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**Computer Aided Engineering Design**

Anupam Saxena Associate Professor Indian Institute of Technology KANPUR

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**We study TOPOLOGY OF SOLIDS to seek the answer TO THIS QUESTION…**

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 We study TOPOLOGY OF SOLIDS to seek the answer TO THIS QUESTION…

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**Surface from the tangent plane: Derivation**

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 n P R n is perpendicular to the tangent plane, ru.n = rv.n = 0 d second fundamental matrix D

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**Classification of points on the surface**

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 P ELLIPTICAL POINT No other point of intersection

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**Classification of points on the surface**

L2+M2+N2 > 0 du = (M/L)dv Case 2: u – u0 = (M/L)(v – v0) tangent plane intersects the surface along this straight line P PARABOLIC POINT two real roots for du Case 3: 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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**Lecture # 35 Gaussian and Mean Curvature of Surfaces**

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**Normal and geodesic curvatures**

kn = nn normal curvature kg = gtg geodesic curvature n t P nc Since n.t = 0 nn tg t nc gtg since kg and n are perpendicular kg .n = 0

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**Normal and geodesic curvatures**

decomposing dr and dn along parametric lengths du and dv Since ru and rv are both perpendicular to n

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**Normal and geodesic curvatures**

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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**Normal and geodesic curvatures**

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

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**Max and Min normal curvatures**

K is the Gaussian curvature… H is the mean curvature

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**Example parametric equation of a Monkey Saddle**

Compute the Gaussian and Mean curvatures

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**Curvature Plots of Monkey Saddle**

minimum principal curvature maximum principal curvature Monkey saddle Gaussian curvature mean curvature

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**Why are these curvatures important ?**

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

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