# Differential Geometry Question

• mXCSNT
In summary, a necessary and sufficient condition for \alpha(I) to lie on a sphere is that R^2 + (R')^2T^2 = const.
mXCSNT

## Homework Statement

Assume that $$\tau(s) \neq 0$$ and $$k'(x) \neq 0$$ for all $$s \in I$$. Show that a necessary and sufficient condition for $$\alpha(I)$$ to lie on a sphere is that $$R^2 + (R')^2T^2 = const$$ where $$R = 1/k$$, $$T = 1/\tau$$, and $$R' = \frac{dr}{ds}$$

## Homework Equations

$$\alpha(s)$$ is a curve in R3 parametrized by arc length
$$k = curvature = |\alpha''|$$
$$\tau = torsion = -\frac{\alpha' \times \alpha'' \cdot \alpha'''}{k^2}$$ (note sign; this is opposite of some conventions)

## The Attempt at a Solution

I've approached this from 2 directions, but I haven't gotten them to meet. First, a necessary and sufficient condition is that $$|\alpha - P|$$ is constant, where P is the center of the circle. Alternatively, $$(\alpha - P) \cdot \alpha' = 0$$.
And I've expanded out $$R^2 + (R')^2T^2 = const$$ to get
$$\frac{(\alpha' \times \alpha'' \cdot \alpha''')^2 + (\alpha'' \cdot \alpha''')^2}{(\alpha'' \cdot \alpha'')(\alpha' \times \alpha'' \cdot \alpha''')^2} = const$$
Also, I'm going to guess that the const on the right hand side is some function of the radius of the sphere, maybe the square of it (which would be $$(\alpha - P) \cdot (\alpha - P)$$), because what else is constant in a sphere?

But I don't know where to go from here. I'm just looking for a hint at an avenue of approach, please nothing specific.

So if I assume that the "const" in question is the square of the radius, that could make R and (R')T the lengths of respective legs of a right triangle whose hypotenuse is the radius. I know that R is the radius of curvature, so drawing a diagram I'm guessing that the center P is
$$P = Rn - (R')Tb$$
(where n is the normal vector $$\alpha''/|\alpha''|$$ and b = $$t \times n$$ is the binormal vector).
So then the sphere radius I want to test, $$(\alpha-P) \cdot (\alpha-P)$$, becomes $$(\alpha - Rn + (R')Tb) \cdot (\alpha - Rn + (R')Tb)$$
$$= \alpha \cdot \alpha - 2 R \alpha \cdot n + 2 R' T \alpha \cdot b + R^2 + (R')^2T^2$$
Now if I assume that $$R^2 + (R')^2T^2$$ is constant I need to show that $$\alpha \cdot \alpha - 2 R \alpha \cdot n + 2 R' T \alpha \cdot b$$ is constant to show the sphere radius is constant. Am I on the right track?

Revised guess for center P:
$$P = \alpha + Rn - (R')Tb$$
so now the constant radius follows immediately and I simply have to show that P itself is constant.

## 1. What is Differential Geometry?

Differential Geometry is a branch of mathematics that deals with the study of curves, surfaces, and higher dimensional objects using the tools of calculus and linear algebra. It combines concepts from both geometry and analysis to understand the properties of these objects and their relationships.

## 2. What are some applications of Differential Geometry?

Differential Geometry has a wide range of applications in various fields such as physics, engineering, computer graphics, and robotics. It is used to model and analyze physical phenomena, design and optimize structures, and develop algorithms for computer simulations.

## 3. How is Differential Geometry different from other branches of mathematics?

Differential Geometry is unique in that it focuses on the study of smooth objects such as curves and surfaces, while other branches of mathematics may deal with discrete or continuous structures. It also uses specialized tools such as differential forms and tensors to describe geometric properties.

## 4. What are some important concepts in Differential Geometry?

Some important concepts in Differential Geometry include curvature, geodesics, and isometries. Curvature measures how much a surface deviates from being flat, geodesics are the shortest paths between points on a surface, and isometries are transformations that preserve distances between points on a surface.

## 5. How can one get started with learning Differential Geometry?

To get started with learning Differential Geometry, it is important to have a strong foundation in calculus, linear algebra, and multivariable analysis. Then, one can begin by studying the basic concepts and definitions, and gradually move on to more advanced topics such as Riemannian Geometry and Manifolds.

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