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Apr
11
accepted Understanding a particular case of modifying curvature and torsion as opposed to modifying the curve
Apr
11
comment Understanding a particular case of modifying curvature and torsion as opposed to modifying the curve
Thanks a lot. The question seems really silly once you look bacak, in fact right after Anthony Carapetis' comments.
Apr
11
comment Understanding a particular case of modifying curvature and torsion as opposed to modifying the curve
@AnthonyCarapetis, On second thoughts, I think I get what you said, the curvature and tosion functions and space curves are in a 1-1 correspondence with each other, so both the ways yield the same curve. Is that right??
Apr
11
comment Understanding a particular case of modifying curvature and torsion as opposed to modifying the curve
@AnthonyCarapetis well from what I understand about the fundamental theorem of space curves, any set of curvature torsion functions uniquely determines a space curve, but I am not able to gather how that helps me establish the relationship between the curves??Can you add something more??
Apr
11
asked Understanding a particular case of modifying curvature and torsion as opposed to modifying the curve
Apr
2
accepted Proof for showing that a set of space curves form a manifold
Apr
2
comment Proof for showing that a set of space curves form a manifold
Thanks for that. I am sorry I guess I was pushing the nitpicking a bit too far. Yeah will look into all that you said. Cheers!!
Apr
2
comment Proof for showing that a set of space curves form a manifold
Aah yes, thats true. Is there some way to circumvent this and still get a meaningful group structure. I was also thinking of inducing a quotient topology on $M_{\alpha}$ using $(0,\infty)$ through the same kind of map $\psi$ or rather the inverse of $\psi$
Apr
2
revised Proof for showing that a set of space curves form a manifold
rectified error
Apr
2
comment Proof for showing that a set of space curves form a manifold
I am sorry for making it all sound convoluted. So a group multiplication cannot be defined for this the way I did?
Apr
2
comment Proof for showing that a set of space curves form a manifold
Yes the single curve $\alpha$ with curvature $\kappa$ and torsion $\tau$ generates the family $M_{\alpha}$, but if $\alpha$ is defined on $I$, then so si every other curve $\dfrac{\alpha}{\mu}$, right??Thats how I defined multiplication. But I guess what you are saying is that at any given point $\dfrac{\alpha}{\mu_1}$ and $\dfrac{\alpha}{\mu_2}$ correspond to different points on the individual curves, so the multiplication is not well defined, am I right?
Apr
2
comment Proof for showing that a set of space curves form a manifold
@Jesse Madnick. I am sorry about the first mistake, you are right. As for the parameters being different, well this might seem a little shady, but I have assumed that the all curves in $M_{\alpha}$ have been defined on the same domain of definition as $\alpha$, so $s$ is the parameter for all curves in $M_{\alpha}$, though $s$ is not the arc length parameter for any other curve apart from $\alpha$. I hope that does not make this problem go kaput. Thanks for your patience.
Apr
1
comment Proof for showing that a set of space curves form a manifold
@Jesse Madnick. I have edited the question to fill the gaps which you pointed out. I hope it fits in logically. Thanks a lot for your response.
Apr
1
revised Proof for showing that a set of space curves form a manifold
added 621 characters in body
Apr
1
asked Proof for showing that a set of space curves form a manifold
Mar
31
comment About multiples of curvature and torsion of a space curve
Thanks, from what I understood by going through your link is that I indeed have different space curves and there is no reparametrisation happening, as curvature is an invariant under reparametrisation. Reparametrisation involves redefining the parameter over a different interval. Am I right?
Mar
31
asked About multiples of curvature and torsion of a space curve
Mar
28
awarded  Yearling
Mar
20
accepted Affine connection, metric and parallel transport and mutual interdependence
Mar
20
comment Affine connection, metric and parallel transport and mutual interdependence
Thanks for your edit.It is much more accessible than before.