parametric - PARAMETRIC SURFACES Math21a, O. Knill HW A)...

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Unformatted text preview: PARAMETRIC SURFACES Math21a, O. Knill HW A) Parametrize the upper half of the ellipsoid x 2 + y 2 + z 2 / 4 = 1 in three different ways: as a graph vector r ( x, y ) = ( x, y, f ( x, y )) (Euclidean coordinates), as a surface of revolution vector r ( θ, z ) = ( g ( z )cos( θ ) , g ( z )sin( θ ) , z ) (cylindrical coordinates) and as a deformed sphere vector r ( φ, θ ) = ( x ( φ, θ ) , y ( φ, θ ) , z ( φ, θ )) (spherical coordinates). B) The curve vector r ( t ) = ( √ t cos( t ) , √ t sin( t ) , t ) is on a surface. Find a parametrization vector r ( t, s ) of this surface. C) Parametrize the surface which has distance 1 from the unit circle in the xy plane. This is a doughnut. Use two angles θ , a rotation angle around the z axes and φ a rotation angle around the circle. D) Parametrize the paraboloid x 2 + y 2 = z in two different ways: as a graph or as a surface of revolution....
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This note was uploaded on 03/29/2008 for the course MATH 251 taught by Professor Skrypka during the Spring '08 term at Texas A&M.

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