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Question 3. The Roman surface was studied by Steiner in 1844. It is defined as the image of the function o(u, v) = (sin 2u cos v, sin 2u sin u, sin² u sin 2v) over all (u, v). To make o injective, and to simplify computations, in this problem we only consider the domain π U = {(u, v): 0 < u < 7/7, 4 π }. 2 (However, you may not assume o is injective.) Show that olu is a regular surface patch, and that its image is a smooth surface. <U< (The Roman surface has self-intersections, and is an immersion of the real projective plane in R³; see the left image. The right image is our restriction.)

Question 3. The Roman surface was studied by Steiner in 1844. It is defined as the image of the function o(u, v) = (sin 2u cos v, sin 2u sin u, sin² u sin 2v) over all (u, v). To make o injective, and to simplify computations, in this problem we only consider the domain π U = {(u, v): 0 < u < 7/7, 4 π }. 2 (However, you may not assume o is injective.) Show that olu is a regular surface patch, and that its image is a smooth surface. <U< (The Roman surface has self-intersections, and is an immersion of the real projective plane in R³; see the left image. The right image is our restriction.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 3. The Roman surface was studied by Steiner in 1844. It is defined as the image of the function o(u, v) = (sin 2u cos v, sin 2u sin u, sin² u sin 2v) over all (u, v). To make o injective, and to simplify computations, in this problem we only consider the domain π U = {(u, v): 0 < u < 7/7, 4 π </}. 2 (However, you may not assume o is injective.) Show that olu is a regular surface patch, and that its image is a smooth surface. <U< (The Roman surface has self-intersections, and is an immersion of the real projective plane in R³; see the left image. The right image is our restriction.)
Transcribed Image Text:Question 3. The Roman surface was studied by Steiner in 1844. It is defined as the image of the function o(u, v) = (sin 2u cos v, sin 2u sin u, sin² u sin 2v) over all (u, v). To make o injective, and to simplify computations, in this problem we only consider the domain π U = {(u, v): 0 < u < 7/7, 4 π </}. 2 (However, you may not assume o is injective.) Show that olu is a regular surface patch, and that its image is a smooth surface. <U< (The Roman surface has self-intersections, and is an immersion of the real projective plane in R³; see the left image. The right image is our restriction.)
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