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Homogeneous coordinates were invented by August Ferdinand Möbius (1790-1868). Today this idea enjoys wide use in applications such as computer graphics. A nice way to think about homogeneous coordinates is as a 2-dimensional projection of a three-dimensional object into the z = 1 plane. The z = 1 plane image of the object captures how it looks to an observer from the origin as it moves in three-dimensional space. Using this way to think about things, plot a triangle in the plane z = 1 with vertices a = (1, 1, 1), b = (1, 3, 1), c = (2, 5, 1). This triangle can be thought of as in space beyond the z = 1 plane, and its image in the z = 1 plane as a projection of the real object. The problem is to find a matrix A3×3 that maps this triangle to a rotated, scaled, and translated triangle with vertices a' = (3, 5, 1), b' = (2, 1, 1), c' = (−1, 1, 1). Round your matrix coefficients appropriately. It is desirable to use a computer algebra system to solve the problem approximately, but show the setup details of your work and the premise of the solution. One can then imagine this rotated, scaled, and translated triangle as having moved in R3, in this case to a new position closer to the observer and hence resulting in its larger apparent size. |
