Does the velocity vector have to have unit speed parametrisation in calculations of the frenet frame?
up vote
1
down vote
favorite
Consider the curvature of a curve $beta$ at a point s.
This is given by $kappa(s):=|T'(s)|$, where $T(s)=beta '(s) $.
similarly we define the fields in the frenet frame ${T,N,B}$ by
$$T(s)=beta'(s)$$
$$N(s):=tfrac{T'(s)}{kappa(s)}$$
$$B(s):=T(s)times N(s)$$
My question is does $beta'(s)$ have to be parametrised to unit speed in all of these calculations ?
differential-geometry frenet-frame
asked Nov 14 at 23:07
exodius
924417
add a comment |
up vote
1
down vote
favorite
Consider the curvature of a curve $beta$ at a point s.
This is given by $kappa(s):=|T'(s)|$, where $T(s)=beta '(s) $.
similarly we define the fields in the frenet frame ${T,N,B}$ by
$$T(s)=beta'(s)$$
$$N(s):=tfrac{T'(s)}{kappa(s)}$$
$$B(s):=T(s)times N(s)$$
My question is does $beta'(s)$ have to be parametrised to unit speed in all of these calculations ?
differential-geometry frenet-frame
asked Nov 14 at 23:07
exodius
924417
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Consider the curvature of a curve $beta$ at a point s.
This is given by $kappa(s):=|T'(s)|$, where $T(s)=beta '(s) $.
similarly we define the fields in the frenet frame ${T,N,B}$ by
$$T(s)=beta'(s)$$
$$N(s):=tfrac{T'(s)}{kappa(s)}$$
$$B(s):=T(s)times N(s)$$
My question is does $beta'(s)$ have to be parametrised to unit speed in all of these calculations ?
differential-geometry frenet-frame
asked Nov 14 at 23:07
exodius
924417
Consider the curvature of a curve $beta$ at a point s.
This is given by $kappa(s):=|T'(s)|$, where $T(s)=beta '(s) $.
similarly we define the fields in the frenet frame ${T,N,B}$ by
$$T(s)=beta'(s)$$
$$N(s):=tfrac{T'(s)}{kappa(s)}$$
$$B(s):=T(s)times N(s)$$
My question is does $beta'(s)$ have to be parametrised to unit speed in all of these calculations ?
differential-geometry frenet-frame
differential-geometry frenet-frame
asked Nov 14 at 23:07
exodius
924417
asked Nov 14 at 23:07
exodius
924417
asked Nov 14 at 23:07
exodius
924417
asked Nov 14 at 23:07
exodius
924417
asked Nov 14 at 23:07
exodius
924417
924417
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
Yes. More precisely, these formulas only work for curves parametrirized by the arclength. Otherwise, you would not have, for instance, that $bigllVert T(s)bigrrVert=1$.
answered Nov 14 at 23:19
José Carlos Santos
140k19111205
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Yes. More precisely, these formulas only work for curves parametrirized by the arclength. Otherwise, you would not have, for instance, that $bigllVert T(s)bigrrVert=1$.
answered Nov 14 at 23:19
José Carlos Santos
140k19111205
add a comment |
up vote
1
down vote
accepted
Yes. More precisely, these formulas only work for curves parametrirized by the arclength. Otherwise, you would not have, for instance, that $bigllVert T(s)bigrrVert=1$.
answered Nov 14 at 23:19
José Carlos Santos
140k19111205
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Yes. More precisely, these formulas only work for curves parametrirized by the arclength. Otherwise, you would not have, for instance, that $bigllVert T(s)bigrrVert=1$.
answered Nov 14 at 23:19
José Carlos Santos
140k19111205
Yes. More precisely, these formulas only work for curves parametrirized by the arclength. Otherwise, you would not have, for instance, that $bigllVert T(s)bigrrVert=1$.
answered Nov 14 at 23:19
José Carlos Santos
140k19111205
answered Nov 14 at 23:19
José Carlos Santos
140k19111205
answered Nov 14 at 23:19
José Carlos Santos
140k19111205
answered Nov 14 at 23:19
José Carlos Santos
140k19111205
140k19111205
add a comment |
add a comment |
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%2f2998943%2fdoes-the-velocity-vector-have-to-have-unit-speed-parametrisation-in-calculations%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
