some differential geometry (1 Viewer)

[RenegadeMx]

part b seems pretty confusing for me..

 

[seanieg89]

What are you finding confusing?

You are given an explicit expression for your smooth curve x, so you just need to compute the Frenet data as per the definitions.

You will end up with something like:

T_2=B_1
N_2=-N_1
B_2=T_1
k_2=t_1
t_2=k_1.

Where T,N,B,k,t denote the tangent, normal, binormal, curvature and torsion respectively.

(Carry out these calculations carefully yourself, there could easily be a small mistake in my answer. Haven't done this kind of stuff in a while.)

 

[RenegadeMx]

What are you finding confusing?

You are given an explicit expression for your smooth curve x, so you just need to compute the Frenet data as per the definitions.

You will end up with something like:

T_2=B_1
N_2=-N_1
B_2=T_1
k_2=t_1
t_2=k_1.

Where T,N,B,k,t denote the tangent, normal, binormal, curvature and torsion respectively.

(Carry out these calculations carefully yourself, there could easily be a small mistake in my answer. Haven't done this kind of stuff in a while.)

well since the curve is parametrised by binormal instead of the usual arc length im used to, but otherwise its still the same?
Overall im just really bad at understanding pure expression I guess, I understand better with a numerical example

 

[seanieg89]

Fair enough, it is pretty much just application of the definitions:

and hence x moves at unit speed in the direction of the Binormal vector of r at time t.

To compute the curvature of x, as usual we differentiate T.

So where , and have unit length, and the first and last equalities follow from the definition of curvature and torsion respectively.

Hence (recall that curvature of a curve is a positive quantity).

This also implies that by normalisation.

Now that we know the tangent and normal vectors of x, we can compute its binormal.



(the last equality can be justified in multiple ways, eg Jacobi's identity for the cross product or by noting that it is clearly a multiple of since it is orthogonal to both and , and also that it has length one since are unit length and orthogonal. Then it is just a matter of checking signs.)

Hence

Since we have already established , this implies .

This completes the calculation.

 

[RenegadeMx]

mostly b and c

struggling quite alot with the geodesic curvature, answer and method dont seem to match up compared to my lecture notes...