Up: Math 53 Selected Course Notes

# Orthographic Views of the Cube:

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The initial frame of this movie shows stereographic projection of the cube looking directly at one of its faces, so we see a square within a square. As the movie progresses, the distance to the light source is increased toward infinity, and at the end, the light source is infinitely far away, so its rays are parallel. Thus the images of both the closest and farthest squares are the same size. We only see one square in the projection as both squares now overlap.
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This movie shows our (orthographic) view of a cube as we move from looking directly at a face to looking directly at an edge. At several points in the movie, we rotate the cube slightly so that you can see all of its faces. This is to help you understand the "sliding squares" image that you are viewing. The front face is colored blue and the back face red. In the orthographic projections, the two squares seem to slide past each other.
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This movie shows the same sequence of motions as the previous one, but this time the two side faces of the cube are colored. In the initial view, the sides are projected as lines (since they are parallel to the light source) so we don't see the color.
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This movie shows the cube rotating from a view that is directly at an edge to a view that is directly at a corner. Again, we tilt the cube slightly at several points in the movie so that you can follow the action a bit better. The final position is looking directly down the long diagonal of the cube.
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This movie shows the cube rotating from a viewpoint were we look directly at a face to where we look direclty at a corner, down the long diagonal of the cube. This corresponds to our earlier view of the cube as two squares moving apart. That original movie was a correct shadow (as the light source moves from directly above to a slanted direction), but it does not represent a view of the cube we could ever actually see, since our image is always perpendicular to the direction of sight. This movie shows a more correct view of the cube.
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This movie shows the same sequence of motions as the previous one, but with a different set of faces colored. We now color the left and right faces rather than the front and back. Notice that in the final view, these faces have the same shape as in the previous movie. In fact, in this view, all the faces of the cube have the same shape in the projection, so this is one of the most symetric views of the cube.
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This movie show the rotation from looking directly at a face, to looking direclty at an edge and then to looking direclty at a corner. Now that you have seen the other ones, this one should make more sense.

Up: Math 53 Selected Course Notes