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Published byGeoffrey Warder Modified about 1 year ago

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Anupam Saxena Associate Professor Indian Institute of Technology KANPUR

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Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1- 2 Transformations and Projections 1-2 Modeling of Curves Representation, Differential Geometry Ferguson Segments Bezier Segments 1-2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

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n P R d n is perpendicular to the tangent plane, r u.n = r v.n = 0 second fundamental matrix D Geometric/PARAMETRIC Modeling Solid Modeling Perception of Solids Topology and Solids Solid Modeling 1-2 Transformati ons and Projections 1-2 Modeling of Curves Representati on, Differential Geometry Ferguson Segments Bezier Segments 1- 2 B-spline curves 1-5 NURBS Modeling of Surfaces (Patches) Differential Geometry Tensor Product Boundary Interpolating Composite NURBS

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tangent plane intersects the surface at all points where d = 0 Case 1:No real value of du P is the only common point between the tangent plane and the surface No other point of intersection P ELLIPTICAL POINT

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Case 2: L 2 +M 2 +N 2 > 0 du = (M/L)dv u – u 0 = (M/L)(v – v 0 ) tangent plane intersects the surface along this straight line P PARABOLIC POINT Case 3: two real roots for du tangent plane at P intersects the surface along two lines passing through P P HYPERBOLIC POINT Case 4: L = M = N = 0 P FLAT POINT

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P t n ncnc ncnc nnnn gtggtg t g t k n = n n normal curvature k g = g t g geodesic curvature Since n.t = 0 since k g and n are perpendicular k g.n = 0

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decomposing dr and dn along parametric lengths du and dv Since r u and r v are both perpendicular to n

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the expression for the normal curvature is where The above equation can be written as For an optimum value of normal curvature Differentiation yields

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Thus This can be simplified to For a non trivial solution, the determinant of the coefficient matrix is zero

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K is the Gaussian curvature… H is the mean curvature

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parametric equation of a Monkey Saddle Compute the Gaussian and Mean curvatures

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Monkey saddle maximum principal curvature minimum principal curvature Gaussian curvaturemean curvature

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To identify a certain class of surface patches e.g. For developable surfaces, the Gaussian curvature is ZERO

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