# let ? be a curve on a sphere. prove that kn is constant.

### James

Guys, does anyone know the answer?

get let ? be a curve on a sphere. prove that kn is constant. from EN Bilgi.

## differential geometry

Let $\alpha (t)$ be a curve such that $|\alpha'(t)|=1$ for all $t\in\mathbb R$. Assume $k(t)\neq 0$, $k'(t)\neq 0$ (whereas $k=|\alpha''(t)|$ is the curvature) and $\tau(s)\neq 0$, whereas $\tau$ i...

## condition for curve on a sphere

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Asked 8 years, 8 months ago

Modified 8 years, 8 months ago

Viewed 5k times 7 Let α(t) α(t)

be a curve such that

| α ′ (t)|=1 |α′(t)|=1 for all t∈R t∈R . Assume k(t)≠0 k(t)≠0 , k ′ (t)≠0 k′(t)≠0 (whereas k=| α ′′ (t)| k=|α″(t)|

is the curvature) and

τ(s)≠0 τ(s)≠0 , whereas τ τ is the torsion. Prove: The trace of α α lies on a sphere ⇔ ⇔ 1 k 2 + 1 ( k ′ ⋅τ ) 2 = 1k2+1(k′⋅τ)2= const. >0 >0 .

I know this somehow works by using the Frenet-Serret-equations, but I don't really know how to do this proof. Can anyone help me out? Thanks!

differential-geometry

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asked Oct 23, 2013 at 14:54

dinosaur 2,04215 15 silver badges 25 25 bronze badges Add a comment

## 1 Answer

14

Your equation is incorrect, the correct condition is

1 κ 2 + ( κ ˙ τ κ 2 ) 2 =constant

1κ2+(κ˙τκ2)2=constant

I will only show the

⇐ ⇐ part here. Let s s

be the arc length parametrization and

t ⃗ (s), n ⃗ (s), b ⃗ (s) t→(s),n→(s),b→(s)

be the vectors appear in Frenet Serret equations. Define

β ⃗ (s)= α ⃗ (s)+ 1 κ(s) n ⃗ (s)− κ ˙ (s) τ(s)κ(s ) 2 b ⃗ (s) (*1)

(*1)β→(s)=α→(s)+1κ(s)n→(s)−κ˙(s)τ(s)κ(s)2b→(s)

Differentiate it with respect to

s s , we get: d ds β ⃗ (s)= = = = = t ⃗ − κ ˙ κ 2 n ⃗ + 1 κ (−κ t ⃗ +τ b ⃗ )− d ds ( κ ˙ τ κ 2 ) b ⃗ − κ ˙ τ κ 2 (−τ n ⃗ ) ( τ κ − d ds ( κ ˙ τ κ 2 )) b ⃗ τ κ 2 κ ˙ ( κ ˙ κ 3 − κ ˙ τ κ 2 d ds ( κ ˙ τ κ 2 )) b ⃗ − τ κ 2 2 κ ˙ d ds ( 1 κ 2 + ( κ ˙ τ κ 2 ) 2 ) b ⃗ 0 ⃗

ddsβ→(s)=t→−κ˙κ2n→+1κ(−κt→+τb→)−dds(κ˙τκ2)b→−κ˙τκ2(−τn→)=(τκ−dds(κ˙τκ2))b→=τκ2κ˙(κ˙κ3−κ˙τκ2dds(κ˙τκ2))b→=−τκ22κ˙dds(1κ2+(κ˙τκ2)2)b→=0→

This implies β ⃗ (s)= β ⃗ (0) β→(s)=β→(0)

is a constant. From this, we get

α ⃗ − β ⃗ (0)=− 1 κ n ⃗ + κ ˙ τ κ 2 b ⃗ ⟹ ∣ ∣ α ⃗ − β ⃗ (0) ∣ ∣ 2 = 1 κ 2 + ( κ ˙ τ κ 2 ) 2 =constant.

α→−β→(0)=−1κn→+κ˙τκ2b→⟹|α→−β→(0)|2=1κ2+(κ˙τκ2)2=constant.

i.e α ⃗ (s) α→(s)

lies on a sphere with

β(0) β(0) as center.

**Motivation of above proof**

You may wonder how can anyone figure out the magic formula in

(∗1) (∗1)

. If you work out the

⇒ ⇒

part of the proof where

α(s) α(s)

lies on a sphere centered at

c ⃗ c→

, you should obtain a bunch of dot products between

α ⃗ (s)− c ⃗ α→(s)−c→ and t ⃗ (s) t→(s) , n ⃗ (s) n→(s) and b ⃗ (s) b→(s)

. In particular, you should get:

⎧ ⎩ ⎨ ⎪ ⎪ ⎪ ⎪ t ⃗ ⋅( α ⃗ − c ⃗ ) n ⃗ ⋅( α ⃗ − c ⃗ ) b ⃗ ⋅( α ⃗ − c ⃗ ) =0 =− 1 κ = κ ˙ τ κ 2

{t→⋅(α→−c→)=0n→⋅(α→−c→)=−1κb→⋅(α→−c→)=κ˙τκ2

Using these, you can express the center

c ⃗ c→ in terms of κ,τ κ,τ

like what we have in

(∗1) (∗1)

. If the curve does lie on a sphere, then the "center" should not move as

s s

changes. The proof of the

⇐ ⇐

part above is really using the given condition to verify the "center" so defined doesn't move.

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edited Oct 24, 2013 at 14:00

answered Oct 23, 2013 at 16:15

achille hui 117k6 6 gold badges 165 165 silver badges 318 318 bronze badges

Thank you for your answer! I tried to do the

⇒ ⇒

part, but I can't even calculate the dot products you pointed out. E.g.

t

Source : **math.stackexchange.com**

## 13.3 Arc length and curvature

Sometimes it is useful to compute the length of a curve in space; for example, if the curve represents the path of a moving object, the length of the curve between two points may be the distance traveled by the object between two times.

Recall that if the curve is given by the vector function

r r then the vector Δr=r(t+Δt)−r(t) Δr=r(t+Δt)−r(t)

points from one position on the curve to another, as depicted in figure 13.2.1. If the points are close together, the length of

Δr Δr

is close to the length of the curve between the two points. If we add up the lengths of many such tiny vectors, placed head to tail along a segment of the curve, we get an approximation to the length of the curve over that segment. In the limit, as usual, this sum turns into an integral that computes precisely the length of the curve. First, note that

|Δr|= |Δr| Δt Δt≈| r ′ (t)|Δt,

|Δr|=|Δr|ΔtΔt≈|r′(t)|Δt,

when Δt Δt

is small. Then the length of the curve between

r(a) r(a) and r(b) r(b) is lim n→∞ ∑ i=0 n−1 |Δr|= lim n→∞ ∑ i=0 n−1 |Δr| Δt Δt= lim n→∞ ∑ i=0 n−1 | r ′ (t)|Δt= ∫ b a | r ′ (t)|dt.

limn→∞∑i=0n−1|Δr|=limn→∞∑i=0n−1|Δr|ΔtΔt=limn→∞∑i=0n−1|r′(t)|Δt=∫ab|r′(t)|dt.

(Well, sometimes. This works if between

a a and b b

the segment of curve is traced out exactly once.)

Example 13.3.1 Let's find the length of one turn of the helix

r=⟨cost,sint,t⟩ r=⟨cost,sint,t⟩

(see figure 13.1.1). We compute

r ′ =⟨−sint,cost,1⟩ r′=⟨−sint,cost,1⟩ and | r ′ |= sin 2 t+ cos 2 t+1 − − − − − − − − − − − − − − √ = 2 – √

|r′|=sin2t+cos2t+1=2

, so the length is ∫ 2π 0 2 – √ dt=2 2 – √ π. ∫02π2dt=22π. □ ◻

Example 13.3.2 Suppose

y=lnx y=lnx

; what is the length of this curve between

x=1 x=1 and x= 3 – √ x=3 ?

Although this problem does not appear to involve vectors or three dimensions, we can interpret it in those terms: let

r(t)=⟨t,lnt,0⟩ r(t)=⟨t,lnt,0⟩

. This vector function traces out precisely

y=lnx y=lnx in the x x - y y plane. Then r ′ (t)=⟨1,1/t,0⟩ r′(t)=⟨1,1/t,0⟩ and | r ′ (t)|= 1+1/ t 2 − − − − − − − √ |r′(t)|=1+1/t2

and the desired length is

∫ 3 √ 1 1+ 1 t 2 − − − − − − √ dt=2− 2 – √ +ln( 2 – √ +1)− 1 2 ln3.

∫131+1t2dt=2−2+ln(2+1)−12ln3.

(This integral is a bit tricky, but requires only methods we have learned.)

□ ◻

Notice that there is nothing special about

y=lnx y=lnx

, except that the resulting integral can be computed. In general, given any

y=f(x) y=f(x)

, we can think of this as the vector function

r(t)=⟨t,f(t),0⟩ r(t)=⟨t,f(t),0⟩ . Then r ′ (t)=⟨1, f ′ (t),0⟩ r′(t)=⟨1,f′(t),0⟩ and | r ′ (t)|= 1+( f ′ ) 2 − − − − − − − √ |r′(t)|=1+(f′)2

. The length of the curve

y=f(x) y=f(x) between a a and b b is thus ∫ b a 1+( f ′ (x) ) 2 − − − − − − − − − − √ dx. ∫ab1+(f′(x))2dx.

Unfortunately, such integrals are often impossible to do exactly and must be approximated.

One useful application of arc length is the **arc length parameterization**. A vector function

r(t) r(t)

gives the position of a point in terms of the parameter

t t

, which is often time, but need not be. Suppose

s s

is the distance along the curve from some fixed starting point; if we use

s s

for the variable, we get

r(s) r(s)

, the position in space in terms of distance along the curve. We might still imagine that the curve represents the position of a moving object; now we get the position of the object as a function of how far the object has traveled.

Example 13.3.3 Suppose

r(t)=⟨cost,sint,0⟩

r(t)=⟨cost,sint,0⟩

. We know that this curve is a circle of radius 1. While

t t

might represent time, it can also in this case represent the usual angle between the positive

x x -axis and r(t) r(t)

. The distance along the circle from

(1,0,0) (1,0,0) to (cost,sint,0) (cost,sint,0) is also

## Bodies of Constant Width: An Introduction to Convex Geometry with Applications

This is the first comprehensive monograph to thoroughly investigate constant width bodies, which is a classic area of interest within convex geometry. It examines bodies of constant width from several points of view, and, in doing so, shows surprising connections between various areas of mathematics. Concise explanations and detailed proofs demonstrate the many interesting properties and applications of these bodies. Numerous instructive diagrams are provided throughout to illustrate these concepts.An introduction to convexity theory is first provided, and the basic properties of constant width bodies are then presented. The book then delves into a number of related topics, which include Constant width bodies in convexity (sections and projections, complete and reduced sets, mixed volumes, and further partial fields)Sets of constant width in non-Euclidean geometries (in real Banach spaces, and in hyperbolic, spherical, and further non-Euclidean spaces)The concept of constant width in analysis (using Fourier series, spherical integration, and other related methods)Sets of constant width in differential geometry (using systems of lines and discussing notions like curvature, evolutes, etc.)Bodies of constant width in topology (hyperspaces, transnormal manifolds, fiber bundles, and related topics)The notion of constant width in discrete geometry (referring to geometric inequalities, packings and coverings, etc.)Technical applications, such as film projectors, the square-hole drill, and rotary enginesBodies of Constant Width: An Introduction to Convex Geometry with Applications will be a valuable resource for graduate and advanced undergraduate students studying convex geometry and related fields. Additionally, it will appeal to any mathematicians with a general interest in geometry.

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## Bodies of Constant Width: An Introduction to Convex Geometry with Applications

Horst Martini, Luis Montejano, Déborah Oliveros

Springer, 16 Mar 2019 - 486 sayfa

0 Eleştiriler

This is the first comprehensive monograph to thoroughly investigate constant width bodies, which is a classic area of interest within convex geometry. It examines bodies of constant width from several points of view, and, in doing so, shows surprising connections between various areas of mathematics. Concise explanations and detailed proofs demonstrate the many interesting properties and applications of these bodies. Numerous instructive diagrams are provided throughout to illustrate these concepts.

An introduction to convexity theory is first provided, and the basic properties of constant width bodies are then presented. The book then delves into a number of related topics, which include

Constant width bodies in convexity (sections and projections, complete and reduced sets, mixed volumes, and further partial fields)

Sets of constant width in non-Euclidean geometries (in real Banach spaces, and in hyperbolic, spherical, and further non-Euclidean spaces)

The concept of constant width in analysis (using Fourier series, spherical integration, and other related methods)

Sets of constant width in differential geometry (using systems of lines and discussing notions like curvature, evolutes, etc.)

Bodies of constant width in topology (hyperspaces, transnormal manifolds, fiber bundles, and related topics)

The notion of constant width in discrete geometry (referring to geometric inequalities, packings and coverings, etc.)

Technical applications, such as film projectors, the square-hole drill, and rotary engines

will be a valuable resource for graduate and advanced undergraduate students studying convex geometry and related fields. Additionally, it will appeal to any mathematicians with a general interest in geometry.

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### Seçilmiş sayfalar

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### İçindekiler

1 Introduction 1 2 Convex Geometry 13

3 Basic Properties of Bodies of Constant Width

59

4 Figures of Constant Width

75

5 Systems of Lines in the Plane

95 6 Spindle Convexity 127

7 Complete and Reduced Convex Bodies

143

8 Examples and Constructions

166 12 Mixed Volumes 278

13 Bodies of Constant Width in Analysis

299

14 Geometric Inequalities

321

15 Bodies of Constant Width in Discrete Geometry

343

16 Bodies of Constant Width in Topology

369

17 Concepts Related to Constant Width

399

18 Bodies of Constant Width in Art Design and Engineering

425 Figure Credits 444

9 Sections of Bodies of Constant Width

197

10 Bodies of Constant Width in Minkowski Spaces

208

11 Bodies of Constant Width in Differential Geometry

247 Bibliography 445 Index 481 Telif Hakkı

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affine assume ball Banach spaces binormal body of constant boundary called centrally symmetric Chapter characterization circle circular closed compact complete conjecture Consequently consider constant width constant width bodies construction contained continuous convex body convex sets Corollary cover curvature curves curves of constant defined denote derived diameter dimension direction discussed disk distance ellipsoid equal example Exercise exists fact Figure figure of constant finite function Furthermore geometry given Hence holds implies inequality interior intersection Lemma length measure minimal Minkowski sum n-dimensional normal normed Note notion obtained origin orthogonal parallel perimeter plane polygons polytope problem projection proof prove reduced refer regular respect Reuleaux triangle rotors segment shown side smooth sphere spherical strictly convex studied subset support hyperplane Suppose surface Theorem translate unit vector volume width h

Guys, does anyone know the answer?