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1、FundamentalsofDifferentialGeometry(Part2)Copyright:DietmarHildenbrand,TUDarmstadt,Nov.2002Whatdothefundamentalformsmean??Length,angle,surfacearea?curvatures(deviationbetweenthesurfaceandthetangentplane)2Copyright:DietmarHildenbrand,TUDarmstadt,Nov.2002Literature?Manfred
2、oP.doCarmo:DifferentialgeometrievonKurvenundFl?chen.Vieweg,1998?http://mathworld.wolfram.com/topics/DifferentialGeometry.html?http://www.mpi-sb.mpg.de/~belyaev/Math4CG/Math4CG.html3Copyright:DietmarHildenbrand,TUDarmstadt,Nov.2002CurvesonsurfacesrCurvesonasurfaceX(u,v)a
3、redefinedbyu=u(t),v=v(t)e.g.u=t,v=constdescribecurveswithconstantv(so-calledparametriclines)4Copyright:DietmarHildenbrand,TUDarmstadt,Nov.2002Curvesonsurfacese.g.cylinderrX(u,v)=(rcosu,rsinu,v);u?[0,2p]u=const:r1.u=0,v=t?X(t)=(r,0,t)pr2.u=,v=t?X(t)=(0,r,t)2v=const:r3.u=
4、t,v=c?X(t)=(rcost,rsint,c)5Copyright:DietmarHildenbrand,TUDarmstadt,Nov.2002Curvesonsurfacese.g.cylinderrX(u,v)=(rcosu,rsinu,v);u?[0,2p]helixr4.u=t,v=t?X(t)=(rcost,rsint,t)6Copyright:DietmarHildenbrand,TUDarmstadt,Nov.2002tangentvectorofcurvesonsurfacesrrletX(t)=X(u(t),
5、v(t))rbeacurveonthesurfaceX(u,v)?accordingtothegeneralchainrulerr&dXrrtangentvectorX(t)==X×u&(t)+X×v&(t)uvdt7Copyright:DietmarHildenbrand,TUDarmstadt,Nov.2002ArclengthofthecurvesonsurfacesArclengthmeanthelengthofaparametriccurvebetweentwopointsdefinedbyitsparametervalue
6、st=aandt=bbbr&arclengths=òX(t)dt=òdsaar&rrrr2222X(t)=X×u&(t)+2XXu&(t)v&(t)+X×v&(t)uuvvbrrrr2222arclengths=X×u&(t)+2XXu&(t)v&(t)+X×v&(t)dtòuuvva8Copyright:DietmarHildenbrand,TUDarmstadt,Nov.2002firstfundamentalformbdss=òdtdtads2rrrr2222=Xu&(t)+2XXu&(t)v&(t)+Xv&(t)2uuvvdt
7、dvdusincev&(t)=,u&(t)=dtdt?rrrr22222ds=Xdu+2XXdudv+Xdvuuvv9Copyright:DietmarHildenbrand,TUDarmstadt,Nov.2002firstfundamentalform()222I=ds=E×du+2×F×dudv+G×dvwithrrE=X×X(g)uu11rrF=X×X(g)uv12rrG=X×X(g)vv22Ideterminesthearclengthofacurveonthesurface10Copyright:DietmarHilden
8、brand,TUDarmstadt,Nov.2002firstfundamentalformb22arclengths=òE×u&(t)+2F×u&(t)v&(t)+G×v&(t)dtaFAngleofparametri
