1
vote
0answers
29 views

PDEs along parametrized curves in $\mathbb R^2$

For Riemannian surfaces in $\mathbb R^n,n\geq 3$, the Laplace-Beltrami operator can be expressed in terms of the Riemannian metric and one can consider evolution equations (e.g. heat diffusion, wave ...
5
votes
1answer
78 views

Relations between curvature and area of simple closed plane curves.

Let $\gamma$ be a simple closed plane curve. We know that a curve with constant curvature $\kappa$ will trace a circle in the plane. The radius of this circle is the inverse of its curvature. Now, ...
2
votes
1answer
44 views

There exists a constant arc length parametrization

I heard that for any curve in the plane that can be given parametrically by $\vec{r}(t)=\langle x(t),y(t)\rangle$ for $a\leq t\leq b$ that there exists a constant arc length parametrization, i.e. ...
1
vote
1answer
42 views

Minimum distance from curve

I was thinking about the following problem: Let $\gamma \subset \mathbb R ^2$ be a curve that admits a $C ^{\infty}$ regular parametrization. Is it always possible to choose an open set $E$ ...
0
votes
2answers
55 views

Curve is a submanifold

In our class today we said today that an 1-times continuously differentiable immersion $\phi:[0,1] \rightarrow \mathbb{R}^n$ is a submanifold of dimension 1 if it is closed such that $\phi(0)=\phi(1)$ ...
1
vote
2answers
364 views

Solving (Frenet-Serret) differential equation system in Matlab

I'm about (trying) to solve the Frenet-Serret equation given by the known formulas, finding $e(s)$, $n(s)$, $b(s)$, where $e'(s) = \kappa(s)v(s)n(s)$ $n'(s) = -\kappa(s)v(s)e(s) + \tau(s)v(s)b(s)$ ...
0
votes
1answer
58 views

$d_pf :T_p(S_1)\to T_{f(p)}(S_2)$ is a linear map for any $p\in S_1$

If $f: S_1\to S_2$ is a smooth map betweentwo regular surface $S_1$ and $S_2$, then $d_pf :T_p(S_1)\to T_{f(p)}(S_2)$ is a linear map for any $p\in S_1$ I cannot prove this statement. I think that ...
0
votes
0answers
149 views

Convex curve in R^2 pass through two point with fixed normal direction

Given two distinct point in $\mathbb{R}^2$ and two distinct normal directions what are some explicit convex $C^2$ curve (e.g. a map $[0,1] \to \mathbb{R}^2$ which satisfy many equivalent property) ...
0
votes
0answers
49 views

Inverse of a pythagorean hodograph curve.

I'm trying to solve this question about pythagorean hodograph curves: If $\alpha$ is a Pythagorean hodograph curve, and $\alpha(t)\neq(0,0)\;\forall t$, then the curve $\frac{1}{\alpha(t)}$ is also a ...
1
vote
1answer
98 views

Show the curve $\alpha$ is differentiable and regular

Consider the map: $$ \alpha(t) = \left\{ \begin{array}{ll} (t,0,e^{-1/t^2}) & t>0 \\ (t,e^{-1/t^2},0) & t<0 \\ (0,0,0) & t=0 \end{array} \right. $$ a. Prove that ...
1
vote
0answers
46 views

Osculating circles intersecting a given point

Well, the problem is a question in Montiel's book. How to prove that a planar curve $\alpha$ such that all osculating circles intersects a given point is actually a circle (or a part of it)? I've ...
0
votes
0answers
42 views

A simple exercice about curves. [duplicate]

Somebody can to give me a hint about this exercise? I don't know how proceed. I try to show that $\alpha$ has curvature zero, but I have no successfully. Prove that a curve $\alpha : I —> ...
3
votes
2answers
148 views

why $x^2 = y^3$ is not smooth?

I read a definition of a smooth curve on the plane: A smooth curve is a map from $[a,b] \to \mathbb R^2: t\mapsto ( f(t),g(t) )$, where $f$ and $g$ are infinitely differentiable functions. According ...
1
vote
2answers
115 views

Finding arc length and binormal vector for a given curve

Can somebody show me the arc length of a curve formula, and the binormal vector formula. The curve C with equation $r(t)=(\sqrt{3}\cos t,t,\sqrt{3}\sin t)$ How do you find the arc length from $t=0$ ...
4
votes
0answers
86 views

Curve genereated by a hanging rope with both ends hold together

So a hanging rope assumes the form of a catenary, but if the ends are together it takes some kind of "drop" shape. I think that if the rope is perfectly flexible the shape is a vertical line but what ...
0
votes
1answer
107 views

Relating the curvature of a plane curve to the curvature of a stretched version

Let $\theta : I \to \mathbb{R}^2$ be a regular plane curve with curvature $ |k_{\theta}|\leq1$ everywhere. We now define a curve $\theta_{d}$ by stretching $\theta$ in one direction, i.e., $\theta = ...
3
votes
0answers
53 views

Why can I write the curve shortening flow system as..

Consider the family of curves $$\gamma:S^1\times [0, T)\rightarrow \mathbb R^2, \gamma(u, t)=(x(u, t), y(u, t)),$$ where $u$ parametrizes the trace of the curve and $t$ parametrizes which curve is ...
1
vote
1answer
53 views

Convex Curve Parametrization

How can I parametrize a convex plane curve using the angle $\theta$ between the tangent line and the $x$-axis?
0
votes
1answer
41 views

Plan curve with zero area has at least two points of zero curvature

Let $\alpha=(x,y)$ be a smooth closed plan curve defined on $[a,b]\subset \mathbb{R}$. We can define the oriented area of $\alpha$ by $A=\int_{a}^{b} x(s)y'(s)ds$. So, if A=0 then there exists $t_1 ...
2
votes
1answer
294 views

What does the definition of curvature mean?

First question, am I right in saying that curvature measure how quickly the direction of a cruve changes? Also, we have been given the "definition": $$T'(t) = \kappa(t) | \gamma'(t) | U(t)$$ where ...
28
votes
3answers
1k views

Ambiguous Curve: can you follow the bicycle?

Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance ...
1
vote
0answers
40 views

Functionals defined on curves

I am looking for classical (real) functionals defined on rectifiable curves (neither necessarily simple nor closed) in either $\mathbb{R}^2$ or $\mathbb{R}^3$. There's length, turning number, total ...
3
votes
1answer
238 views

What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid?

What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid? If possible, show some reference please?
3
votes
2answers
567 views

Reconstructing space curves from its curvature and torsion

I have to write a program which gets two functions (curvature and torsion) and 3 vectors of the Frenet-Serret "frame" at the starting point - and I have to reconstruct the space curve from this given ...
1
vote
0answers
174 views

Plane curve: orhogonal projection and closest points

I am reading 'Differential Geometry of curves and surfaces' by Banchoff and Lovett and I'm confused about the following statement on page 28: "Let two curves $C_1$ and $C_2$ have a regular point $P$ ...
10
votes
2answers
297 views

A question about curves in $\mathbb{R}^2$

I need to show this result: Let $\alpha :I\rightarrow \mathbb{R}^2$ a smooth curve, where $I$ is a compact interval of the real line. If $\lVert \alpha (s) - \alpha (t) \rVert$ depends only on ...
3
votes
0answers
371 views

Reconstructing a curve from its curvature

I'm with a problem in an exercise form Do Carmo's Differential Geometry of Curves and Surfaces; it is number 9 section 1.6. I have a differentiable real function $k(s)$, $s\in I$, and I need to show ...
0
votes
2answers
118 views

Parametrise curve by angle and convex curves

Can one parametrise any closed curve by the angle its tangent makes to the $x$-axis? I seem to remember that this is only possible for convex curves. Could anyone tell me why, please? Also is ...
1
vote
1answer
80 views

Computing the gradient of the function $\psi(u) = f[ \phi(u)]$

This question is based on section 6 of the paper Kriging and splines with derivative information. A parametric curve $\phi(u)$ in three dimensions is deformed by the function $f$ to a new curve ...
2
votes
2answers
108 views

What is the limit distance to the base function if offset curve is a function too?

I asked a question about parallel functions in here . I understood that offset curves that are the parallels of a function may not be functions after J.M.'s answer. I got new questions after that ...
2
votes
0answers
146 views

Reversing a roulette on a straight line - solving for a parameterization?

(See below for update.) I would like to reverse the following equations. Here, the path is traced out by the polar origin of the wheel. You can put the trace point anywhere on or off of the wheel ...
0
votes
1answer
237 views

what is the total curvature of the logarithmic spiral?

given the parameterization: exp(t)*(cos(t), sin(t)) t $\in [0, 2\pi$] how do I calculate the total curvature?
2
votes
2answers
358 views

Ways to define a curve

I'm trying to give shapes in my physics engine roundness/ curvature. I am aware of various methods for mathematically defining curvature such as bezier-curves, ellipses, etc; but I'm not sure which ...
0
votes
1answer
98 views

Non-intersecting smooth paths in the plane and the relation to curvature

I'm interested in a problem and have no idea how to approach solving it. Could you please point me in the right direction. Given 3 smooth paths $\varphi_{1}:[0,1]\to \mathbb{R}^{2}$, ...
1
vote
0answers
71 views

Approximation in the Plane of Constant Curvature

In Euclidean space, There is a classic Theorem claims that: The length of every rectifiable curve can be approximated by sufficiently small straight line segment with ends on the curve. Now, The ...
1
vote
0answers
118 views

Vector equation and curvature

Vector equation $r(t)=2\cos(t)\mathbf i + 3\sin(t)\mathbf j\ \ (0 \le t \le 2\pi)$ represents ellipse. I need to find curvature of this ellipse on endpoints of x and y axis that are given with ...
1
vote
0answers
163 views

Is the extra condition in this definition superfluous?

I am learning Differential Geometry and someone told me that the second condition of a definition provided in books follows from the first and is hence superfluous. I cannot dispute it, so convince me ...
1
vote
1answer
276 views

rolling wheel problem

To achieve this: http://en.wikipedia.org/wiki/Square_wheel, what should $L$ be?
1
vote
2answers
334 views

Parameterizing a curve

Edited... I have a Cartesian equation of a cycloid: $$\arcsin\left(k\sqrt{y(x)}\right) - k\sqrt{y(x)-k^2y(x)^2} + c = x$$ where $k$ and $c$ are constants. How might I parameterize it so that I get ...
0
votes
3answers
340 views

points toward the center of the osculating circle (second derivate in a arc length parameter curve)

Is it true that if I have a planar curve parametrized by its arc length, then the second derivative points towards to the center of the osculating circle? I can´t see it, but the book says that it´s ...
2
votes
2answers
623 views

Direction of the second derivative of an arclength parametrized curve

I have a question, it's so simple and stupid ._. If I have a planar curve parametrized by arc length, it's easy to show that the second derivate is orthogonal to the first derivate vector (tangent ...
2
votes
1answer
805 views

The signed curvature of the Catenary

Now I want to show that the signed curvature of the catenary, with parameterization $$(t,\cosh(t))$$ is $k(t)=\frac{1}{\cosh^2(t)}$ Now what I have done (and presumably went astray), is first ...
2
votes
1answer
290 views

Does anyone know the name of this curve?

I have come upon the curve with the following parametric equations: $$x(t)=\log(2+2\cos(t))/2$$ $$y(t)=t/2$$ for $-\pi<t<\pi$. It gives the image in the complex plane under $\log(1+z)$ of the ...
6
votes
3answers
271 views

Can closed curves have small curvature?

Let $\gamma$ be a smooth curve in Euclidean space of length $2\pi$ whose curvature function satisfies $-1 < k(t) < 1$. Can $\gamma$ be closed? This seems like it should be an easy exercise, at ...
1
vote
2answers
707 views

Tractrix tangent segment (from Baby Do-Carmo)

I need some help with question 4 in section 1.3 in Baby Do-Carmo textbook in DG. The question asks: Let $\alpha(t):(0,\pi)\rightarrow R^2 $ be given by: $$ \alpha(t)= (\cos(t), \cos(t) ...
1
vote
2answers
710 views

definition of length of non-rectifiable curves

Following the usual definition of length of a curve, the curve have to be rectifiable to have finite length. However for example the curve $ \gamma \colon [0,1] \to \mathbb{R}, t \to t \cos^2(\pi/t) ...
5
votes
4answers
403 views

Curvature of planar implicit curves

I am trying to understand how the curvature equation $$\kappa = -\frac{f_{xx} f_y^2-2f_{xy} f_x f_y + f_x^2 f_{yy}}{(f_x^2+f_y^2)^{3/2}}$$ for implicit curves is derived. These curves arise from ...
6
votes
2answers
391 views

The Dido problem with an arclength constraint

It is well known that the solution to the classical Dido problem is a semicircle, and that the solution to the classical isoperimetric problem is a circle. It's also reasonably obvious that the ...
1
vote
3answers
331 views

Parametric equations of curves

Is there a way to produce parametric equations for a curve?(If we do know cartesian coordinates of course)
3
votes
3answers
1k views

The function that draws a figure eight

I'm trying to describe a counterexample for a theorem which includes the figure eight or "infinity" symbol, but I'm having trouble finding a good piecewise function to draw it. I need it to be the ...