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.

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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 some ...
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53 views

Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
6
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118 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
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275 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 of ...
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37 views

Does a plane curve with polar equation $r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$ have a name?

Does a plane curve with polar equation $$r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$$ where both $\lambda_i>0$ have a name? It's very similar to hippopede, also known as lemniscate of Booth, ...
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321 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.
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67 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 ...
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165 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 ...
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160 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 $t\in\mathbb{R}$...
4
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103 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 ...
3
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46 views

Polar forms of algebraic curves & surfaces

A paper I'm reading says the following ... With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let $F(\mathbf{x}) = 0$ be the equation of a surface of degree $n$. The first polar form of $F(\...
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27 views

2D trajectory in minimum amount of time given min/max acceleration per axis

I am having a little problem with determining a trajectory. I have a 2D curve, say $\{x(\lambda), y(\lambda)\}$ where $\lambda$ is not time; it is only a parameter that describes the evolution of the ...
3
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75 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 $P\...
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141 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 $a_x,a_y,b\in\mathbb{R}^...
3
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281 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 ...
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507 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 ...
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74 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 \mathbb{...
3
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80 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 ...
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36 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 ...
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594 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 ...
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36 views

Deviation of a plane curve from the $x$-axis

I have a smooth plane curve $\gamma$ enclosed in a circle of radius $R$. See Figure 1. I'm interested in measuring the deviation of this curve from the $x$ axis, denoted $\Delta(t)$ in the figure, as ...
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36 views

Blowing up a model for a plane curve

Let $R = \mathbb{C}[[t]]$ and let $\mathcal{X} \hookrightarrow \mathbb{P}_R^2$ be the arithmetic surface defined by the equation $$(X^2 - 2Y^2 + Z^2)(X^2 - Z^2) + tY^3Z = 0.$$ The generic fiber ...
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75 views

Parametric Interpolation in the Plane

Given $i+j$ points in the plane, when can we find $x(t),y(t)$, polynomials of degree $i$ and $j$ respectively such that the parametric curve $(x(t),y(t))$ goes through each point? We can do this ...
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40 views

Integers characterizing singularities of algebraic curves

The problem in the following : given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field $\...
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47 views

What's so special about involute curves??

An involute curve (specifically, an involute of a circle) is very commonly used to define the shape of the teeth on a gear. Apparently this idea goes back to Euler. Why is this? What special ...
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79 views

Logarithmic spiral characterized by signed curvature and arc length parameter.

This is a homework problem I am having trouble with: Show that if a planar unit speed curve $q(s)$ satisfies $$\kappa_s = \frac{1}{es+f}$$ for constants $e, f >0$, then the curve is a logarithmic ...
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49 views

Show that $|\alpha'(t)|^2-1=0$ for any arbitrary parameter $t$

There is this supposed to be not so hard question but concept of re-parametrisation by arc length bothers me a bit. Let $\alpha:I\to R^2$ be a regular paramatrised plane curve (arbitrary parameter) ...
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90 views

Existence of a Rectifiable Piecewise Smooth Path

Suppose you have $\gamma(t):[0,1]\rightarrow \mathbb{C}$ simple piecewise smooth, $\gamma(0) = 0$ and $\gamma(1)=1$. Does there exist $\eta:[0,1]\rightarrow \mathbb{C}$, another simple piecewise ...
2
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78 views

Intersecting convex figures

Take in the real plane a finitely long horizontal line segment and connect the two endpoints by a convex path, above the segment, with the property that the only extreme points of the convex hull of ...
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338 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 ...
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71 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: ...
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41 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 t_0}\...
2
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69 views

A nonplanar closed curve such that the plane curve with the same curvature as function of the arclength is not closed

Find a non plane, closed curve such that the plane curve with the same curvature as function of the arclength is not closed. Been thinking a lot in this problem and haven't got a clue. Any ...
2
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43 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
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81 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
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433 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
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48 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 ...
2
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78 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
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53 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 ...
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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
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321 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 $P$ ...
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109 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 ...
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177 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 ...
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41 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 ...
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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 ...
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35 views

A curve with Lebesgue measure non zero

In this Continuously Differentiable Curves in $\mathbb{R}^{d}$ and their Lebesgue Measure the domain of the curve is a compact set. I want know if the same answer holds for curves with non-compact ...
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35 views

Roots of the Taylor approximation of the exponential

While answering another question, I looked at the roots of the $n^{th}$ degree Taylor approximation of the exponential. $$e^x\approx E_n(x):=\sum_{k=0}^n\frac{x^k}{k!}.$$ Apparently, these root are ...
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26 views

A space curve with non-vanishing curvature is planar iff its torsion is 0

Intuitively this is simple and to prove the backwards direction: $\tau = 0 \Rightarrow \mathbf{b'}=0 \Rightarrow \mathbf{b} $ is constant. Then letting $\gamma$ be the parametrisation of the curve ...
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11 views

Exercise concerning areas inside closed curves

Let $\alpha (s)$, $s\in[0,l]$, be a closed, convex, plane curve with $\kappa >0$. Let $r$ be a positive constant and define $\beta (s)=\alpha (s)-rn(s)$, where $n(s)$ is the normal vector of $\...
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28 views

Normal and tangent vectors to a curve

Assume we have a displacement field $u=u(x,z)=(u_x(x,z), u_z(x,z)))$ which takes the point $R=(x,z)$ to $r=R+u=(x+u_x(x,z), z+u_z(x,z))$. We want to find the tangent and normal to the curve which was $...