Robot Rossie moves within a square room ABCD. Rossie moves along straight segments, never leaving that...

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Robot Rossie moves within a square room ABCD. Rossie moves along straight segments, never leaving that room. When Rossie encounters a wall she stops, makes a right-angle turn (with direction cho- sen to face into the room), and continues in that new direction. If Rossie comes to one of the room's corners, she rotates through two right angles, and moves back along her previous path. Suppose Rossie starts at point P on AB and her path begins as a line segment of slope s. We hope to describe Rossie's path. For some values of P and s, Rossie's path will be a tilted rectangle with one vertex on each wall of the room. (Often, this inscribed rect- angle is itself a square.) In this case, Rossie repeatedly traces that stable rectangle. D A P C A P B Figure 1: Stable rectangle when s = 1. D C B Figure 2: Stable rectangle when 0 < s < 1. (a) Suppose s = 1 so that the path begins at a 45 degree angle. For every starting point P, show: Rossie's path is a stable rectangle. (If P is a corner point, the path degenerates to a line segment traced back and forth.) Now draw some examples with various P and s. Given P and s, does Rossie's path always converge to a stable rectangle? Robot Rossie moves within a square room ABCD. Rossie moves along straight segments, never leaving that room. When Rossie encounters a wall she stops, makes a right-angle turn (with direction cho- sen to face into the room), and continues in that new direction. If Rossie comes to one of the room's corners, she rotates through two right angles, and moves back along her previous path. Suppose Rossie starts at point P on AB and her path begins as a line segment of slope s. We hope to describe Rossie's path. For some values of P and s, Rossie's path will be a tilted rectangle with one vertex on each wall of the room. (Often, this inscribed rect- angle is itself a square.) In this case, Rossie repeatedly traces that stable rectangle. D A P C A P B Figure 1: Stable rectangle when s = 1. D C B Figure 2: Stable rectangle when 0 < s < 1. (a) Suppose s = 1 so that the path begins at a 45 degree angle. For every starting point P, show: Rossie's path is a stable rectangle. (If P is a corner point, the path degenerates to a line segment traced back and forth.) Now draw some examples with various P and s. Given P and s, does Rossie's path always converge to a stable rectangle?

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The problem outlines a scenario where the robot Rossie within a square room moves along straight lines and reacts to encountering walls and corners by ... View the full answer