# Computer Aided Engineering Design

## Presentation on theme: "Computer Aided Engineering Design"— Presentation transcript:

Computer Aided Engineering Design
Anupam Saxena Associate Professor Indian Institute of Technology KANPUR

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…

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

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

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

Lecture # 35 Gaussian and Mean Curvature of Surfaces

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

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

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

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

Max and Min normal curvatures
K is the Gaussian curvature… H is the mean curvature

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