Computer Aided Engineering Design

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## Computer Aided Engineering Design

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**Computer Aided Engineering Design**AnupamSaxena Associate Professor Indian Institute of Technology KANPUR 208016**Curves on a surface**c(t)=r(u(t), v(t)) r(u, v) tangent to the curve**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**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**Curves on a surface …**length of the curve segment in t0tt1 c(t1) and c(t2) as two curves on the surface r(u, v) that intersect the angle of intersection is given by**Curves on a surface …**If ut1 and vt2 two curves are orthogonal to each other if**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)**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**Second fundamental matrix**L, M and N are called the second fundamental form coefficients use**Second fundamental matrix …**ruu = xuui+ yuuj + zuuk ruv = xuvi+ yuvj + zuvk rvv = xvvi+ yvvj + zvvk**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**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