Points are represented as column vectors throughout.
Calculators and single-page formula sheets are permitted.
Answer any 3 of the 5 questions.
You have 45 minutes.
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Show that a reflection across the yz plane, followed by a reflection across the zx plane, is equivalent to a rotation about the z axis.
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Let Q be the matrix of endpoints of an arbitrary figure in space, let L be the matrix of endpoints of an arbitrary line in space, and let
be an arbitrary angle. Write a MatLab function that rotates Q about L by
and returns the result. Use 3D homogeneous
coordinates. -
Let T(a, b, c) be the matrix that translates from the origin to point (a, b, c) and let S(s) be the matrix that scales by a factor of s in all 3 dimensions. Is it true that T(a, b, c) S(s) = S(s) T(a/s, b/s, c/s)? Prove your answer.
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Let Q be the matrix of endpoints of an arbitrary figure lying in the xy plane and let L be the matrix of endpoints of an arbitrary line lying in the xy plane. Write a MatLab function that reflects Q across L and returns the result. You may use 2D or 3D homogeneous coordinates.
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Derive the perspective transformation that looks at the origin from the point (0, 3r, -4r).