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

2. 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…

3. 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

4. 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

5. 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

6. Lecture # 35 Gaussian and Mean Curvature of Surfaces

7. 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

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

9. 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

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

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

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

13. Curvature Plots of Monkey Saddle
minimum principal
curvature
maximum principal
curvature
Monkey saddle
Gaussian curvature
mean curvature

14. Why are these curvatures important ?
To identify a certain class of surface patches
e.g. For developable surfaces, the Gaussian curvature is ZERO