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Isometries: Traveling Triangles
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Points, lines and polygons are the building blocks out of which most computer animations are built. Thus, in order to make an animations, the place to start is figuring out how to animate a polygon, say a triangle.

Perhaps the easiest way of drawing a triangle is to start with the three corners or vertices, and connect them with line segments. Computers keep track of the location of points by recording their coordinates. Therefore, one might use something like the instructions on the left to tell a computer to draw the triangle on the right:
 Triangle instructions: draw line from (0,0) to (1,0) draw line from (1,0) to (1,1) draw line from (1,1) to (0,0)
So far so good, but how do we tell the computer to move the triangle? Well, we need to give the computer coordinates for three more points, and intruct it to erase the old triangle, and draw a new one at the new location. Sometimes, this isn't so complicated. For example, to move the triangle up and to the right, we might add 1 to the first coordinate and 2 to the second coordinate of each point:
 Shifted triangle instructions: draw line from (1,2) to (2,2) draw line from (2,2) to (2,3) draw line from (2,3) to (1,2)
But, if we want to do anything more complicated, like rotating our triangle, we are going to end up doing some math. If you just try to do it by hand, picking coordinates by trial and error, it will a) take forever, and b) come out bad. Here, for example, is a first attempt to do it by hand:

Next: Tricky Transformations in "Toe the Line"

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