# 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.

259 questions
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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 of ...
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### 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 ...
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### 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 ...
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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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### 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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### 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 $|t_1-t_2|$. ...
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### 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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### 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 ...
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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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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), ...
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### 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 ...
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### 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) ...
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### 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 (...
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### 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 $(121')$ ...
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### 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 ...
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 ...