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Rotations

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Calculator: TI-83 Plus.

A matrix [B] = [[cos(A), -sin(A)][sin(A), cos(A)]] represents a counterclockwise rotation by angle A, around a center of coordinates on a plane.  Remember that if A is negative, the direction is actually clockwise (opposite to counterclockwise).

Here is a program that rotates a 2-dimensional picture counterclockwise by angle A around the origin.  It uses (calls) the program DRAW, which is in the unit Pictures. You need to prepare matrices [I] and [J] beforehand.  [I] holds the coordinates of the points in your drawing, and [J] tells which points are connected.  The program keeps all the pictures, overlaying the next one on the previous ones.  (I named it ROTATETW (for “rotate in 2 dimensions”), but you may choose your own name!)

Drawing Program	Explanation
PROGRAM:ROTATETW
:PrgmDRAW	Draws picture in original orientation
:Prompt A :While 1 :[[cos(A), -sin(A)][sin(A),cos(A)]]→[B]	User enters angle A in degrees
:[B]*[I]->[I]	Rotate matrix [I] counterclockwise by angle A
:PrgmDRAW	Draw the new picture
:Pause	Press enter to get the next picture
:End

How would you change the program if you wanted to erase the previous picture before drawing the next one?

The question is: Given a 2 by 2 matrix [B], does it represent a rotation?  If so, how to find its angle A?

The answer to the first question.

If                      det([B]) ≠ 1, then [B] is not a rotation,

else if                [B]*[B]^T is the 2 by 2 identity matrix [[1,0][0,1]] then [B] is a rotation, 

else                  [B] is not a rotation.

The answer to the second question.

If [B] is a rotation, compute, cos^-1([B](1,1)), (it is always positive); then if [B](2,1) is negative then A = -cos^-1([B](1,1)), else A = cos^-1([B](1,1))

Task.

Students work in pairs using one calculator. One student prepares a matrix which either is or is not a rotation, and the other checks it (and finds A if appropriate).

(Set: MODE to Float 2.)

Example.

First student:

[[cos(27),-sin(27)][sin(27),cos(27)]]→[B]

        [[.89   -.45]

         [.45   .89 ]]

CLEAR

Second student:

det([B])

                1.00            So far it is okay.

[B]*[B]^T

        [[1     0]

         [0     1]]             It is a rotation.

cos^-1([B](1,1))

                27.00

[B](2,1)

                .45             it is positive.

The angle is A = 27 degrees.

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Webpage and Flash Animation Developed by Aous Manshad
Last Modified: September 22, 2005