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Mathematics
Differential Geometry
Natural parametrization of a curve
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[QUOTE="anuttarasammyak, post: 6432518, member: 674023"] Hi. The surface is made by rotating the graph of ##z=x^2## around z axis. The shortest curve between the Origin and Point ##(X,Y,Z)## on the surface is presented by parameter ##\xi## as ##(X \xi,Y \xi,Z\xi^2)## where ##Z=X^2+Y^2,0<\xi<1##. [tex]ds^2=X^2 d\xi^2 + Y^2 d\xi^2 + (X^2+Y^2)^2 4\xi^2 d\xi^2=(X^2 + Y^2)[ (X^2+Y^2)4\xi^2+1 ]d\xi^2[/tex] [tex]s(\xi)=\sqrt{X^2 + Y^2} \int_0^\xi \sqrt{ (X^2+Y^2)4\eta^2+1}\ d\eta[/tex] After calculating ##s(\xi)## you can get its inverse ##\xi(s)## and thus get ##(X\xi(s),Y\xi(s),Z\xi(s)^2)## as geodesic from Origin to end point (X,Y,Z) expressed by parameter s, the length from Origin. [/QUOTE]
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Mathematics
Differential Geometry
Natural parametrization of a curve
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