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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
asked Aug 5 at 22:28
S.Alfaro
12
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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
asked Aug 5 at 22:28
S.Alfaro
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
add a comment |Â
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
asked Aug 5 at 22:28
S.Alfaro
12
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
asked Aug 5 at 22:28
S.Alfaro
12
edited Aug 6 at 6:59
asked Aug 5 at 22:28
S.Alfaro
12
asked Aug 5 at 22:28
S.Alfaro
12
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
add a comment |Â
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
add a comment |Â
1 Answer
1
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.
answered Aug 6 at 0:07
Szeto
4,2161521
And there are plenty more!
– Lubin
Aug 6 at 2:18
add a comment |Â
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.
answered Aug 6 at 0:07
Szeto
4,2161521
And there are plenty more!
– Lubin
Aug 6 at 2:18
add a comment |Â
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.
answered Aug 6 at 0:07
Szeto
4,2161521
And there are plenty more!
– Lubin
Aug 6 at 2:18
add a comment |Â
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.
answered Aug 6 at 0:07
Szeto
4,2161521
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.
answered Aug 6 at 0:07
Szeto
4,2161521
answered Aug 6 at 0:07
Szeto
4,2161521
answered Aug 6 at 0:07
Szeto
4,2161521
answered Aug 6 at 0:07
Szeto
4,2161521
4,2161521
And there are plenty more!
– Lubin
Aug 6 at 2:18
add a comment |Â
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
add a comment |Â
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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