Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [plane-curves]

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.

12
votes
0answers
547 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 some of ...
8
votes
0answers
180 views

A variation of the square peg problem

This question spawned from this recent thread. The notorious square peg problem states that any continuous, simple and closed curve $\gamma$ in the plane contains the vertices of some square. It has ...
7
votes
0answers
108 views

Spaces That Have Uncountably Many Disjoint Copies in $\mathbb{R}^2$

There is a theorem by Moore that says there are not uncountably many disjoint copies of the simple triod in the plane (the simple triod is the space by adjoining one end point from three copies of $[0,...
7
votes
0answers
106 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 ...
7
votes
0answers
131 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, ...
7
votes
0answers
121 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 ...
6
votes
0answers
613 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.
5
votes
0answers
49 views

Must a curve $\eta \colon [a, b] \to \mathbb{R}^2$ intersect the curves $\eta + \frac{\eta(b) - \eta(a)}{n}$ ($n \geqslant 1$)?

Must a curve $\eta \colon [a, b] \to \mathbb{R}^2$ intersect the curves $\eta + \frac{\eta(b) - \eta(a)}{n}$ ($n \geqslant 1$)? This is the fourth - and with any luck, the last! - in a series of ...
5
votes
0answers
85 views

Show that a closed curve formed by two disjoint paths contains the corner of a unit square?

I am looking for a reference, or a topologically/analytically rigorous way of showing the following: Consider two injective paths in $\mathbb{R}^2$ parametrized as $p_1(t)$, $p_2(t)$, $0 \leq t \...
5
votes
0answers
142 views

Coordinate ring of the complement to the theta divisor

Let $C$ be a smooth projective curve over $\mathbb{C}$ and let $\Theta$ be the theta divisor in $J^{g-1}(C)$. The theta divisor is ample, so $J^{g-1}\setminus \Theta$ is affine. What is coordinate ...
5
votes
0answers
265 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 ...
5
votes
0answers
580 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 ...
4
votes
0answers
97 views

A curve is a circle or a line

Let $\gamma(t):\mathbb{R}\to\mathbb{R}^2$ be a continuous curve in the plane such that for every $t_1,t_2\in\mathbb{R}$ the euclidean distance $d(\gamma(t_1),\gamma(t_2))$ depends only on $|t_1-t_2|$. ...
4
votes
0answers
108 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\...
4
votes
0answers
194 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
votes
0answers
124 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
148 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 ...
4
votes
0answers
40 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 ...
3
votes
0answers
47 views

A question on plane curves.

Consider a closed, simple, smooth curve $C$. Can someone help me prove the following: For all point in $C$ and their respective neighborhoods, the radius of curvature cannot be infinite. I ...
3
votes
0answers
87 views

Genus of projective curves

I have the projective curve in $\mathbb{P}^2$ given by \begin{align} F(X,Y,Z)=Y^2 Z^2-X^4-Y^4. \end{align} I want to calculate the genus of the curve. My approach would be to calculate the partial ...
3
votes
0answers
75 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(\...
3
votes
0answers
76 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) ...
3
votes
0answers
51 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
votes
0answers
99 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 $C^1$ ...
3
votes
0answers
146 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
votes
0answers
342 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
97 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 ...
3
votes
0answers
657 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
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
votes
0answers
151 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 ...
3
votes
0answers
2k 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 ...
2
votes
0answers
29 views

Finding Line tangent to surface and parallel to plane.

I had an exam today and I wanted to know if I solved this question correct. It asked me to, given a surface curve, like $z=x^2+y^2+10$, find the line tangent to it at point $(2,2,1)$ and parallel ...
2
votes
0answers
20 views

Can a Jordan convex curve be rewritten as the image of 4 monotone real functions?

Is it true that a smooth Jordan curve $C \subseteq \mathbb{R}^2$ that is convex (in the sense that the region bounded by this curve is convex) can be rewritten as the union of the image sets of 4 ...
2
votes
0answers
31 views

An observation made on Lemma 4.1.1 of “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 observation did by the authors in parenthesis, i.e., the Gauss map injective ...
2
votes
0answers
121 views

Calculating the area of cardioid with trisectrix with green's theorem: Will the area of the loop be added twice? See picture inside

I have a cardioid with a trisectrix, making a loop inside. This is what the cardioid looks like: . My question is the following, if we use Green's theorem to calculate the area of C, which is the ...
2
votes
0answers
62 views

Good references to start studying the curve shortening flow

I want to study the curve shortening flow on curves in $\mathbb{R^2}$. I have knowledge of the local theory of plane and space curves (obviously that comes with linear algebra and calculus as well), ...
2
votes
0answers
63 views

Finding a quartic with some prescribed multiplicities

Let $F(X_0,X_1,X_2)=-X_1^4+X_1^3X_2+X_0^3X_2$ and $C:=\{F=0\}\subset\mathbb{C}\mathbb{P}^2$. I am asked to find a quartic $D\subset\mathbb{C}\mathbb{P}^2$ subject to the following conditions (here $I(...
2
votes
0answers
146 views

Describing the plane curve $α(θ)$ that has the following property: the area of the triangle given by $cQT$ is constant (details below)

A plane curve, $α(θ)$, has the following property: if $c(θ)$ is the center of curvature of $α$ in $θ$, $Q(θ)$ is the projection of $α(θ)$ on the x axis and $T(θ)$ is the intersection point of the ...
2
votes
0answers
165 views

I found a proof of the Jordan curve theorem for $C^1$ curves. Can anyone check correctness or generalize method to full JCT?

Bear with me, this is a little long. Let $\gamma: S^1 \to \Bbb{R}^2$ be a continuously differentiable simple closed curve. We will often refer to $S^1$ as $[0, 1]$ with the endpoints identified. Let $...
2
votes
0answers
43 views

The first lap of an Archmidean spiral

In polar coordinates, the Archimedean spiral is $r=\theta,\ \theta\ge0.$ Or maybe also $r= c\,\theta,\ \theta\ge0,$ where $c$ does not change as $\theta$ changes. I found myself referring to that ...
2
votes
0answers
226 views

Random non-intersecting cubic bezier curves between prescribed anchor points

I am given 4 pairs of red-green points in 2D. each pair corresponds to end points of a cubic bezier curve. My objective is to generate random control points for 4 curves (one going through each pair) ...
2
votes
0answers
43 views

Derive equation for plane from two parallel curves

I have seven linear regression lines, each with a different value for $z$, i.e. $$\text{eq1}: y = 3x + 2,\, z = 0,\qquad \text{eq2}: y = 2.5x + 3,\, z = 2$$ All of the lines have the same $x$ range (...
2
votes
0answers
74 views

Introduce a generalization of the Pappus theorem and the Pascal theorem

The theorem proved in A chain of six circles associated with a conic Let $1, 2, 3, 4, 5, 6$ be six arbitrary points in a hyperbola Let $1'$ be arbitrary point in the hyperbola. The circle $(121')$ ...
2
votes
0answers
48 views

Minimal surface having a Jordan curve as boundary

I'm trying to prove that if $\gamma$ is a Jordan curve (contained in the plane $\pi$) and $\Sigma\subset\mathbb{R}^3$ is a minimal surface with $\partial\Sigma=\gamma$, then $\Sigma$ must be the plane ...
2
votes
0answers
327 views

Dehomogenization and finding multiplicities

I'm trying to find the simple/multiple points of the projective curve with the defining polynomial $F(X,Y,Z)=XZ-Y^2$. As far as I understand, I need to dehomogenize $F$ in each of the three standart ...
2
votes
0answers
51 views

Is this curve known?

A median of a triangle through mid-point of $(-c,0),(c,0) $ is such so that ratio of cosines of angles between sides/median $$ \cos \phi/\cos \psi =e $$ is a constant. Is the curve known? $$ \frac ...
2
votes
0answers
26 views

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 ...
2
votes
0answers
156 views

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 ...
2
votes
0answers
74 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 ...
2
votes
0answers
153 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 ...