Mixing
Surface area of 3D self-intersecting curve (tangent indicatrix)
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
I am looking to find the spherical area enclosed by a (closed) tangent indicatrix that has multiple self- intersections. For example, intersections such as those found in a lemniscate, except in a curve that does not allow me to exploit symmetry. Because it is a non-simple curve, intuition tells me I would have to split the surface into its individual enclosed regions created by the self intersections in order to use Stoke’s theorem. I am wondering if there is any easier way to do this that would allow me to not have to split up the curve by regions as this would become very complex and tedious.
Also, I am wondering if standard formulas for arc-length in 3-space apply for self-intersecting curves. I am looking to find the arc length of my tangent indicatrix in order to determine curvature.
geometry differential-geometry differential-topology
asked Jul 26 at 19:15
Ztan
61
add a comment |Â
up vote
1
down vote
favorite
I am looking to find the spherical area enclosed by a (closed) tangent indicatrix that has multiple self- intersections. For example, intersections such as those found in a lemniscate, except in a curve that does not allow me to exploit symmetry. Because it is a non-simple curve, intuition tells me I would have to split the surface into its individual enclosed regions created by the self intersections in order to use Stoke’s theorem. I am wondering if there is any easier way to do this that would allow me to not have to split up the curve by regions as this would become very complex and tedious.
Also, I am wondering if standard formulas for arc-length in 3-space apply for self-intersecting curves. I am looking to find the arc length of my tangent indicatrix in order to determine curvature.
geometry differential-geometry differential-topology
asked Jul 26 at 19:15
Ztan
61
Yes, the arclength integral formula holds even if the curve intersects itself (of course, if segments of the curve overlap, you count with multiplicities). I'm not sure how you intend to apply Stokes's Theorem to the little pieces (worrying about orientations, of course).
– Ted Shifrin
Jul 26 at 21:22
I'm not quite sure either. Perhaps I would have to reparameterize each little piece on its own, although I'm sure there must be an easier way.
– Ztan
Jul 27 at 16:40
There's a standard computation to do such things for the normal indicatrix by applying Gauss-Bonnet. I've never computed the geodesic curvature of the tangent indicatrix.
– Ted Shifrin
Jul 27 at 22:40
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am looking to find the spherical area enclosed by a (closed) tangent indicatrix that has multiple self- intersections. For example, intersections such as those found in a lemniscate, except in a curve that does not allow me to exploit symmetry. Because it is a non-simple curve, intuition tells me I would have to split the surface into its individual enclosed regions created by the self intersections in order to use Stoke’s theorem. I am wondering if there is any easier way to do this that would allow me to not have to split up the curve by regions as this would become very complex and tedious.
Also, I am wondering if standard formulas for arc-length in 3-space apply for self-intersecting curves. I am looking to find the arc length of my tangent indicatrix in order to determine curvature.
geometry differential-geometry differential-topology
asked Jul 26 at 19:15
Ztan
61
I am looking to find the spherical area enclosed by a (closed) tangent indicatrix that has multiple self- intersections. For example, intersections such as those found in a lemniscate, except in a curve that does not allow me to exploit symmetry. Because it is a non-simple curve, intuition tells me I would have to split the surface into its individual enclosed regions created by the self intersections in order to use Stoke’s theorem. I am wondering if there is any easier way to do this that would allow me to not have to split up the curve by regions as this would become very complex and tedious.
Also, I am wondering if standard formulas for arc-length in 3-space apply for self-intersecting curves. I am looking to find the arc length of my tangent indicatrix in order to determine curvature.
geometry differential-geometry differential-topology
asked Jul 26 at 19:15
Ztan
61
asked Jul 26 at 19:15
Ztan
61
asked Jul 26 at 19:15
Ztan
61
asked Jul 26 at 19:15
Ztan
61
61
Yes, the arclength integral formula holds even if the curve intersects itself (of course, if segments of the curve overlap, you count with multiplicities). I'm not sure how you intend to apply Stokes's Theorem to the little pieces (worrying about orientations, of course).
– Ted Shifrin
Jul 26 at 21:22
I'm not quite sure either. Perhaps I would have to reparameterize each little piece on its own, although I'm sure there must be an easier way.
– Ztan
Jul 27 at 16:40
There's a standard computation to do such things for the normal indicatrix by applying Gauss-Bonnet. I've never computed the geodesic curvature of the tangent indicatrix.
– Ted Shifrin
Jul 27 at 22:40
add a comment |Â
Yes, the arclength integral formula holds even if the curve intersects itself (of course, if segments of the curve overlap, you count with multiplicities). I'm not sure how you intend to apply Stokes's Theorem to the little pieces (worrying about orientations, of course).
– Ted Shifrin
Jul 26 at 21:22
I'm not quite sure either. Perhaps I would have to reparameterize each little piece on its own, although I'm sure there must be an easier way.
– Ztan
Jul 27 at 16:40
There's a standard computation to do such things for the normal indicatrix by applying Gauss-Bonnet. I've never computed the geodesic curvature of the tangent indicatrix.
– Ted Shifrin
Jul 27 at 22:40
Yes, the arclength integral formula holds even if the curve intersects itself (of course, if segments of the curve overlap, you count with multiplicities). I'm not sure how you intend to apply Stokes's Theorem to the little pieces (worrying about orientations, of course).
– Ted Shifrin
Jul 26 at 21:22
Yes, the arclength integral formula holds even if the curve intersects itself (of course, if segments of the curve overlap, you count with multiplicities). I'm not sure how you intend to apply Stokes's Theorem to the little pieces (worrying about orientations, of course).
– Ted Shifrin
Jul 26 at 21:22
I'm not quite sure either. Perhaps I would have to reparameterize each little piece on its own, although I'm sure there must be an easier way.
– Ztan
Jul 27 at 16:40
I'm not quite sure either. Perhaps I would have to reparameterize each little piece on its own, although I'm sure there must be an easier way.
– Ztan
Jul 27 at 16:40
There's a standard computation to do such things for the normal indicatrix by applying Gauss-Bonnet. I've never computed the geodesic curvature of the tangent indicatrix.
– Ted Shifrin
Jul 27 at 22:40
There's a standard computation to do such things for the normal indicatrix by applying Gauss-Bonnet. I've never computed the geodesic curvature of the tangent indicatrix.
– Ted Shifrin
Jul 27 at 22:40
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2863718%2fsurface-area-of-3d-self-intersecting-curve-tangent-indicatrix%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Yes, the arclength integral formula holds even if the curve intersects itself (of course, if segments of the curve overlap, you count with multiplicities). I'm not sure how you intend to apply Stokes's Theorem to the little pieces (worrying about orientations, of course).
– Ted Shifrin
Jul 26 at 21:22
I'm not quite sure either. Perhaps I would have to reparameterize each little piece on its own, although I'm sure there must be an easier way.
– Ztan
Jul 27 at 16:40
There's a standard computation to do such things for the normal indicatrix by applying Gauss-Bonnet. I've never computed the geodesic curvature of the tangent indicatrix.
– Ted Shifrin
Jul 27 at 22:40