13472J/1.128J/2158J/16940J COMPUTATIONAL GEOMETRY Lecture 3 Kwanghee Ko T Maekawa N.M. Patrikalakis Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA Copyright 2003 Massachusetts Institute of Technology Contents 3 Differential geometry of surfaces 3.1 Definition of surfaces 3.2 Curves on a surface 3.3 First fundamental form(arc length 3.4 Tangent plane 3.5 Normal vector 2234668 3.6 Second fundamental form II (curvature) 3.7 Principal curvatures Bibliography Reading in the Textbook Chapter 3, pp. 49-pp 72
13.472J/1.128J/2.158J/16.940J COMPUTATIONAL GEOMETRY Lecture 3 Kwanghee Ko T. Maekawa N. M. Patrikalakis Massachusetts Institute of Technology Cambridge, MA 02139-4307, USA Copyright c 2003 Massachusetts Institute of Technology Contents 3 Differential geometry of surfaces 2 3.1 Definition of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.2 Curves on a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.3 First fundamental form (arc length) . . . . . . . . . . . . . . . . . . . . . . . . 4 3.4 Tangent plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.5 Normal vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.6 Second fundamental form II (curvature) . . . . . . . . . . . . . . . . . . . . . . 8 3.7 Principal curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Bibliography 13 Reading in the Textbook • Chapter 3, pp.49 - pp.72 1
Lecture 3 Differential geometry of surfaces 3.1 Definition of surfaces Implicit surfaces F(r,,a)=0 Example: 22+6+2=1 Ellipsoid, see Figure 3.1 Figure 3.1: Ellipsoid · Explicit surfaces If the implicit equation F(, y, a)=0 can be solved for one of the variables as a function of the other two, we obtain an explicit surface, as shown in Figure 3.2. Example: 2= 2(ax+ By2) Parametric surfaces x=ru, v),y=y(u, u), t=z(u, v) Here functions z(u, u), y(u, u), x(u, u) have continuous partial derivatives of the r th order, and the parameters u and v are restricted to some intervals(i.e.,u1≤u≤u2,n≤t≤v2) leading to parametric surface patches. This rectangular domain D of u, v is called parametric space and it is frequently the unit square, see Figure 3.3. If derivatives of the surface are continuous up to the rth order, the surface is said to be of class r, denoted
Lecture 3 Differential geometry of surfaces 3.1 Definition of surfaces • Implicit surfaces F(x, y, z) = 0 Example: x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 Ellipsoid, see Figure 3.1. x y z Figure 3.1: Ellipsoid. • Explicit surfaces If the implicit equation F(x, y, z) = 0 can be solved for one of the variables as a function of the other two, we obtain an explicit surface, as shown in Figure 3.2. Example: z = 1 2 (αx2 + βy 2 ) • Parametric surfaces x = x(u, v), y = y(u, v), z = z(u, v) Here functions x(u, v), y(u, v), z(u, v) have continuous partial derivatives of the r th order, and the parameters u and v are restricted to some intervals (i.e., u1 ≤ u ≤ u2, v1 ≤ v ≤ v2) leading to parametric surface patches. This rectangular domain D of u, v is called parametric space and it is frequently the unit square, see Figure 3.3. If derivatives of the surface are continuous up to the r th order, the surface is said to be of class r, denoted C r . 2
Figure 3.2: Explicit quadratic surfaces z=2(ax+ By").(a)Left: Hyperbolic paraboloid (a=-3, B=1).(b) Right: Elliptic paraboloid(a=l, B=3) In vector notation r=r(u,U where r=(a, g, a),r(u,v)=((u,u),y(u, u), 2(u, u) x=+ 1 y=u-}→ eliminate,→2=5(x2+y2) paraboloid 3.2 Curves on a surface Let r=r(u, v) be the equation of a surface, defined on a domain D(i.e, uI <usu U1 <U<U2). Let B(t=(u(t), u(t)be a curve in the parameter plane. Then r=ru(t), v(t)) is a curve lying on the surface, see Figure 3. 3. a tangent vector of curve B(t)is given by B(t=(i(t),i(t)) a tangent vector of a curve on a surface is given by By using the chain rule dr(u(t), v(t)) ar du ar du rui(t)+ri(t
Figure 3.2: Explicit quadratic surfaces z = 1 2 (αx2 + βy 2 ). (a) Left: Hyperbolic paraboloid (α = −3, β = 1). (b) Right: Elliptic paraboloid (α = 1, β = 3). In vector notation: r = r(u, v) where r = (x, y, z), r(u, v) = (x(u, v), y(u, v), z(u, v)) Example: r = (u + v, u − v, u 2 + v 2 ) x = u + v y = u − v z = u 2 + v 2 ⇒ eliminate u, v ⇒ z = 1 2 (x 2 + y 2 ) paraboloid 3.2 Curves on a surface Let r = r(u, v) be the equation of a surface, defined on a domain D (i.e., u1 ≤ u ≤ u2, v1 ≤ v ≤ v2). Let β(t) = (u(t), v(t)) be a curve in the parameter plane. Then r = r(u(t), v(t)) is a curve lying on the surface, see Figure 3.3. A tangent vector of curve β(t) is given by β˙(t) = (u˙(t), v˙(t)) A tangent vector of a curve on a surface is given by: dr(u(t), v(t)) dt (3.1) By using the chain rule: dr(u(t), v(t)) dt = ∂r ∂u du dt + ∂r ∂v dv dt = ruu˙(t) + rvv˙(t) (3.2) 3
(u(t),v(t)) 阝(t)=(u(t),v(t)yx Figure 3.3: The mapping of a curve in 2D parametric space onto a 3D biparametric surface 3.3 First fundamental form(arc length) Consider a curve on a surface r=ru(t), v(t)). The arc length of the curve on a surface is dt dt=r t dt V(rui+rui).(rui+rui)dt +(ry ru) √Ea2+2Faud+Ga2 The first fundamental form is defined as I dr dr=(rudu + rudu).(rudu+rudu) Edu2+2Fdudv+Gd2 E, F, G are called first fundamental form coefficients Note that E= ru. ru>0 and G ryry>0if ru#0 and ru+0. The first fundamental form I is positive definite. That is I>0 and I=0 if and only if du =0 and du =0 since e du+F Edu2 and EG-F=ru xTu2>0 I depends only on the surface and not on the parametrization The area of the surface can be derived as follows:
u v x y z D r(u,v) r(u(t),v(t)) β(t)=(u(t),v(t)) Parametric Space D 3D Space Figure 3.3: The mapping of a curve in 2D parametric space onto a 3D biparametric surface . 3.3 First fundamental form (arc length) Consider a curve on a surface r = r(u(t), v(t)). The arc length of the curve on a surface is given by ds = | dr dt|dt = |ru du dt + rv dv dt |dt = q (ruu˙ + rvv˙) · (ruu˙ + rvv˙)dt = q (ru · ru)du2 + 2rurvdudv + (rv · rv)dv2 = p Edu2 + 2Fdudv + Gdv2 (3.3) where E = ru · ru, F = ru · rv, G = rv · rv (3.4) The first fundamental form is defined as I = dr · dr = (rudu + rvdv) · (rudu + rvdv) = Edu2 + 2Fdudv + Gdv2 (3.5) E, F, G are called first fundamental form coefficients Note that E = ru · ru > 0 and G = rv · rv > 0 if ru 6= 0 and rv 6= 0. The first fundamental form I is positive definite. That is I ≥ 0 and I = 0 if and only if du = 0 and dv = 0 since I = 1 E (E du + F dv) 2 + EG − F 2 E dv2 and EG − F 2 = |ru × rv| 2 > 0. I depends only on the surface and not on the parametrization. The area of the surface can be derived as follows: 4
了 (uo+8 式(uvo+6v)-(uo,vo) y r(uo+8u vo)-=(uo, vo r(uo, vo) Figure 3.4: Area of an infinitessimal surface patch r 6A=|ra6×r2ov=|ru×rl|66v rn×rl2=(rn×r2)·(r1×rn) Using the vector identity (a x b). (cx d)=(a c(. d)-(a d)(b c), we get uIu (36) SA=VEG-F2 dudu A eg-F2 dudu Example: For the hyperbolic paraboloid r(u, u)=(u, 0,u2-u2), let us derive an expression for the area of a region of its surface corresponding to a the circle u2+u20 √1+4a2+42dude
r(u0,v0+δv)−r(u0,v0) r(u0+δu,v0)−r(u0,v0) r(u0,v0+δv) r(u0+δu,v0) r(u0,v0) δA Figure 3.4: Area of an infinitessimal surface patch. r(u0, v0 + δv) − r(u0, v0) ' ∂r ∂v δv r(u0 + δu, v0) − r(u0, v0) ' ∂r ∂u δu δA = |ruδu × rvδv| = |ru × rv|δuδv |ru × rv| 2 = (ru × rv) · (ru × rv) Using the vector identity (a × b) · (c × d) = (a · c)(b · d) − (a · d)(b · c), we get |ru × rv| 2 = (ru · ru)(rv · rv) − (ru · rv) 2 (3.6) = EG − F 2 (3.7) δA = p EG − F2 δuδv, A = Z Z p EG − F2 dudv (3.8) Example: For the hyperbolic paraboloid r(u, v) = (u, v, u 2−v 2 ), let us derive an expression for the area of a region of its surface corresponding to a the circle u 2 +v 2 ≤ 1 in the parametric domain D. We begin by forming expressions for the derivatives of the position vector r and the first fundamental form coeffients. ru = (1, 0, 2u) rv = (0, 1, −2v) E = ru · ru = 1 + 4u 2 F = ru · rv = −4uv G = rv · rv = 1 + 4v 2 Using Equation (3.8), we find EG − F 2 = (1 + 4u 2 )(1 + 4v 2 ) − 16u 2 v 2 = 1 + 4u 2 + 4v 2 > 0 A = Z Z D p 1 + 4u 2 + 4v 2dudv 5
To compute the area, we need to evaluate the double integral over the unit disk u+u in the parametric domain D A 1+42+42dud To perform the integration, let us change variables u= rcos(), v=rsin(0), and du do= r dr de √1+4r2rdrd x(5√5-1) 3.4 Tangent plane Tangent plane at a point r(uo, vo) is the union of tangent vectors of all curves on the surface pass through r(uo, vo), as shown in Figure 3.5. Since the tangent vector of a curve on a parametric surface is given by d =r, d+r d, the tangent plane lies on the plane of the vectors Tu and ru. The equation of the tangent plane is Tp(u, v)=r(u, v)+ Aru(u, v)+ur(u, u) where a and u are real variables parameterizing the plane 式+ Figure 3.5: The tangent plane at a point on a surface 3.5 Normal vector The surface normal is the vector at point r(uo, vo) perpendicular to the tangent plane, see Figure 3.6. And therefore (3.10) Note that ru and ru are not necessarily perpendicular
To compute the area, we need to evaluate the double integral over the unit disk u 2 +v 2 ≤ 1 in the parametric domain D; A = Z Z u2+v 2≤1 p 1 + 4u 2 + 4v 2 du dv. To perform the integration, let us change variables. u = r cos(θ), v = r sin(θ), and du dv = r dr dθ A = Z Z r≤1 p 1 + 4r 2 r dr dθ = Z 2π 0 Z 1 0 p 1 + 4r 2 r dr dθ = π 6 (5√ 5 − 1) 3.4 Tangent plane Tangent plane at a point r(uo, vo) is the union of tangent vectors of all curves on the surface pass through r(uo, vo), as shown in Figure 3.5. Since the tangent vector of a curve on a parametric surface is given by dr dt = r u du dt + rv dv dt , the tangent plane lies on the plane of the vectors ru and rv. The equation of the tangent plane is Tp(u, v) = r(u, v) + λru(u, v) + µrv(u, v) (3.9) where λ and µ are real variables parameterizing the plane. x y z r=ruu+rvv r(u0,v0) Tp Figure 3.5: The tangent plane at a point on a surface. 3.5 Normal vector The surface normal is the vector at point r(uo, vo) perpendicular to the tangent plane, see Figure 3.6. And therefore N = ru × rv |ru × rv| (3.10) Note that ru and rv are not necessarily perpendicular. 6
Figure 3.6: The normal to the point on a surface A regular (ordinary) point P on the surface is defined as one for which ru xr+0. A point point P the vectors ru and r, do not vanish and have different directions equires that at that here ru x ru=0 is called a singular point. The condition ru x rufore Example: Elliptic Paraboloid r(u,v)=(u+v,u-U,u2+u) (1,1,2u) (1,-1,20) (+)er+2(u-)ey-2e≠ (x+u)2+( 2√2u2+202+1>0→ Regular! (2(+v),2(u-t),-2) 2V2u2+2211 at(t,v)=(0,0),N=(0,0,-1) Example: Circular Cone r(u, u)=(usin a cos U, u sin a sin v, u cos a), see Figure 3.7 (sin a cos v, sin a sin v, cos a) 0) u sin a sin uu sin a cosu 0
x y z Tp N ru rv Figure 3.6: The normal to the point on a surface. A regular (ordinary) point P on the surface is defined as one for which ru ×rv 6= 0. A point where ru × rv = 0 is called a singular point. The condition ru × rv 6= 0 requires that at that point P the vectors ru and rv do not vanish and have different directions. Example: Elliptic Paraboloid r(u, v) = (u + v, u − v, u 2 + v 2 ) ru = (1, 1, 2u) rv = (1, −1, 2v) ru × rv = ex ey ez 1 1 2u 1 −1 2v = 2(u + v)ex + 2(u − v)ey − 2ez 6= 0 |ru × rv| = 2 q (u + v) 2 + (u − v) 2 + 1 = 2 p 2u 2 + 2v 2 + 1 > 0 ⇒ Regular ! N = (2(u + v), 2(u − v), −2) 2 √ 2u 2 + 2v 2 + 1 = (u + v, u − v, −1) √ 2u 2 + 2v 2 + 1 at (u, v) = (0, 0), N = (0, 0, −1) Example: Circular Cone r(u, v) = (u sin α cos v, u sin α sin v, u cos α), see Figure 3.7 ru = (sin α cos v,sin α sin v, cos α) rv = (−u sin αsinv, u sin αsinv, 0) ru × rv = ex ey ez sinα cos v sin α sin v cos α −u sinα sin v u sin α cos v 0 7
us⊥n us1nds]nv us]nacos Figure 3.7: Ci Ircular cone -u sin a cos a cos ver-u sin a cos a sin vey t usin" aez At the origin n=0 Therefore, the apex of the cone is a singular point 3.6 Second fundamental form II(curvature) k Figure 3. 8: Definition of normal curvature In order to quantify the curvatures of a surface S, we consider a curve C on S which passes through point P as shown in Figure 3.8. t is the unit tangent vector and n is the unit normal vector of the curve C at point P ds where kn is the normal curvature vector normal to the surface, kg is the geodesic curvature vector tangent to the surface, and k= kn is the curvature vector of the curve C at point P kin is called the normal curvature of the surface at p in the direction t
p α v u y x z usinαcosv usinαsinv singular α u usinα Figure 3.7: Circular cone. = −u sinα cos α cos vex − u sin α cos α sin vey + u sin2 αez At the origin n = 0, ru × rv = 0 Therefore, the apex of the cone is a singular point. 3.6 Second fundamental form II (curvature) S P N kg k kn n C t Figure 3.8: Definition of normal curvature In order to quantify the curvatures of a surface S, we consider a curve C on S which passes through point P as shown in Figure 3.8. t is the unit tangent vector and n is the unit normal vector of the curve C at point P. dt ds = κn = kn + kg (3.11) kn = κnN (3.12) where kn is the normal curvature vector normal to the surface, kg is the geodesic curvature vector tangent to the surface, and k = κn is the curvature vector of the curve C at point P. κn is called the normal curvature of the surface at P in the direction t. 8
Meusnier's Theorem: All curves lying on a surface s passing through a given point pE S with the same tangent line have the same normal curvature at this point Since n.t=0. differentiate w r.t. t N′.t+N,t dt dr dN d Recoginizing that ds.ds=dx2+dy 2+dz2=dr, dr, we can rewrite Equation 3.13 as ter of center of curvature Figure 3. 9: Definition of positive normal:(a)kn. N=Kn;(b)kn.N=-Kn II = -dr dN=-(rudu+rudu)(Nudu Ndu Dudu+ nd (314) r MEN.r N=N,r (315) Therefore the normal curvature is given by ⅠIL+2MA+NX2 I E+2FA+GA2 (3.16) Suppose P is a point on a surface and Q is a point in the neighborhood of P, as in Figure 3.10. Taylor's expansion gives r(u+du, v+du)=ru, v)+rudu +r,du+fruudu-+ 2ruududv +roudu )+HOT( 3.17)
Meusnier’s Theorem : All curves lying on a surface S passing through a given point p ∈ S with the same tangent line have the same normal curvature at this point. Since N · t = 0, differentiate w.r.t. s d ds(N · t) = N0 · t + N · t 0 dt ds · N = −t · dN ds = − dr ds · dN ds (3.13) Recoginizing that ds · ds = dx2 + dy2 + dz2 = dr · dr, we can rewrite Equation 3.13 as: dt ds · N = = − dr · dN dr · dr while dt ds · N = κn · N ≡ κn center of curvature center of curvature N N P P (a) (b) Figure 3.9: Definition of positive normal: (a) κn · N = κn; (b) κn · N = −κn. II = −dr · dN = −(rudu + rvdv) · (Nudu + Nvdv) = Ldu2 + 2Mdudv + Ndv2 (3.14) where L = N · ruu, M = N · ruv, N = N · rvv (3.15) Therefore the normal curvature is given by κn = II I = L + 2Mλ + Nλ 2 E + 2Fλ + Gλ2 (3.16) where λ = dv du . Suppose P is a point on a surface and Q is a point in the neighborhood of P, as in Figure 3.10. Taylor’s expansion gives r(u + du, v + dv) = r(u, v) + rudu + rvdv + 1 2 (ruudu2 + 2ruvdudv + rvvdv2 ) + H.O.T. (3.17) 9
Figure 3.10: Geometrical illustration of the second fundamental form Therefore PQ=r(u+du,u+dv)-r(u, u)=rudu+rudy+5(ruudu+2ruududu+rudu2)+HOT Thus, the projection of PQ onto N d=PQ N=(rudu +rudu)N+II 0, we get d==lI=s(Ldu+ 2Mdudv+ndu We want to observe in which situation d is positive and negative. When d=0 Ldu+2Mdudu+ Ndu= 0 d M±√(Md)2-D nda,2 M±√M2-LN (318) Figure 3.11:(a) Elliptic point;(b)Parabolic point;(c) Hyperbolic point If M2-LN 0, there are two roots. The surface intersects its tangent plane with two M+vAR-LNdv, which intersect at point P. P is called hyperbolic point (Figure 3.11(c))
N P Q d Tp r=r(u,v) Figure 3.10: Geometrical illustration of the second fundamental form. Therefore PQ = r(u + du, v + dv) − r(u, v) = rudu + rvdv + 1 2 (ruudu2 + 2ruvdudv + rvvdv2 ) + H.O.T. Thus, the projection of PQ onto N d = PQ · N = (rudu + rvdv) · N + 1 2 II and since ru · N = rv · N = 0, we get d = 1 2 II = 1 2 (Ldu2 + 2Mdudv + Ndv2 ) We want to observe in which situation d is positive and negative. When d = 0 Ldu2 + 2Mdudv + Ndv2 = 0 Solve for du du = −M ± p (Mdv) 2 − LNdv2 L = −M ± √ M2 − LN L dv (3.18) N N N P P Tp P Tp Tp Figure 3.11: (a) Elliptic point; (b) Parabolic point; (c) Hyperbolic point. • If M2−LN 0, there are two roots. The surface intersects its tangent plane with two lines du = −M± √ M2−LN L dv, which intersect at point P. P is called hyperbolic point (Figure 3.11(c)). 10