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Show David R. Gallup School of Computing / Sponsor ter Shirley COMPUTATION AND APPLICATION OF SURFACE CURVATURE David R. Gallup, (Peter Shirley) School of Computing When dealing with curves and surfaces it is often useful to have an estimation of their local shape. Curvature is a quantitative measurement that approximates the shape of a curve or surface to second order. This paper explains the mathematic principles that define curvature and describes in detail methods for computing it. The performance of these methods is evaluated, and several concrete applications of curvature are discussed. Curvature can be understood by considering several intuitive definitions. For curves, it is the amount of bend per unit distance. In other words, at any point along the curve, curvature measures how much the curve is turning. Curvature is equivalent^ defined as the reciprocal of the radius of the best fit circle to the curve. For surfaces, curvature can be measured by fitting circles perpendicular to the surface in all possible directions. This gives a set of curvature values, and surface curvature is defined by the maximum and minimum of the set. For 2D functions and height fields, formulas for curvature can be solved for analytically. However, surfaces often take the form of a polygonal mesh, which can be difficult or impossible to parameterize. One technique for measuring the curvature of a mesh is that of fitting a quadric surface to the vertices of the mesh. The curvature of the quadric is then used to estimate of the curvature of the mesh. The effectiveness of this technique is evaluated using several experiments, and limitations are discussed. This paper also discusses some specific applications of curvature in computer graphics and computer vision. |
