Math 111 2016-17 Assignment 6 Due Thursday Dec 1, 2016 1. a) Find an affine transformation T which maps the black triangle V onto the grey triangle W with vertex vi mapping to vertex wi. 4 (b) Now imagine that each point x in the v-triangle moves in a straight line at constant speed (though the speed will be different for different points) so that it arrives at its image T(x) at time t = 1. Let Tt(x) be its position at time t (0t1). Find an expression for the transformation Tt. In particular, this will allow you to see that it will be affine for all t. 2 v1 v2 w2 0 v0 0 w0 w1 2 4 6 (c) Since the transformation Tt is affine for all t, the image of the black triangle at any time t will be a triangle, though its shape will change with time. In fact there will be some moment at which it will in fact be a line segment, that is, the three points Tt(vi) will be collinear. [Why is this?--it is because the sequence v0- v1- v2 runs clockwise around the black triangle but w0- w1- w2 runs counterclockwise around the grey triangle, so at some point the v2image will have to cross the line containing the v0-image and the v1image, and at that moment, the triangle will turn into a line segment.] At what time t does this occur? 2. Both circles at the right have radius 1 and the shaded circle is centered at the origin. At time t = 0 (seconds) the flower circle begins to roll (without slipping) around the outside of the shaded circle at constant speed, rotating counterclockwise so that it travels counterclockwise around the shaded circle. Its speed is such that it takes exactly 4 seconds to travel around the circumference and return to its starting position. (a) Construct a family of transformations Tt(x) (0 t 4) which map the starting small circle to its position at any time. (b) Check the correctness of your formula by using the geometry to track the point (2, 1) over the first second, and then 2 calculating T1 . 1 math 111 assignment 6 fall 2016 1