Mixing
Relationships between curvature, torsion, unit tangent vector, and binormal vector of a curve
$begingroup$
Homework has already been collected and graded (but no explanation given) for these problems. I'm curious how to approach the problem.
Assume that the vector space we're in is $Re^{3}$. Prove that
$$
begin{eqnarray*}
(1) &;;;;;;;;& (vec{mathbf{tau}} cdot vec{mathbf{beta}} cdot vec{mathbf{beta^{'}}}) &=& kappa , \
(2)&&(vec{mathbf{beta}} cdot vec{mathbf{beta^{'}}} cdot vec{mathbf{beta^{''}}}) &=& kappa^{2}(k / kappa)^{'} ,\
(3)&&(vec{mathbf{tau}} cdot vec{mathbf{tau^{'}}} cdot vec{mathbf{tau^{''}}})&=& k^{3}(kappa/k)^{'} ,
end{eqnarray*}
$$
where $tau$ is the unit tangent vector, $beta$ is the binormal vector, $kappa$ is torsion, and $k$ is curvature.
I started to attempt these proofs by starting from the vector form of the curve
$$vec{r}(t) = x(t)vec{i} +y(t)vec{j} +z(t)vec{k}$$
and differentiating with respect to $t$ (and so on ...), but the algebra got really messy very quickly. Are there simpler relations between these mathematical objects that I'm missing or will I simply have to "grind out" the algebra?
differential-geometry
edited Feb 10 '12 at 13:27
Willie Wong
56k10111212
asked Feb 10 '12 at 3:50
JubblesJubbles
2511414
$endgroup$
|
show 2 more comments
$begingroup$
Homework has already been collected and graded (but no explanation given) for these problems. I'm curious how to approach the problem.
Assume that the vector space we're in is $Re^{3}$. Prove that
$$
begin{eqnarray*}
(1) &;;;;;;;;& (vec{mathbf{tau}} cdot vec{mathbf{beta}} cdot vec{mathbf{beta^{'}}}) &=& kappa , \
(2)&&(vec{mathbf{beta}} cdot vec{mathbf{beta^{'}}} cdot vec{mathbf{beta^{''}}}) &=& kappa^{2}(k / kappa)^{'} ,\
(3)&&(vec{mathbf{tau}} cdot vec{mathbf{tau^{'}}} cdot vec{mathbf{tau^{''}}})&=& k^{3}(kappa/k)^{'} ,
end{eqnarray*}
$$
where $tau$ is the unit tangent vector, $beta$ is the binormal vector, $kappa$ is torsion, and $k$ is curvature.
I started to attempt these proofs by starting from the vector form of the curve
$$vec{r}(t) = x(t)vec{i} +y(t)vec{j} +z(t)vec{k}$$
and differentiating with respect to $t$ (and so on ...), but the algebra got really messy very quickly. Are there simpler relations between these mathematical objects that I'm missing or will I simply have to "grind out" the algebra?
differential-geometry
edited Feb 10 '12 at 13:27
Willie Wong
56k10111212
asked Feb 10 '12 at 3:50
JubblesJubbles
2511414
$endgroup$
4
$begingroup$
What is that dot product of three vectors you use in each of the equations?
$endgroup$
– Henning Makholm
Feb 10 '12 at 3:55
2
$begingroup$
@Henning: Maybe the the box product (scalar triple product) was intended... Jubbles, have you looked up proofs for Frenet-Serret in textbooks?
$endgroup$
– J. M. is not a mathematician
Feb 10 '12 at 3:56
1
$begingroup$
Maybe this? I've never seen the notation before, though.
$endgroup$
– Dylan Moreland
Feb 10 '12 at 3:57
4
$begingroup$
Tau is the unit tangent while kappa is the torsion and k is the curvature? Holy switcharoo, Batman! That's confusing.
$endgroup$
– anon
Feb 10 '12 at 3:57
1
$begingroup$
@anon: Correct. My professor has remarked that the textbook chose uncommon notation for curvature and torsion.
$endgroup$
– Jubbles
Feb 10 '12 at 4:00
|
show 2 more comments
$begingroup$
Homework has already been collected and graded (but no explanation given) for these problems. I'm curious how to approach the problem.
Assume that the vector space we're in is $Re^{3}$. Prove that
$$
begin{eqnarray*}
(1) &;;;;;;;;& (vec{mathbf{tau}} cdot vec{mathbf{beta}} cdot vec{mathbf{beta^{'}}}) &=& kappa , \
(2)&&(vec{mathbf{beta}} cdot vec{mathbf{beta^{'}}} cdot vec{mathbf{beta^{''}}}) &=& kappa^{2}(k / kappa)^{'} ,\
(3)&&(vec{mathbf{tau}} cdot vec{mathbf{tau^{'}}} cdot vec{mathbf{tau^{''}}})&=& k^{3}(kappa/k)^{'} ,
end{eqnarray*}
$$
where $tau$ is the unit tangent vector, $beta$ is the binormal vector, $kappa$ is torsion, and $k$ is curvature.
I started to attempt these proofs by starting from the vector form of the curve
$$vec{r}(t) = x(t)vec{i} +y(t)vec{j} +z(t)vec{k}$$
and differentiating with respect to $t$ (and so on ...), but the algebra got really messy very quickly. Are there simpler relations between these mathematical objects that I'm missing or will I simply have to "grind out" the algebra?
differential-geometry
edited Feb 10 '12 at 13:27
Willie Wong
56k10111212
asked Feb 10 '12 at 3:50
JubblesJubbles
2511414
$endgroup$
Homework has already been collected and graded (but no explanation given) for these problems. I'm curious how to approach the problem.
Assume that the vector space we're in is $Re^{3}$. Prove that
$$
begin{eqnarray*}
(1) &;;;;;;;;& (vec{mathbf{tau}} cdot vec{mathbf{beta}} cdot vec{mathbf{beta^{'}}}) &=& kappa , \
(2)&&(vec{mathbf{beta}} cdot vec{mathbf{beta^{'}}} cdot vec{mathbf{beta^{''}}}) &=& kappa^{2}(k / kappa)^{'} ,\
(3)&&(vec{mathbf{tau}} cdot vec{mathbf{tau^{'}}} cdot vec{mathbf{tau^{''}}})&=& k^{3}(kappa/k)^{'} ,
end{eqnarray*}
$$
where $tau$ is the unit tangent vector, $beta$ is the binormal vector, $kappa$ is torsion, and $k$ is curvature.
I started to attempt these proofs by starting from the vector form of the curve
$$vec{r}(t) = x(t)vec{i} +y(t)vec{j} +z(t)vec{k}$$
and differentiating with respect to $t$ (and so on ...), but the algebra got really messy very quickly. Are there simpler relations between these mathematical objects that I'm missing or will I simply have to "grind out" the algebra?
differential-geometry
differential-geometry
edited Feb 10 '12 at 13:27
Willie Wong
56k10111212
asked Feb 10 '12 at 3:50
JubblesJubbles
2511414
edited Feb 10 '12 at 13:27
Willie Wong
56k10111212
asked Feb 10 '12 at 3:50
JubblesJubbles
2511414
edited Feb 10 '12 at 13:27
Willie Wong
56k10111212
edited Feb 10 '12 at 13:27
Willie Wong
56k10111212
edited Feb 10 '12 at 13:27
Willie Wong
56k10111212
56k10111212
asked Feb 10 '12 at 3:50
JubblesJubbles
2511414
asked Feb 10 '12 at 3:50
JubblesJubbles
2511414
asked Feb 10 '12 at 3:50
JubblesJubbles
2511414
2511414
4
$begingroup$
What is that dot product of three vectors you use in each of the equations?
$endgroup$
– Henning Makholm
Feb 10 '12 at 3:55
2
$begingroup$
@Henning: Maybe the the box product (scalar triple product) was intended... Jubbles, have you looked up proofs for Frenet-Serret in textbooks?
$endgroup$
– J. M. is not a mathematician
Feb 10 '12 at 3:56
1
$begingroup$
Maybe this? I've never seen the notation before, though.
$endgroup$
– Dylan Moreland
Feb 10 '12 at 3:57
4
$begingroup$
Tau is the unit tangent while kappa is the torsion and k is the curvature? Holy switcharoo, Batman! That's confusing.
$endgroup$
– anon
Feb 10 '12 at 3:57
1
$begingroup$
@anon: Correct. My professor has remarked that the textbook chose uncommon notation for curvature and torsion.
$endgroup$
– Jubbles
Feb 10 '12 at 4:00
|
show 2 more comments
4
$begingroup$
What is that dot product of three vectors you use in each of the equations?
$endgroup$
– Henning Makholm
Feb 10 '12 at 3:55
2
$begingroup$
@Henning: Maybe the the box product (scalar triple product) was intended... Jubbles, have you looked up proofs for Frenet-Serret in textbooks?
$endgroup$
– J. M. is not a mathematician
Feb 10 '12 at 3:56
1
$begingroup$
Maybe this? I've never seen the notation before, though.
$endgroup$
– Dylan Moreland
Feb 10 '12 at 3:57
4
$begingroup$
Tau is the unit tangent while kappa is the torsion and k is the curvature? Holy switcharoo, Batman! That's confusing.
$endgroup$
– anon
Feb 10 '12 at 3:57
1
$begingroup$
@anon: Correct. My professor has remarked that the textbook chose uncommon notation for curvature and torsion.
$endgroup$
– Jubbles
Feb 10 '12 at 4:00
4
4
$begingroup$
What is that dot product of three vectors you use in each of the equations?
$endgroup$
– Henning Makholm
Feb 10 '12 at 3:55
$begingroup$
What is that dot product of three vectors you use in each of the equations?
$endgroup$
– Henning Makholm
Feb 10 '12 at 3:55
2
2
$begingroup$
@Henning: Maybe the the box product (scalar triple product) was intended... Jubbles, have you looked up proofs for Frenet-Serret in textbooks?
$endgroup$
– J. M. is not a mathematician
Feb 10 '12 at 3:56
$begingroup$
@Henning: Maybe the the box product (scalar triple product) was intended... Jubbles, have you looked up proofs for Frenet-Serret in textbooks?
$endgroup$
– J. M. is not a mathematician
Feb 10 '12 at 3:56
1
1
$begingroup$
Maybe this? I've never seen the notation before, though.
$endgroup$
– Dylan Moreland
Feb 10 '12 at 3:57
$begingroup$
Maybe this? I've never seen the notation before, though.
$endgroup$
– Dylan Moreland
Feb 10 '12 at 3:57
4
4
$begingroup$
Tau is the unit tangent while kappa is the torsion and k is the curvature? Holy switcharoo, Batman! That's confusing.
$endgroup$
– anon
Feb 10 '12 at 3:57
$begingroup$
Tau is the unit tangent while kappa is the torsion and k is the curvature? Holy switcharoo, Batman! That's confusing.
$endgroup$
– anon
Feb 10 '12 at 3:57
1
1
$begingroup$
@anon: Correct. My professor has remarked that the textbook chose uncommon notation for curvature and torsion.
$endgroup$
– Jubbles
Feb 10 '12 at 4:00
$begingroup$
@anon: Correct. My professor has remarked that the textbook chose uncommon notation for curvature and torsion.
$endgroup$
– Jubbles
Feb 10 '12 at 4:00
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
You may consider the curve parametrized for arc length.
The formulas for curvature and torsion become
k(t)=||r''(t)||, tau(t)=(r'(t),r''(t),r'''(t))/k^2(t)
You can substitute these in the right hand of the equation and obtain the left hand side.
Sources:
http://mathhelpforum.com/differential-geometry/258751-relation-between-curvature-torsion.html
http://yourhomeworkhelp.org/math-tests/geometry-tests/
answered Jul 25 '16 at 8:59
Piter H.Piter H.
1
$endgroup$
add a comment |
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f107725%2frelationships-between-curvature-torsion-unit-tangent-vector-and-binormal-vect%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You may consider the curve parametrized for arc length.
The formulas for curvature and torsion become
k(t)=||r''(t)||, tau(t)=(r'(t),r''(t),r'''(t))/k^2(t)
You can substitute these in the right hand of the equation and obtain the left hand side.
Sources:
http://mathhelpforum.com/differential-geometry/258751-relation-between-curvature-torsion.html
http://yourhomeworkhelp.org/math-tests/geometry-tests/
answered Jul 25 '16 at 8:59
Piter H.Piter H.
1
$endgroup$
add a comment |
$begingroup$
You may consider the curve parametrized for arc length.
The formulas for curvature and torsion become
k(t)=||r''(t)||, tau(t)=(r'(t),r''(t),r'''(t))/k^2(t)
You can substitute these in the right hand of the equation and obtain the left hand side.
Sources:
http://mathhelpforum.com/differential-geometry/258751-relation-between-curvature-torsion.html
http://yourhomeworkhelp.org/math-tests/geometry-tests/
answered Jul 25 '16 at 8:59
Piter H.Piter H.
1
$endgroup$
add a comment |
$begingroup$
You may consider the curve parametrized for arc length.
The formulas for curvature and torsion become
k(t)=||r''(t)||, tau(t)=(r'(t),r''(t),r'''(t))/k^2(t)
You can substitute these in the right hand of the equation and obtain the left hand side.
Sources:
http://mathhelpforum.com/differential-geometry/258751-relation-between-curvature-torsion.html
http://yourhomeworkhelp.org/math-tests/geometry-tests/
answered Jul 25 '16 at 8:59
Piter H.Piter H.
1
$endgroup$
You may consider the curve parametrized for arc length.
The formulas for curvature and torsion become
k(t)=||r''(t)||, tau(t)=(r'(t),r''(t),r'''(t))/k^2(t)
You can substitute these in the right hand of the equation and obtain the left hand side.
Sources:
http://mathhelpforum.com/differential-geometry/258751-relation-between-curvature-torsion.html
http://yourhomeworkhelp.org/math-tests/geometry-tests/
answered Jul 25 '16 at 8:59
Piter H.Piter H.
1
answered Jul 25 '16 at 8:59
Piter H.Piter H.
1
answered Jul 25 '16 at 8:59
Piter H.Piter H.
1
answered Jul 25 '16 at 8:59
Piter H.Piter H.
1
1
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f107725%2frelationships-between-curvature-torsion-unit-tangent-vector-and-binormal-vect%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
4
$begingroup$
What is that dot product of three vectors you use in each of the equations?
$endgroup$
– Henning Makholm
Feb 10 '12 at 3:55
2
$begingroup$
@Henning: Maybe the the box product (scalar triple product) was intended... Jubbles, have you looked up proofs for Frenet-Serret in textbooks?
$endgroup$
– J. M. is not a mathematician
Feb 10 '12 at 3:56
1
$begingroup$
Maybe this? I've never seen the notation before, though.
$endgroup$
– Dylan Moreland
Feb 10 '12 at 3:57
4
$begingroup$
Tau is the unit tangent while kappa is the torsion and k is the curvature? Holy switcharoo, Batman! That's confusing.
$endgroup$
– anon
Feb 10 '12 at 3:57
1
$begingroup$
@anon: Correct. My professor has remarked that the textbook chose uncommon notation for curvature and torsion.
$endgroup$
– Jubbles
Feb 10 '12 at 4:00