Natural parametrization of a curve

In summary, the surface z=x^2+y^2 with x(s=0)=0, y(s=0)=0, dx/ds (s=0)=cos(a) and dy/ds(s=0)=sin(a), with a constant a, is expressed as a function of arc length r(s).
  • #1
eva_92
3
0
Hello,
I need the natural parametrization or a geodesic curve contained in the surface z=x^2+y^2, that goes through the origin, with x(s=0)=0, y(s=0)=0, dx/ds (s=0)=cos(a) and dy/ds(s=0)=sin(a), with "a" constant, expressed as a function of the arc length, i.e., I need r(s)=r(x(s),y(s)).
Thank you very much!
Eva
 
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  • #2
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.
 
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  • #3
Hello, thank you for the answer. I have tried with the parametrization x=rcos(a), y=rsin(a), z=r^2, reaching to a similar expression: ds^2=1+4*r^2 (similar to yours with X=cos(a), Y=sin(a)). The solution is s = (1/4)*ln⁡(2*r+sqrt(1+4*r^2))+(1/2)*r*sqrt(1+4*r^2), which is an equation in which I can not calculate r as a function of s to perform the natural parametrization. This is the reason of my question, maybe another parametrization could work, buy it does not happen to me. Could you help me with this?
 
  • #4
You show points on geodesic
[tex](r \cos a, r \sin a, r^2 )[/tex]
where ##0<r<R## corresponding to the end point and
[tex]s = \frac{1}{4}ln⁡(2r+\sqrt{1+4r^2})+\frac{1}{2}r\sqrt{1+4r^2} [/tex]
I would like to know what more you want.
 
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  • #5
The idea is to write x,y,x as a function of s, i.e., x(s),y(s),z(s). If I can calculate r(s), then I can write x(r(s)),y(r(s)),z(r(s)), but I can't calculate r(s) from that equation
 
  • #6
I am afraid it is a tough thing to get the formula of r(s) from s(r) formula.
 

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  • Screenshot_2020-12-20 inverse function of s = (1 4) ln⁡(2 r+sqrt(1+4 r^2))+(1 2) r sqrt(1+4 r^...png
    Screenshot_2020-12-20 inverse function of s = (1 4) ln⁡(2 r+sqrt(1+4 r^2))+(1 2) r sqrt(1+4 r^...png
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Related to Natural parametrization of a curve

1. What is a natural parametrization of a curve?

A natural parametrization of a curve is a way of describing the points on a curve using a single variable, such as time or arc length, in a way that is intuitive and reflects the geometric properties of the curve. It is often used in mathematics and physics to simplify calculations and make them more meaningful.

2. How is a natural parametrization different from other parametrization methods?

A natural parametrization is unique in that it is based on the intrinsic properties of the curve, rather than external factors such as the choice of coordinate system. This means that it is independent of the specific representation of the curve and can be used to describe the curve in a consistent and meaningful way.

3. What are the benefits of using a natural parametrization?

Using a natural parametrization can make calculations and interpretations of a curve much simpler and more intuitive. It also allows for a more direct comparison between different curves, as they are all described using the same variable. Additionally, it can reveal important geometric properties of the curve that may not be apparent in other parametrizations.

4. How is a natural parametrization determined?

The determination of a natural parametrization depends on the specific curve and its properties. In general, it involves finding a variable that reflects the intrinsic properties of the curve, such as its curvature or arc length. This variable is then used to define the parametrization of the curve.

5. Can any curve be described using a natural parametrization?

Yes, any curve can be described using a natural parametrization. However, the method for determining the natural parametrization may differ depending on the properties of the curve. In some cases, a natural parametrization may not be unique, and multiple variables may be used to describe the curve in different ways.

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