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Differential Geometry - Lecture 11 Differential Geometry c...

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Lecture 11: Differential Geometry c Bryan S. Morse, Brigham Young University, 1998–2000 Last modified on February 28, 2000 at 8:45 PM Contents 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 11.1.1 What Differential Geometry Is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 11.1.2 Assumptions of Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 11.1.3 Normals and Tangent Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 11.1.4 Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 11.1.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 11.2 First Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 11.2.1 The Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 11.2.2 First-order Gauge Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 11.3 Second Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 11.3.1 The Hessian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 11.3.2 Principal Curvatures and Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 11.3.3 Second-order Gauge Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 11.4 Using Principal Curvatures and Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 11.4.1 Gaussian Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 11.4.2 Mean Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 11.4.3 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 11.4.4 Deviation From Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 11.4.5 Patch Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 11.4.6 Parabolic Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 11.4.7 Umbilics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 11.4.8 Minimal Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 11.4.9 Shape Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 11.5 Other Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 11.5.1 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 11.5.2 Genericity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 11.6 Applications (Examples) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 11.6.1 Corners and Isophote Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 11.6.2 Ridges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 11.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Reading These notes. 11.1 Introduction 11.1.1 What Differential Geometry Is Differential Geometry is very much what the name implies: geometry done using differential calculus. In other words, shape description through derivatives.
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11.1.2 Assumptions of Differentiability The first major assumption made in differential geometry is, of course, that the curve, surface, etc. is everywhere differentiable—there are no sharp corners in a differential-geometry world. (Of course, once one accepts the notion of scaled measurements, there are no infinitely-precise corners anyway.) 11.1.3 Normals and Tangent Planes The first obvious derivative quantity is the curve or surface tangent . For curves, this is a vector; for surfaces, this is a tangent plane . For differential geometry of 3D shapes, there is no “global” coordinate system. All measurements are made relative to the local tangent plane or normal. For images, though, we’re going to use a coordinate system that defines the image intensity as the “up” direction. Be aware that differential geometry as a means for analyzing a function (i.e., an image) is quite different from differential geometry on general surfaces in 3D. The concepts are similar, but the means of calculation are different. 11.1.4 Disclaimer If a true differential geometer were to read these notes, he would probably cringe. This is a highly condensed and simplified version of differential geometry. As such, it contains no discussion of forms (other than the Second Fun- damental Form), covectors, contraction, etc. It also does not attempt to address non-Euclidean aspects of differential geometry such as the bracketing, the Levi-Civita tensor, etc. 11.1.5 Notation For this lecture, we’ll use the notation common in most of the literature on the subject. This notation uses subscripts to denote derivatives in the following fashion. If we let L ( x, y ) be our image’s luminance function (pixel values), the derivative with respect to x is L x , and the derivative with respect to y is L y . In fact, we can denote the derivative of L with respect to any direction (unit vector) ¯ v as L v .
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