Generic Curve in R^n: What We Know

In summary, the conversation discusses the identification of curves with constant curvatures in R^n, with the exercise asking for a description of these curves. It is suggested that this is a homework problem and the person is directed to post it in the appropriate forum with the full problem statement, equations, and their own thoughts and progress.
  • #1
diffgeo4life
3
1
TL;DR Summary
κ1,κ2,...,κn-1 is constant
What do we know of a curve(/what can it look like) in R^n if we know that κ1,κ2,...,κn-1 is constant?
 
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  • #2
If you don't tell us what the ##\kappa##s are, not a lot.
 
  • #3
Precisely the exercise asks: "Describe those curves of general type in R^n which have
constant curvatures."
 
  • #4
So this is a homework type problem then. It should therefore be posted in the appropriate homework forum using the homework template - including the full problem statement, relevant equations, and your work/thoughts so far.
 
  • #5
Orodruin said:
your work/thoughts so far.
Indeed. @diffgeo4life - a little bit of Googling will get you at least part way there even if your textbook isn't helpful.
 
  • #6
Orodruin said:
So this is a homework type problem then. It should therefore be posted in the appropriate homework forum using the homework template - including the full problem statement, relevant equations, and your work/thoughts so far.
Sorry then, I guess then i will try posting in the Homework- Calculus and beyond forum, that looks most appropriate
 
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1. What is a generic curve in R^n?

A generic curve in R^n is a mathematical concept that refers to a curve that exhibits certain properties that are true for most curves in n-dimensional space. These properties can include smoothness, non-self-intersection, and non-degeneracy.

2. How is a generic curve different from a regular curve?

A regular curve is a specific type of curve that is defined by a set of equations or parametric equations. A generic curve, on the other hand, is a broader concept that encompasses a larger set of curves that share certain properties. A regular curve can be considered a subset of a generic curve.

3. What is the significance of studying generic curves in R^n?

Studying generic curves in R^n allows us to gain a deeper understanding of the properties and behavior of curves in higher dimensions. This can have applications in fields such as geometry, physics, and computer science.

4. Can generic curves be visualized in lower dimensions?

Yes, generic curves in R^n can be visualized in lower dimensions by projecting them onto a lower-dimensional space. For example, a 3-dimensional curve can be projected onto a 2-dimensional plane for visualization.

5. Are there any real-world examples of generic curves in R^n?

Yes, there are many real-world examples of generic curves in R^n. For instance, the path of a planet orbiting around a star can be approximated as a generic curve in 3-dimensional space. Similarly, the trajectory of a projectile can be modeled as a generic curve in 2-dimensional space.

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