7/27/2019 Parameterizing Curves Surfaces

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Created by Harrison Gammon (Fall 2008)

Parameterizing Curves and Surfaces

The ability to parameterize arbitrary curves and surfaces is an essential skill inmultivariable calculus. This template acts as a guide to parameterizing different types of curves and surfaces.

Curves:

A parameterization for a curve in 3space is a set of functions depending only on aparameter, such as x=f(t), y=g(t), z=h(t), along with bounds for the parameter. Theparameter is simply a dummy variable that is introduced to facilitate latercomputation. It is important to remember, when representing a curve as a set of parametric equations, to maintain the initial relationships that define the curve. Forexample, if the curve is the unit circle, one must ensure that no matter what functions are chosen for x and y, it is true that x(t) 2+y(t) 2=1.

The orientation of a curve must also be considered when creating aparameterization. As the parameter increases, the position vectorr(t)=(x(t),y(t),z(t)) moves in space. The direction of this movement determines theorientation of the parameterization. Often it is of interest to maintain a counterclockwise orientation, such as in applications of Greens Theorem.

Examples:

Consider when one of the variables can be written as an explicit function of the other variable. For example, say y=f(x); that is, y is a function of the othervariable x. Then we parameterize this curve by x=t, y=f(t), using theappropriate bounds for x.

Another common curve to consider is a line. Here, we consider a line in 3dimensional space. Given two points P 1,P2 on the line, this curve isparameterized by finding the vector v=P 2P1 and definingr(t)=(x(t),y(t))=P 1+tv.

An interesting class of curves to consider are curves that can be defined bythe implicit function x a+yb=C, where a,b,C are all nonzero constants. Thesecurves can always be parameterized using a combination of trigonometricfunctions. Let x = (C cos 2 t )1/ a and y = (C sin 2 t )1/ b . Notice that any circlecentered at the origin is a member of this class of curves. Circles centered

elsewhere can be parameterized using a similar method, with an appropriateshift applied to x and y. The last example represents a common case in parameterization of curves in

3dimensional space. If one of the coordinates is allowed to take on anynumerical value we often simply assign this variable the function f(t)=t. Forexample, the helix can be parameterized by x(t)=cos(t), y(t)=sin(t), andz(t)=t.

7/27/2019 Parameterizing Curves Surfaces

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Created by Harrison Gammon (Fall 2008)

Surfaces:

Parameterizing surfaces is similar in structure to parameterizing curves. Aparameterization for a surface is a set of functions depending on two parameters,usually u and v, where x=f(u,v), y=g(u,v), and z=h(u,v). We must also providebounds for the two parameters u and v. As before, the most important concept tounderstand is that the initial relationships that define the surface must be observed.For example, if the surface under consideration is the unit sphere, we must havethat x(u,v) 2+y(u,v) 2+z(u,v) 2=1. Any surface can be parameterized in many different ways and the best choice for a surface depends greatly on context. The skillsnecessary to identify a good parameterization can be developed with practice.

Examples:

The easiest class of surfaces to consider include surfaces in which onevariable can be written as an explicit function of the other variables. Forexample, in the case where z=f(x,y) we can write z as an explicit function of the other variables x and y. Then the most simple parameterization would bex=u, y=v, z=f(u,v), with appropriate bounds for u and v.

Planes can be parameterized by the method in the first example.Remembering that an explicit equation for a plane can be found bygenerating a normal vector from two vectors in the plane.

Cones can be defined by the implicit function z 2=C(x2+y2). In this case, thesurface would include cones above and below the origin. Since this implicit function cannot be solved explicitly a different method must be used thanabove, though if only the upper cone is of interest we can write z = C ( x2 + y 2 explicitly. However, if both parts of the cone are considered

we can use the parameterization x = C 1/ 2

v cos u , y = C 1/ 2

v sin u , z=v. Noticethat we have maintained the initial requirement that z 2=C(x2+y2).

Another important class of examples can be seen by examining surfaces that are defined by the equation x a+yb+zc=D, for nonzero constants a,b,c,D. In thiscase, we use the parameterization x = ( D cos 2 (u)sin 2(v))1/ a , y = ( D sin 2(u)sin 2 (v))1/ b , z = ( D cos 2 (v))1/ c . Spheres centered at the origin canbe parameterized in this manner.

Finally, tetrahedrons and cubes often arise in practical applications requiringparameterization. These surfaces can be parameterized in multiple pieces byfirst parameterizing the planes that comprise these surfaces, then usingappropriate bounds to extract the correct piece of the plane.

Reasons for Parameterizing Curves and Surfaces:

By parameterizing, we can treat curves and surfaces as vector valued functions . Doing so gives rise to the ability to perform vector based algebraic operations . Applications include line integrals, surface area, surface integrals, Stokes Theorem,Greens Theorem, and the Divergence Theorem.

7/27/2019 Parameterizing Curves Surfaces

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Created by Harrison Gammon (Fall 2008)

This template can assist students with a concept that often hinders theirunderstanding of multivariable calculus. The ability to parameterize curves andsurfaces is often an implicit skill that is assumed but never emphasized. It becomesessential to achieve proficiency in parameterizing curves and surfaces in order tosucceed in multivariable calculus. I believe that this template best embodies UDI

Principle 1: Equitable Use . Instruction is made more by useful by making explicit some of the implicit prerequisites required for success.