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Understanding Parameterized Surfaces and Curves in 3-Space

This section explores the concept of parameterized surfaces and curves in 3-dimensional space. It begins with a review of parametric equations in rectangular and spherical coordinates, demonstrating how these can describe geometric figures like helices and toruses. We analyze parameterized surfaces derived from two variable equations and how to manipulate parameters to create shapes. The section also covers surfaces of revolution, illustrating the concept with examples such as cones and toruses, providing a thorough understanding of their structure and applications in various coordinate systems.

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Understanding Parameterized Surfaces and Curves in 3-Space

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Presentation Transcript

  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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