Computer Aided Engineering Design

1. Computer Aided Engineering Design AnupamSaxena Associate Professor Indian Institute of Technology KANPUR 208016

3. Lecture #34Differential Geometry of Surfaces

4. Curves on a surface c(t)=r(u(t), v(t)) r(u, v) tangent to the curve

5. Curves on a surface c(t) =r(u(t), v(t)) r(u, v) differential arc ds length of the curve Symmetric G is called the first fundamental matrixof the surface

6. Curves on a surface … unit tangent t to the curve for t to exist G should be always be positive definite G11G22 – G12G21 > 0 implies thatG is always positive definite

7. Curves on a surface … length of the curve segment in t0tt1 c(t1) and c(t2) as two curves on the surface r(u, v) that intersect the angle of intersection  is given by

8. Curves on a surface … If ut1 and vt2 two curves are orthogonal to each other if

9. Area of the surface patch v = v0 + dv u = u0 + du r(u0, v0 + dv) r(u0 + du, v0) v = v0 u = u0 rudu rvdv r(u0, v0)

10. Surface from the tangent plane: Derivation n P R d

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

12. Second fundamental matrix L, M and N are called the second fundamental form coefficients use

13. Second fundamental matrix … ruu = xuui+ yuuj + zuuk ruv = xuvi+ yuvj + zuvk rvv = xvvi+ yvvj + zvvk

14. Classification of pointson 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

15. Classification of pointson 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