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Chapter 6 : Perspective and Animation
Study Questions and Projects
Introduction
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What characteristics of dimensions and viewing allow Dalí's three-dimensional model of a horse to seem at times to be a horse, or at least to have the same effect as a skillful two-dimensional representation?
Viewing in Perspective
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Why, if you hold your thumb up next to (from your point of view) a faraway person, does your thumb appear to be the same size as the person? Why might parallel lines appear to converge as they move farther away from the observer?
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What is the four-dimensional analogy to the distortions we see in the three-dimensional cube due to perspective? (Begin by thinking of the cube with one vanishing point -- seen as a "square within a square.")
Perspective Views of the Hypercube
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Why does one portion of the projection of the hypercube appear to be "inside" of another?
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How does central projection work?
Schlegel Diagrams of Polyhedra
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How are Schlegel diagrams generated?
Animating the Hypercube
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When a computer creates an animation of the hypercube, what is the minimum number of points whose movements must be determined?
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Why do we need to see an animation of the hypercube in order to appreciate its symmetries, which may not be immediately apparent from viewing only one projection?
The Polyhedral Torus in the Hypercube
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Why can we not simply fill in all of the squares in a hypercube?
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Why can the polyhedral torus be constructed in four-space, but not three-space?
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Why is the torus useful in filling in the hypercube?
Stereographic Projection
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How is stereographic projection performed?
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Why does this type of projection give a distorted map of the world, and what can be done to counter the original distortion? (What further distortions does this cause?)
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What is the image of a parallel of latitude in stereographic projection? Of a circle?