• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Differential Geometry Question

  • Thread starter Dahaka14
  • Start date
1. Homework Statement
Find an explicit unit-speed non-degenerate space curve [tex]\vec{r}:(0,\infinity)\rightarrow\Re^{3}[/tex] whose curvature and torsion [tex]\kappa,\tau:(0,\infinity)\rightarrow\Re[/tex] are given by the functions [tex]\kappa(s)=\tau(s)=\frac{1}{s}[/tex].

2. Homework Equations
the only thing that I can think of that would help us here are the Frenet equations:
[tex]t'=\kappa n[/tex]
[tex]n'=-\kappa t -\tau b[/tex]
[tex]b'=\tau n[/tex]

3. The Attempt at a Solution
If we are to have [tex]\kappa(s)=\tau(s)=\frac{1}{s}[/tex], then we must have
[tex]t'=\frac{1}{s} t[/tex] and
[tex]b'=\frac{1}{s} t[/tex], thus
[tex]t'=b'[/tex]. I'm not sure what to do after this point, as I messed with these equations for awhile to no avail.
 

lanedance

Homework Helper
3,304
2
hi dahaka14

from your frenet equations you have
[tex]\textbf{t}'=\kappa \textbf{n}[/tex]
[tex]\textbf{b}'=-\tau \textbf{n}[/tex]

write down a vector a, with some constants c & d we will choose
[tex]\textbf{a}= c\textbf{t} + d \textbf{n}[/tex]

differentiating
[tex]\textbf{a}'= c.\textbf{t}' + d .\textbf{b}'= c .\kappa .\textbf{n} - d.\tau .\textbf{n} = \frac{1}{s} (c-d) \textbf{n}[/tex]

so choose c=d and the vector a is constant, might as well make a a unit vector so set:
[tex]c = d = \frac{1}{\sqrt{2}}[/tex]

now think about the dot product of a with t and what this means...
hopefully this helps you get started...
 
The dot product should give
[tex]\textbf{a}\cdot\textbf{t}=c\textbf{t}\cdot\textbf{t}=\frac{\textbf{t}\cdot\textbf{t}}{\sqrt{2}}[/tex]

I'm not sure where to go from here. The only thing that I have been able to think of is that perhaps the curve should be a helix, since a helix is such that [tex]\frac{\tau}{\kappa}[/tex] is constant.

Edit: that LaTeX image should have:
torsion/curvature=constant
 

lanedance

Homework Helper
3,304
2
yeah i think you are on the right track, as i understand it a general helix is defined as when [tex]\frac{\tau}{\kappa}[/tex] is constant, which is equivalent to the tangent vector making a constant angle with some vector, say a, which is what your dot product shows as t.t = 1

Not 100% where to go, but picking an aribtrary (a), then for s=0, a starting t which matches your dot product could be a good place to start
 

Related Threads for: Differential Geometry Question

  • Posted
Replies
2
Views
1K
  • Posted
Replies
7
Views
1K
  • Posted
Replies
2
Views
2K
  • Posted
Replies
0
Views
705
Replies
4
Views
2K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top