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.
#Composition of transformations plus
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. Composite Transformations In geometry, a transformation is an operation that moves, flips, or changes a shape to create a new shape. In a composition, one transformation produces an image upon which the other transformation is then performed. Identify the two translations of triangle ABC. When two or more transformations are combined to form a new transformation, the result is called a composition of transformations, or a sequence of transformations. Clearly the blue equations aren't the same as the red equations. Practice Composite Transformations Look at the following diagram. 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. However, the effective translation amount is 20 and x and 12 and y. 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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