# Tagged Questions

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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### 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 ...
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### 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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### 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 $S$. (...
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### 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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### 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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### 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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### 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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### 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 $t\in\mathbb{R}$...
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### 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 ...
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### 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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### 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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### 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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### What does it mean to say a point is uniquely mapped?

I am looking at space filling curves. Essentially their is a mapping $f: I \to \mathcal{Q}$ where I is an interval in $\mathbb{R}$ such as $[0,1]$ and $\mathcal{Q}$ is a square $[0,1]^2$. For the ...
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### Homotopy of closed curves is also a closed curve?

I'm trying to prove the following statement: Let $\gamma_1$ and $\gamma_2$ be two closed curves from $[a, b]$ to $\mathbb{C}$, and let $h: [a,b]\times[0,1] \to \mathbb{C}$ be a homotopy between ...
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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 $C^1$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 $$y^2 + (2-k)xz +(... 0answers 323 views ### Plane curve: orhogonal projection and closest points I am reading 'Differential Geometry of curves and surfaces' by Banchoff and Lovett and I'm confused about the following statement on page 28: "Let two curves C_1 and C_2 have a regular point P ... 0answers 110 views ### Generalizations of equi-oscillation criterion When constructing minimax (sup-norm) polynomial approximations of real-valued functions, well-known results say (roughly speaking) that optimal solutions are characterized by the fact that they have ... 0answers 178 views ### Reversing a roulette on a straight line - solving for a parameterization? (See below for update.) I would like to reverse the following equations. Here, the path is traced out by the polar origin of the wheel. You can put the trace point anywhere on or off of the wheel ... 0answers 41 views ### How do I use k-dimensional planes as bounds for generating k-dimensional vectors? I am an essentially self-trained programmer with little mathematical background. I do not quite know where to start for this problem, and do not know the terminology to help me get conclusions ... 0answers 1k views ### How to find all intersection points of two splines? 2D-Cubic splines are given in parametric form (X(t), Y(t) and X(s), Y(s)). Every segment has it's own X and Y expression. And I want to find all intersection points. Some segments are intersecting ... 0answers 36 views ### A curve with Lebesgue measure non zero In this Continuously Differentiable Curves in \mathbb{R}^{d} and their Lebesgue Measure the domain of the curve is a compact set. I want know if the same answer holds for curves with non-compact ... 0answers 35 views ### Roots of the Taylor approximation of the exponential While answering another question, I looked at the roots of the n^{th} degree Taylor approximation of the exponential.$$e^x\approx E_n(x):=\sum_{k=0}^n\frac{x^k}{k!}. Apparently, these root are ...
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 ...