Composition of transformations1/9/2024 ![]() ![]() In the next exercise, you'll be ask to verify that this general form for composite transformation consisting of scales and translations always holds, no matter how many scales and translations are combined, and no matter what the order. When two or more transformations are combined we call it a composite transformation. Where t x stands for the effective, or final translation amount in x, and t y is the effective translation amount in y. But in either case, we can write the results of combining scaling and translation in the form x two equals s times x zero plus t x, and y two equals s times y zero plus t y. Clearly the blue equations aren't the same as the red equations. ![]() So algebraically, x two equals x one plus five, which equals four times x zero plus five and y two equals y one plus three which equals four times y zero plus three. X one equals four times x 0, and y one equals four times y zero. For comparison, let's do the operations in the opposite order. Isometry: Another word for rigid transformation, a transformation that does not change the shape or size of a figure. Rigid: A transformation that preserves size and shape. However, the effective translation amount is 20 and x and 12 and y. Transformation: An operation that moves, flips, or changes a figure to create a new figure. So the effective scale factor is still four. Y two is equal to four times y zero plus three which equals four times y zero plus 12. X two equals four times x zero plus five which equals four times x zero plus four times five, which equals four times x zero plus 20. Substitute our expressions for x one and y one. Scaling says, x two equals four times x one, and y two equals four times y one. Where does x one y one go? Let's call the point it goes to, x two y two. Now, suppose we scale about the origin by a factor of four. That point goes to a point, x one, y one, given by, x one equals x zero plus five, y one equals y zero plus three. Pick a point, x zero y zero, in the image we're translating. Suppose we translate by an amount of five and x and three and y. ![]() Let's see if we can get a better understanding of what's going on using some algebra. (crunch) Earlier, we saw that translation and scaling don't commute. Did you get final approval from the director? Congratulations. ![]()
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