Plane curves are continuous (or smooth) functions $\gamma\colon I\to\mathbb R^2$ from a real interval to the plane. Sometimes also the image $\gamma(I)$ is called curve.

learn more… | top users | synonyms

12
votes
0answers
178 views

Maximum number of intersection points of two different Bernoulli lemniscates

What is the maximum number of intersection points of two different Bernoulli lemniscates in the real plane? (Of course two identical lemniscates share an infinite number of points.) Here are ...
6
votes
0answers
97 views

Shortest closed loop containing all extreme points of a convex set

Suppose $S\subset \mathbb{R}^2$ is compact and convex. Suppose $\Gamma:[0,1]\to\mathbb{R}^2$ is a continuous map with $\Gamma(0)=\Gamma(1)$. Suppose $\Gamma$ passes through all extreme points of $S$. ...
6
votes
0answers
188 views

curve which crosses itself at every point

Does there exist a continuous curve $C$ on $\mathbb{R}^2$ that crosses itself at every point on $C$? For two pieces of curve to cross and not just touch, each must have some length either side ...
5
votes
0answers
219 views

What is the parametric equations for the following closed curves?

First case Second case For the sake of simplicity, let the circle of radius $R$ be at the origin, the rectangle width and height be $w$ and $h$, respectively.
4
votes
0answers
63 views

Are closed simple curves with that property necessarily circles?

This is a more interesting follow-up to the question Are closed simple curves with this property necessarily circles? Let $\gamma:[0,1]\to \mathbb R^2 $ be a closed simple $C^1$ convex curve and ...
4
votes
0answers
120 views

Newton's Investigation of Cubics: Generalization?

I recently read about Newton's investigation of cubic curves, and how, like for quadratic curves we can classify them into parabolas, ellipses and hyperbolas, Newton was able to classify cubic curves ...
4
votes
0answers
147 views

In how few points can a continuous curve meet all lines?

Inspired by this (currently closed) question, I'm wondering about a related topic: given that we have a continuous parametric curve (that is, a curve of the form $(x=x(t), y=y(t))$ for ...
4
votes
0answers
98 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 ...
4
votes
0answers
510 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 ...
3
votes
0answers
50 views

Cycles non-homologous but with the same winding number at each point outside?

Problem (Fulton's Algebraic Topology: A First Course, Problem 6.28) Can we find a closed set $X\subseteq\mathbb R^2$, such that there's a cycle $\gamma$ in $X$ not homologous to zero but for each ...
3
votes
0answers
130 views

Intersections of two exponential curves in a plane

I am struggling to show that two exponential curves in $\mathbb{R}^n$ do not overlap except finite distinct points. Could anyone help me? Let $A\in\mathbb{R}^{n\times n}$ and ...
3
votes
0answers
247 views

Question on Moment of inertia about center of mass of a smooth plane curve.

This question is in continuation to this and motivated to answer this question. If I have an answer to it, then I can prove a special case of this question. Consider two smooth plane curves $C \equiv ...
3
votes
0answers
435 views

Guilloché security printing — can it be cracked?

Money uses Security printing, and often uses Guilloché patterns. These curves are inscribed by wheels on wheels on wheels, ten wheels deep in some cases. For example, the back of the US $1 bill has ...
3
votes
0answers
72 views

How to extend an interval to a circle in $\mathbb{R}^2$

Let $$\gamma : [0,1] \rightarrow \mathbb{R}^2$$ be a $C^0$ imbedding. How can I show that there exists another imbedding $$\eta : [0,1] \rightarrow \mathbb{R}^2$$ with $\eta ((0,1)) \subset ...
3
votes
0answers
73 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 ...
3
votes
0answers
33 views

Characterize a large class of shapes using a finite number of parameters

I am doing some numerical computations searching for an optimal shape for a certain functional. In my particular case, the shape $\Omega$ is a 2 dimensional star shaped domain by the origin, which ...
2
votes
0answers
26 views

The heat equation shrinking convex plane curves

In the article "The heat equation shrinking convex plane curves" by M. Gage and R. S. Hamilton, I didn't understand the following theorem: ...
2
votes
0answers
32 views

Reparametrization with non-vanishing lateral derivatives

Let $\mathbf{r}:I\subseteq\mathbb{R}\to\mathbb{R}^2$ be a $C^{1}(I)$ function, non-constant on any subinterval, such that $\forall\ t_0\in I$, the following limits exist: $\lim\limits_{t\nearrow ...
2
votes
0answers
41 views

Tractable indefinite integral of the exponentiation of some function

Consider the function $z(s)\in\mathbb{C}$ defined as $z(s)=\int_0^s \exp\left[i(q u+\lambda(u))\right]du$ for some $q\in\mathbb{Q}-\mathbb{Z}$ and $\lambda(s)$ a $2\pi$-periodic real differentiable ...
2
votes
0answers
74 views

A variational strategy for finding a family of curves?

In a recent question, I asked for examples of families of distinct smooth curves with fixed area and perimeter (which for this question I will dub as doubly-isometric). That wording allows $C^1$ ...
2
votes
0answers
227 views

Equation for the Maximum-Area Rotor in a Wankel Engine

I am attempting to construct a specialized Wankel Engine in Autodesk Inventor. For my particular project, the rotor must take up the maximum area, in order to separate the air spaces of the top and ...
2
votes
0answers
71 views

Should a “good” equation divide the plane?

At the question Is there any equation for triangle? (MSE) the answer given by Henning Makholm received the most upvotes. Therefore let's define the following triangle function $H$ with (a,b,c) the ...
2
votes
0answers
51 views

Jordan curves, its interiors and the existence of a continuous function.

Let $L_t(s):S^1 \rightarrow \mathbb{R}^2(t\in [0,1])$ is a Jordan curve, $O(t)$ is its interior and $H(t,s)=L_t(s)$. If $H$ is a homotopy from $L_0(s)$ to$L_1(s)$, is there exists a continuous ...
2
votes
0answers
52 views

Solving the algebraic equations .

I am working with an equation to find the singular points in $\mathbb P^2 (\mathbb C)$ . Basically after taking the partial derivatives and doing some manipulations it reduces to $$y^2 + (2-k)xz ...
2
votes
0answers
100 views

Generalizations of equi-oscillation criterion

When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ...
2
votes
0answers
163 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 ...
2
votes
0answers
39 views

How do I use k-dimensional planes as bounds for generating k-dimensional vectors?

I am an essentially self-trained programmer with little mathematical background. I do not quite know where to start for this problem, and do not know the terminology to help me get conclusions ...
2
votes
0answers
1k views

How to find all intersection points of two splines?

2D-Cubic splines are given in parametric form (X(t), Y(t) and X(s), Y(s)). Every segment has it's own X and Y expression. And I want to find all intersection points. Some segments are intersecting ...
1
vote
0answers
44 views

How to tell whether a curve has a regular parametrization?

A parametrization of a 1-dimensional curve is called regular if its velocity is always positive. For example, the following parametrization: $$x(t)=t^3, y(t)=t^6$$ is not regular because its ...
1
vote
0answers
37 views

The curvature blows up

A curve evolves according to the evolution equation $\displaystyle\frac{\partial X}{\partial t}= k \times N$ where $k$ is the curvature of the curve and $N$ is the inward unit normal vector. Then ...
1
vote
0answers
26 views

Plane and Ellipse Intersection

Short Version: If some can solve the easier to read form as follows, I would be thankful. \begin{equation} B = \frac{1 - d^{T}Bd}{ K_{1} } A \end{equation} \begin{equation} B^{T}d = \frac{1 - ...
1
vote
0answers
35 views

Why Are Fresnel Functions Used For Splines?

Why are Fresnel functions still used in the research and implementation of clothoid splines? They cannot represent curves of constant curvature, which has led to a lot of research/implementation ...
1
vote
0answers
52 views

Direction of outward normal differential $\hat n\ ds$

In vector calculus, why is the outward normal differential $\hat n\ ds$ around a closed curve $dy\ \hat i - dx\ \hat j$? Why isn't it $-dy\ \hat i + dx\ \hat j$ or something else?
1
vote
0answers
76 views

Embedding a $deg \geq 4$ curve in $\mathbb{P}^2$

Reading the paper by Kollar "The structure of algebraic threefolds", among some examples he does talking about intrinsic and extrinsic geometry over $\mathbb{C}$, he mentions that "an irreducible ...
1
vote
0answers
25 views

Gradient Vectors as Normal Vectors: How to Find D in Form Ax + By + Cz = D?

I've got a question that looks like this: Use the normal gradient vector to determine the equation of the line/plane tangent to the given curve/surface at point P. $x^4 + xy + y^2 = 19$, P(2,-3) I ...
1
vote
0answers
78 views

Tangent vector for a curve defined by a discrete set of points

I have a curve defined by a discrete set of points (x,y). How can I approximate the tangent vector at a point for such a curve?
1
vote
0answers
51 views

How to derive Jordan curve theorem for three arcs from the two arc version?

The Jordan curve theorem can be stated as follows: Let $p,q\in\mathbb R^2$, $p\ne q$ and $a,b$ be arcs between $p$ and $q$ intersecting only in the endpoints. Then $a\cup b$ divides $\mathbb R^2$ ...
1
vote
0answers
103 views

Harris Exercise 5.13 (points in general position)

I have a question about the second part of Exercise 5.13 in Harris' Algebraic Geometry: A First Course: Given $n \leq 2d + 1$ points in $\mathbb{P}^2$, characterize the subset of $(\mathbb{P}^2)^n$ ...
1
vote
0answers
142 views

How to estimate/determine surface normals and tangent planes at points of a depth image (point cloud)?

I have depth image, that I've generated using 3D CAD data. This depth image can also be taken from a depth imaging sensor such as Kinect or a stereo camera. So basically it is depth map of points ...
1
vote
0answers
38 views

$r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ where $a>0$

What does the following equation represent? $r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ where $a>0$ My approach: I factorized the equation and it became $(a+r\cos\theta)(a-r)=0$ I feel that ...
1
vote
0answers
91 views

Parameterization of simple closed curve

A curve $Z$ in a two dimensional space is parametrized by $0\leq t < 1$ , and satisfies $Z(t) = Z(t+1)$. If it is sufficiently well behaved, it can be represented using a Fourier series with a ...
1
vote
0answers
43 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 ...
1
vote
0answers
35 views

Generalisation of circumscribed circle for polygones instead of triangles.

Given a triangle, there exists a unique circle passing through its $3$ vertices. I wonder if more generally for any $n\geq3$ we can define a family $\mathcal{C}_n$ of convex closed curves in the ...
1
vote
0answers
162 views

How to see and proof that the hyperbola as a constant difference of distances holds for $\frac{1}{x}$?

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, which is the distance ...
1
vote
0answers
374 views

Definition of multiplicity of a point (in a plane curve)

In the book "Basic Agebraic Geometry I (third edition, 2013)" at page 14 Shafarevich says, about plane curves, what it follows: If $P=(0,0)$ and the leading terms (note:by leading terms I suppose ...
1
vote
0answers
187 views

Canonical form of a curve (geometry)

I am bothering with this geometric problem more than half a day and couldn't understand it yet. Here it is: In orthonormal coordinate system K=Oxy we have a curve C: $9x^2 - 4xy + 6y^2 + 6x - 8y + ...
1
vote
0answers
45 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 ...
1
vote
0answers
80 views

Addition law on moduli space of curves

Dislaimer: I know very little about this, so if parts of my question don't make sense, please feel free to edit in a way that does, or ask me to clarify. Let $\mathcal{M}_{1,2}$ be the moduli space ...
1
vote
0answers
259 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 $P$ ...
1
vote
0answers
75 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 ...