2-D Transformations

2-D Transformations Local/Modelling Coordinates y Object descriptions • Often defined in model coordinates • Must be mapped to world coordinates • Groups of objects are combined; complete image is formed by combining primitives x World Coordinates CS-321Dr. Mark L. Hornick

2-D Transformations Local/Modelling Coordinates y Problem statement: • Convert points from coordinates in one system to a second coordinate system x World Coordinates CS-321Dr. Mark L. Hornick

y2 x2 Combining Rotation and Translation • T(,p) can be expressed in terms of submatrices as • The inverse of T(,p) is given by y*2 x1 v1 v2  x*2 p y1 CS-321Dr. Mark L. Hornick

Scaling Scaling affects every coordinate in the shape; e.g. doubling each value when the scale factor=2 Sx and Sy usually have the same values; this Is called Uniform Scaling Before scaling CS-321Dr. Mark L. Hornick After scaling by 2

Combining Rotation, Translation and Scaling to convert from coordinates in x2y2 to x1y1 CS-321Dr. Mark L. Hornick

Inverse transformation: converting from coordinates in x1y1 to x2y2 Typically, you just take the inverse of TS, rather than inverting each component matrix. Here we’re just showing what the inverses of the component matrices are. CS-321Dr. Mark L. Hornick

2-D Transformations