Is there only one natual parametrization per curve?

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Asumming a curve has a natural parametrization (is simple and regular), such natural parametrization is unique or are more parametrizations equivalent to it?



I should add that by natural I mean by arc length, I was taught both terms were equivalent.




calculus integration curves parametrization




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  • 2




    The only classification of parameterisations that I know of which is (basically) unique is parameterisation by arc length.
    – Arthur
    Aug 5 at 22:36











  • Your problem is that there is no universally accepted definition of "natural." One "natural" parameterization of a circle is by means of angle $theta$. Another is by arc length. Another is by $x$ component...
    – David G. Stork
    Aug 5 at 23:08










  • I meant by arc length, edited my question, sorry if it caused any confusion.
    – S.Alfaro
    Aug 6 at 7:01











  • If you fix which point has parameter zero, and you fix which of the two directions is the increasing direction, then is the parameterization by arc length not unique?
    – GEdgar
    Aug 6 at 10:25














up vote
0
down vote

favorite












Asumming a curve has a natural parametrization (is simple and regular), such natural parametrization is unique or are more parametrizations equivalent to it?



I should add that by natural I mean by arc length, I was taught both terms were equivalent.




calculus integration curves parametrization




share|cite|improve this question

















  • 2




    The only classification of parameterisations that I know of which is (basically) unique is parameterisation by arc length.
    – Arthur
    Aug 5 at 22:36











  • Your problem is that there is no universally accepted definition of "natural." One "natural" parameterization of a circle is by means of angle $theta$. Another is by arc length. Another is by $x$ component...
    – David G. Stork
    Aug 5 at 23:08










  • I meant by arc length, edited my question, sorry if it caused any confusion.
    – S.Alfaro
    Aug 6 at 7:01











  • If you fix which point has parameter zero, and you fix which of the two directions is the increasing direction, then is the parameterization by arc length not unique?
    – GEdgar
    Aug 6 at 10:25












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Asumming a curve has a natural parametrization (is simple and regular), such natural parametrization is unique or are more parametrizations equivalent to it?



I should add that by natural I mean by arc length, I was taught both terms were equivalent.




calculus integration curves parametrization




share|cite|improve this question













Asumming a curve has a natural parametrization (is simple and regular), such natural parametrization is unique or are more parametrizations equivalent to it?



I should add that by natural I mean by arc length, I was taught both terms were equivalent.





calculus integration curves parametrization





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 6 at 6:59
























asked Aug 5 at 22:28









S.Alfaro

12




12







  • 2




    The only classification of parameterisations that I know of which is (basically) unique is parameterisation by arc length.
    – Arthur
    Aug 5 at 22:36











  • Your problem is that there is no universally accepted definition of "natural." One "natural" parameterization of a circle is by means of angle $theta$. Another is by arc length. Another is by $x$ component...
    – David G. Stork
    Aug 5 at 23:08










  • I meant by arc length, edited my question, sorry if it caused any confusion.
    – S.Alfaro
    Aug 6 at 7:01











  • If you fix which point has parameter zero, and you fix which of the two directions is the increasing direction, then is the parameterization by arc length not unique?
    – GEdgar
    Aug 6 at 10:25












  • 2




    The only classification of parameterisations that I know of which is (basically) unique is parameterisation by arc length.
    – Arthur
    Aug 5 at 22:36











  • Your problem is that there is no universally accepted definition of "natural." One "natural" parameterization of a circle is by means of angle $theta$. Another is by arc length. Another is by $x$ component...
    – David G. Stork
    Aug 5 at 23:08










  • I meant by arc length, edited my question, sorry if it caused any confusion.
    – S.Alfaro
    Aug 6 at 7:01











  • If you fix which point has parameter zero, and you fix which of the two directions is the increasing direction, then is the parameterization by arc length not unique?
    – GEdgar
    Aug 6 at 10:25







2




2




The only classification of parameterisations that I know of which is (basically) unique is parameterisation by arc length.
– Arthur
Aug 5 at 22:36





The only classification of parameterisations that I know of which is (basically) unique is parameterisation by arc length.
– Arthur
Aug 5 at 22:36













Your problem is that there is no universally accepted definition of "natural." One "natural" parameterization of a circle is by means of angle $theta$. Another is by arc length. Another is by $x$ component...
– David G. Stork
Aug 5 at 23:08




Your problem is that there is no universally accepted definition of "natural." One "natural" parameterization of a circle is by means of angle $theta$. Another is by arc length. Another is by $x$ component...
– David G. Stork
Aug 5 at 23:08












I meant by arc length, edited my question, sorry if it caused any confusion.
– S.Alfaro
Aug 6 at 7:01





I meant by arc length, edited my question, sorry if it caused any confusion.
– S.Alfaro
Aug 6 at 7:01













If you fix which point has parameter zero, and you fix which of the two directions is the increasing direction, then is the parameterization by arc length not unique?
– GEdgar
Aug 6 at 10:25




If you fix which point has parameter zero, and you fix which of the two directions is the increasing direction, then is the parameterization by arc length not unique?
– GEdgar
Aug 6 at 10:25










1 Answer
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No, parametrizations are not unique.



Both
$$t:[0,2pi],z=e^it$$
and
$$t:[-pi,pi],z=e^i(t+textany real number)$$
parametrize a unit circle on the complex plane.






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  • And there are plenty more!
    – Lubin
    Aug 6 at 2:18










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
2
down vote













No, parametrizations are not unique.



Both
$$t:[0,2pi],z=e^it$$
and
$$t:[-pi,pi],z=e^i(t+textany real number)$$
parametrize a unit circle on the complex plane.






share|cite|improve this answer





















  • And there are plenty more!
    – Lubin
    Aug 6 at 2:18














up vote
2
down vote













No, parametrizations are not unique.



Both
$$t:[0,2pi],z=e^it$$
and
$$t:[-pi,pi],z=e^i(t+textany real number)$$
parametrize a unit circle on the complex plane.






share|cite|improve this answer





















  • And there are plenty more!
    – Lubin
    Aug 6 at 2:18












up vote
2
down vote










up vote
2
down vote









No, parametrizations are not unique.



Both
$$t:[0,2pi],z=e^it$$
and
$$t:[-pi,pi],z=e^i(t+textany real number)$$
parametrize a unit circle on the complex plane.






share|cite|improve this answer













No, parametrizations are not unique.



Both
$$t:[0,2pi],z=e^it$$
and
$$t:[-pi,pi],z=e^i(t+textany real number)$$
parametrize a unit circle on the complex plane.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Aug 6 at 0:07









Szeto

4,2161521




4,2161521











  • And there are plenty more!
    – Lubin
    Aug 6 at 2:18
















  • And there are plenty more!
    – Lubin
    Aug 6 at 2:18















And there are plenty more!
– Lubin
Aug 6 at 2:18




And there are plenty more!
– Lubin
Aug 6 at 2:18












 

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