Surfaces and Transformations

Summary and Review

In this chapter, we learned to use derivatives and vectors together in order to study the geometry of surfaces, coordinate systems, and coordinate transformations.  We began by showing that the gradient of a level surface is normal to the surface. Often, however, surfaces are studied parametrically, where a parameterization of a surface is a function of the form

r( u,v) = á x( u,v) ,y(u,v) ,z( u,v) ñ

The principal vectors r_u = áx_u,y_u,z_u ñ and r_v = áx_v,y_v,z_v ñ are tangent to the surface, thus allowing us to parameterize the tangent plane at a given point r(p,q) on the surface by

L( du,dv) = r( p,q) +ru( p,q) du+rv( p,q) dv

Also, the cross product r_u×r_v is normal to the surface, thus allowing us to define the unit surface normal n to an oriented surface to be the unit vector in the direction of the normal to the surface at each point.

Parameterizations in which z( u,v) = 0 for all u and v are called coordinate transformations and are of the form

T( u,v) = á x( u,v) ,y( u,v) ñ

The derivative of a coordinate transformation is called the Jacobian of the transformation and is defined

J( u,v) = é ê ë xu xv yu yv ù ú û

As a transformation, the Jacobian J( u,v) maps tangent vectors to a curve C in the uv-plane to tangent vectors to the image of C in the xy-plane. It follows that the determinant of the transformation closely scales the area of a small region in the uv-plane to the area of the image of the region in the xy-plane. We capture this idea with the area differential, which is given by

dA = ê ê  ¶( x,y) ¶( u,v) ê ê dudv

Jacobians, normals, parameterizations, and coordinate transformations are instrumental in the study of coordinate systems other than the usual Euclidean coordinate systems. In the plane, polar coordinates are often used to locate points, where polar coordinates are defined by the relationships

x = rcos( q) y = rsin( q)

which immediately imply that x²+y² = r². If polar coordinates are applied to the xy-coordinates in 3 dimensional space, then the result is known as cylindrical coordinates. Also, spherical coordinates form an alternative coordinate system in 3-dimensional space and are defined by 2 sets of relationships:

x = rcos( q) r = rsin( f) y = rsin( q) z = rcos( f)

These lead to the fundamental idea x²+y²+z² = r², and once combined, lead also to the more direct expression of the Spherical coordinate transformation:

x = rsin( f) cos( q) y = rsin( f) sin( q) z = rcos( f)

Finally, given a parametric surface r( u,v) , we define g₁₁ = r_u·r_u, g₁₂ = r_u·r_v, and g₂₂ = r_v·r_v. It then follows that short distances ds on the surface satisfy

ds2 = g11du2+2g12dudv+g22dv2

which is known as the fundamental form of the surface. If g₁₂ = r_u·r_v = 0, then the parameterization is said to be orthogonal and the fundamental form resembles the Pythagorean theorem. Properties of the surface which can be derived from the fundamental form are said to be intrinsic to the surface. Intrinsic properties include the length of a curve and geodesics of the surface, which are curves whose acceleration is normal to the surface at each point on the curve. The curvature of a geodesic is said to be a normal curvature of a surface, and the maximum and minimum curvatures at a given point are said to be the principal curvatures of the surface at that point. The sum of the principal curvatures is said to be the mean curvature of the surface, and the product of the curvatures is said to be the Gaussian Curvature of the surface. Gauss' Theorem Egregium shows that the Gaussian curvature of a surface is intrinsic to the surface.

There are other ideas and concepts introduced in this chapter, and you should re-read the individual sections in addition to this summary.  The review materials are based on both the ideas above and some of those in the chapter not mentioned here.  Review questions and solutions are in web page form on the left and in pdf form on the right.  For maximum benefit, you should attempt to answer the questions before you look at the solutions.   

You will need Acrobat reader in order to open and print the pdf files.

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