Computer Graphics The Geometry of 2 x 2 Matrices Since a 2 x 2 matrix corresponds uniquely to a linear transformation from R^{2} to R^{2}, we can think of a matrix as transforming a planar figure into a new planar figure. Example Consider the matrix
and the triangle with vertices (0,0), (12), (5,3). We have
It is a property of linear transformations that if the matrix is nonsingular, then line segments map onto line segments. Hence triangles map onto triangles. The picture below shows the original triangle.
Some Basic Transformations There are certain basic transformation that are building blocks for general transformations. Example Reflection With Respect to the x axis. To find the matrix for this transformation, we consider where the vectors e_{1} and e_{2} are mapped. The reflection with respect to the xaxis makes the ycoordinate negative and leaves the xcoordinate constant. We have L(1, 0) = (1, 0) L(0, 1) = (0, 1) These vectors are the column vectors for the matrix. We have
Example Reflection About the Line y = x We see that L(1,0) = (0,1) L(0,1) = (1,0) so that
Example Rotation About an Angle q The point (0,1) rotated about this angle is on the unit circle at radian angle q. The point (1,0) rotated about this angle is on the unit circle at radian angle p/2 + q. We have L(1,0) = (cos q, sin q) L(0,1) = (cos(p/2 + q), sin(p/2 + q) = (sin q, cos q) We have
Example Shear in the ydirection Another transformation that is common in computer graphics is a shear in the x or y direction. The picture below gives and example
The matrix that makes this happen is
for some constant k. You can find an interactive applets that lets you play with computer graphics and matrices at http://www.ltcconline.net/greenl/java/Other/MatrixAnimation/MatrixAnimation.html
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