Geometry of Surfaces
In Section 7.1 we consider the basic notions of a parametric surface and a regular surface, and use a number of MATLAB ® commands to produce surfaces by various methods. (Similar definitions for curves were studied in Section 5.1). In Section 7.2 we calcu
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Geometry of Surfaces
In Section 7.1 we consider the basic notions of a parametric surface and a regular R surface, and use a number of MATLAB commands to produce surfaces by various methods. (Similar definitions for curves were studied in Section 5.1). In Section 7.2 we calculate and plot the tangent planes and normal vectors of a surface. As an application we solve conditional extremum problems in space. In Section 7.3 we consider parametric and implicitly defined surfaces with singularities. In Section 7.4 we use changes in coordinates and linear transformations in space to calculate and plot the osculating paraboloid at a point of a surface. This elementary approach is given only for methodical reasons. In Section 7.5 we calculate characteristics related to the first and second fundamental forms, the Gaussian and mean curvatures, write down the equations of geodesics using the M-file (program) of Section A.8, and plot geodesics on surfaces.
7.1 Regular Surface 7.1.1 What is a surface? Euclid defines a surface intuitively, as a two-dimensional figure either swept by the path of a moving curve, or bounding a solid body. Planes, some polyhedra, and curved surfaces, each defined in some practical way, are studied in school. The mathematically correct definition of a surface is based on notions from topology, but it starts from the key notion of an elementary surface, which can be imagined as an elementary domain of the plane after continuous deformation that is stretched in and out in space. We call G ⊂ Rn (n = 2, 3) an open set if for each point P ∈ G there exists ε > 0 such that the ball B(P, ε ) of radius ε with center P (the disk, when n = 2) lies inside of G. The complement F = R3 \ G of an open set G is called a closed set. Open sets R V. Rovenski, Modeling of Curves and Surfaces with MATLAB , Springer Undergraduate Texts in Mathematics and Technology 7, DOI 10.1007/978-0-387-71278-9 7, c Springer Science+Business Media, LLC 2010
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7 Geometry of Surfaces
(or domains) are convenient for working with continuous and differentiable functions defined on such sets. An arbitrary open set is the union of a finite or countable number of disjoint domains. Recall that a homeomorphism is a one-to-one correspondence between points in two geometric figures that is continuous in both directions. As in the definition of an elementary curve, we use the notion of a homeomorphism from one geometric figure to another, i.e., a one-to-one map that is continuous and has a continuous inverse. A set M in space is called an elementary surface if it is the image of a planar elementary domain G under the homeomorphism r : G → R3 . An open set in the plane (in space) that is homeomorphic to a disk (respectively, a ball) is called an elementary domain. The following two elementary domains in the plane with coordinates u, v are often used: the disc u2 + v2 < R2 of radius R and the rectangle with sides a, b parallel to the coordinate axes. Since we will also be interested in self-intersecting surfaces, we give the following
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