: Rotation Matrices about points, not origin

Hi Jon,

I think the idea you're looking for is called "conjugation", which
means to do something, then do something else, then undo the first
thing.  For instance, to leave a room when the door is shut,
you open the door, leave the room, then un-open the door (that is,
close it).  Now you have magically gotten out of a room with a closed

Similarly, to rotate about a point that is not the origin, first
you move all the points so the center is the origin, use the usual
rotation matrix, and then move all the points back to where you found

For instance, if the center is (3,5) you first subtract (3,5) from all
the coordinates, then use your matrix, then add the (3,5) back to

Does that help clear things up?

- Doctor Schwa, The Math Forum 

Date: 12/19/2006 at 15:04:56
From: Jon
Subject: Thank you (Rotation Matrices about points, not origin)

You guys make it look so easy!  Thank you a TON for your help, it
makes sense now!
Associated Topics:
College Linear Algebra
High School Linear Algebra

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