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lecnotes3 - 13.472J/1.128J/2.158J/16.940J COMPUTATIONAL...

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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 Copyrigh t c ± 2003 Massachusetts Institute o f Technology Contents 3 Differential geometry of surfaces 2 3.1 DeFnition of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.2 Curves on a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.3 ±irst 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
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Lecture 3 Diferential geometry oF surFaces 3.1 Defnition 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 ( αx 2 + 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., u 1 u u 2 , v 1 v v 2 ) 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
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Explicit quadratic surfaces z = 1 2 ( αx 2 + 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, de±ned on a domain D (i.e., u 1 u u 2 , v 1 v v 2 ). 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: d r ( u ( t ) , v ( t )) dt (3.1) By using the chain rule: d r ( u ( t ) , v ( t )) dt = r ∂u du dt + r ∂v dv dt = r u ˙ u ( t ) + r
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lecnotes3 - 13.472J/1.128J/2.158J/16.940J COMPUTATIONAL...

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