1 / 15

Computer Aided Engineering Design

Computer Aided Engineering Design. Anupam Saxena Associate Professor Indian Institute of Technology KANPUR 208016. Lecture #34 Differential Geometry of Surfaces. Curves on a surface. c ( t )= r ( u ( t ), v ( t )). r ( u , v ). tangent to the curve. Curves on a surface.

louise
Download Presentation

Computer Aided Engineering Design

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


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

  2. Lecture #34Differential Geometry of Surfaces

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

  4. 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

  5. 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

  6. 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

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

  8. 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)

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

  10. 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

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

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

  13. 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

  14. 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

More Related