PROBLEM 4. A CONE IS FLAT. Consider a cone with axis along the z-axis, given in...

  

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PROBLEM 4. A CONE IS FLAT. Consider a cone with axis along the z-axis, given in spherical coordinates by = 40, 0 < 40 < a. Give a parametrization of the cone of the form Y(p, 0), where p and are the spherical coordinate parameters. Does this parametrization preserve the distance along either coordinate directions? (You could calculate, or you could reason geometrically.) b. We want to figure out how to parametrize the cone without distortion. So imagine a region D in the (u, v)-plane which in polar coordinates is given by 0 ≤ 0 ≤B, where 0 < 3 < 2π. We want to fold this wedge up into a cone. So we want the radial rays from the origin to map to generators of the cone, preserving distance. Make a sketch of the region D and the cone; for fun take π <ß < 2π. (i) In terms of polar coordinates (r, 0) for D, an arc at fixed r = a with 0 ≤ 0 ≤ B has to map to a cross-sectional circle on the cone. We also want to preserve distance. Show that for this to work, we need 3 = 2π sin yo. Show that the corresponding parametrization of the cone is given by W(r, 0) = (r sin po cos r sin po sin (i),1 ,r cos po 0 340). 2 sin 0 sin 40 (ii) For the same reason in part a., this parametrization W does not preserve parameter distance along the 0-direction. Of course, it should preserve distances from the (u, v)-plane. To see this, express the parametrization W in terms of the Cartesian coordinates (u, v) on D, and call the result X(u, v). The polar angle 0 = 0(u, v) is a smooth function of u and v in this domain (at least away from the origin). Express (u, v) explicitly for (u, v) in the first quadrant 0 ≤ 0 <. (iii) Find the first fundamental form of the cone in the parametrization X(u, v). Is is what you expected? Explain. HINT: First, before you try this, maybe you should check with me to make sure your X(u, v) is correct. One it's correct, you have to take partials. To make the the dot products easier to compute, show that Xu = u²4 v² X(u, v) + √u²+v² ª(u, v), where a(u, v) is a unit vector orthogonal to X(u, v). Xv = u² + v² X (u, v) — √² + v² a(u, v) PROBLEM 4. A CONE IS FLAT. Consider a cone with axis along the z-axis, given in spherical coordinates by = 40, 0 < 40 < a. Give a parametrization of the cone of the form Y(p, 0), where p and are the spherical coordinate parameters. Does this parametrization preserve the distance along either coordinate directions? (You could calculate, or you could reason geometrically.) b. We want to figure out how to parametrize the cone without distortion. So imagine a region D in the (u, v)-plane which in polar coordinates is given by 0 ≤ 0 ≤B, where 0 < 3 < 2π. We want to fold this wedge up into a cone. So we want the radial rays from the origin to map to generators of the cone, preserving distance. Make a sketch of the region D and the cone; for fun take π <ß < 2π. (i) In terms of polar coordinates (r, 0) for D, an arc at fixed r = a with 0 ≤ 0 ≤ B has to map to a cross-sectional circle on the cone. We also want to preserve distance. Show that for this to work, we need 3 = 2π sin yo. Show that the corresponding parametrization of the cone is given by W(r, 0) = (r sin po cos r sin po sin (i),1 ,r cos po 0 340). 2 sin 0 sin 40 (ii) For the same reason in part a., this parametrization W does not preserve parameter distance along the 0-direction. Of course, it should preserve distances from the (u, v)-plane. To see this, express the parametrization W in terms of the Cartesian coordinates (u, v) on D, and call the result X(u, v). The polar angle 0 = 0(u, v) is a smooth function of u and v in this domain (at least away from the origin). Express (u, v) explicitly for (u, v) in the first quadrant 0 ≤ 0 <. (iii) Find the first fundamental form of the cone in the parametrization X(u, v). Is is what you expected? Explain. HINT: First, before you try this, maybe you should check with me to make sure your X(u, v) is correct. One it's correct, you have to take partials. To make the the dot products easier to compute, show that Xu = u²4 v² X(u, v) + √u²+v² ª(u, v), where a(u, v) is a unit vector orthogonal to X(u, v). Xv = u² + v² X (u, v) — √² + v² a(u, v)

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Given equations Cone is flat Consider a cone with axis along thezaxis given in sp... View the full answer