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Section 17.5 Parameterized Surfaces

Section 17.5 Parameterized Surfaces. Recall Parametric curves in 3-space. In rectangular coordinates ( x , y , z ) we have We used these to make a helix. Parameterized curves in spherical coordinates. In spherical coordinates we have ( ρ , , θ )

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Section 17.5 Parameterized Surfaces

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  1. Section 17.5Parameterized Surfaces

  2. Recall Parametric curves in 3-space • In rectangular coordinates (x,y,z) we have • We used these to make a helix

  3. Parameterized curves in spherical coordinates • In spherical coordinates we have (ρ, , θ) • As parametric equations we let C be the curve • We used these to make a spherical helix

  4. Now we are going to move on to parameterized surfaces • What would we get from the following set of parametric equations? • Let’s take a look with Maple

  5. In the first two cases we have a parameterized curve in 3 space • In the third case we have a parameterized surface as our parametric equations were in terms of two variables • In rectangular coordinates, parameterized surfaces are given by where a ≤ u ≤ b and c ≤ v ≤ d • Both u and v are object parameters • Can do in cylindrical and spherical also

  6. Let’s see how we can parameterize a sphere • In rectangular coordinates • In spherical coordinates • How about a Torus • One way of thinking about a Torus is a circle rotated around the z-axis • Let’s take a look in maple

  7. Generalized Torus • Say we want to create a Torus that is not circular • Essentially we want to rotate some region in a plane around the third axis (8,14) z (12,6) (5,1) x

  8. Generalized Torus • We need to make parametric equations for the triangular region • This will be easier to do in cylindrical coordinates (8,14) z (12,6) (5,1) x

  9. Parameterizing Planes • The plane through the point with position vector and containing the two nonparallel vectors and is given by • So if • The parametric equations are

  10. Parameterizing Surfaces of Revolution • We can create surfaces that have an axis rotational symmetry and circular cross sections to that axis • For example, how about a cone that has a base that is a circle of radius 3 in the xy-plane and a height of 10. • The following structure can be used to revolve a curve around the z axis • This can be modified to revolve around other axes as well

  11. Parameter Curves • Parameter curves are obtained by setting one of the parameters to a constant and letting the other vary • Take the following parametric equations • What do they give us? • What would we get if z is held constant? • What would we get if t is held constant? • These parameter curves are cross sections of our parameterized surface

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