Can Differential Geometry Solve This Challenging Curve Containment Problem?

In summary, the problem asks to prove a statement regarding a non-degenerate unit speed curve on a sphere with a given radius. The statement states that there exists a center point on the sphere and an angle that satisfies a specific equation involving the curvature and torsion of the curve. Using Frenet equations, it can be shown that the normal vector of the curve points towards the center of the circle as the radius decreases, and the torsion relates to the change in radius. This provides a starting point for solving the problem.
  • #1
Dahaka14
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0

Homework Statement



Let [tex]\sigma:I\rightarrow R^{3}[/tex] be a non-degenerate unit speed curve, and [tex]R[/tex] be a real number [tex]>0[/tex]. Fix a value [tex]s_{0}\in I[/tex]. Prove that:

(There exists a center [tex]\vec{p}\in R^{3}[/tex] such that [tex]\sigma(I)\subset S_{R}(p)[/tex])[tex]\iff[/tex] (There exists an angle [tex]\phi\in R[/tex] such that, for all [tex]s\in I[/tex], [tex]\frac{1}{\kappa(s)}=R\cos(\phi+\int_{s_{0}}^{s}\tau(\lambda)d\lambda)[/tex]).

Homework Equations



I know all of the equations for Frenet, but I'm not sure how to apply them.

The Attempt at a Solution



No idea where to start...I have been staring at this problem for many days now, and I haven't a clue what to do. Please help!
 
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  • #2
might be a start that the curve is constrained to a sphere so t will be tangent to the sphere

define
r = sigma-p

then
t.r = 0

differentiating and some frenet substitution gets to
1/k(s) = -n.r

this is a step closer to the equation...
 
  • #3
also worth thinking about this physically.. when the torsion is zero, the curve can be contained in a plane, and a plane intersecting a sphere gives a circle

the largest circle is a great circle of the sphere, & on this path the normal will point towards the centre of the circle

so where does the normal point as the radius of the circle is made smaller? and how does the torsion relate to a change in radius?
 

Related to Can Differential Geometry Solve This Challenging Curve Containment Problem?

What is differential geometry?

Differential geometry is a branch of mathematics that studies the properties of curves and surfaces using calculus and linear algebra. It also deals with the geometric properties of higher-dimensional spaces.

What are some applications of differential geometry?

Differential geometry has various applications in fields such as physics, engineering, computer graphics, and robotics. It is used to model and analyze the motion of objects, understand the curvature of space and time in general relativity, and design efficient algorithms for computer graphics and robotic motion planning.

What are some common differential geometry problems?

Some common problems in differential geometry include finding the curvature and torsion of a curve, studying the properties of surfaces such as Gaussian and mean curvature, and solving geodesic equations for shortest paths on curved surfaces.

What are some techniques used to solve differential geometry problems?

Some techniques used in solving differential geometry problems include differential and integral calculus, linear and multilinear algebra, differential equations, and manifold theory. Numerical methods are also commonly used to approximate solutions to complex problems.

Are there any real-world applications of differential geometry problems?

Yes, differential geometry has numerous real-world applications. For example, in physics, it is used to understand the curvature of space and time in general relativity and to model the motion of particles in curved space. In engineering, it is used to design optimal paths for robots and vehicles. In computer graphics, it is used to create realistic 3D models of surfaces and objects. In economics, it is used to model and analyze consumer preferences and utility functions.

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