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.
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Is my rule about the number of $x$- and $y$- intercepts of hyperbolas with equation sof the form $y=\frac a {b(x+c)}+d$ correct?
I developed this rule as a shorthand for determining whether, and how many, $x$- or
$y$-intercepts there are by looking only at the equation of a hyperbola of the form $y=\frac a{b(x+c)}+d$.
If $b=0$,...
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locus of centers of deltoids [closed]
It appears to be challenging to find a curve which is a locus of the centers of deltoids passing through $3$ given points.
Maybe someone can count degree of this curve, or find number of deltoids ...
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What is an equation for rounded square with smooth corner and no flat sides? What is this attempt of mine?
What are
intuition or geometric explanations
simple examples
plots or references
for an equation (any form — implicit, parametric, or polar) for a curve that
has overall shape similar to a square
...
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How to prove that each point on a Jordan curve $J$ can be connected with each point not on $J$ by a Jordan arc intersecting $J$ in a single point?
Let $J \subset \mathbb R^2$ be a Jordan curve (i.e. a homeomorphic copy of the unit circle $S^1 \subset \mathbb R^2$).
Theorem. For any two points $x,y \in \mathbb R^2$ such that $x \in J$ and $y \...
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Theta Curve Theorem
Let $a$ and $b$ be distinct points in the real plane $\mathbb{R}^2$ and let $P_1, P_2, P_3$ be (the images of) Jordan curves from $a$ to $b$ that don't have any points in common except their endpoints....
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Formula to generate N-sided, smoothly rounded regular polygons
I am seeking a formula for generating an $N$-sided regular polygon with an additional parameter that controls curvature near the vertices, with a range of results between a perfect polygon and a ...
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Down-to-earth definition of "genus".
Cubic Curve to Weierstrass Form
For the cubic curve $C$ in general form with rational coefficients:$$ax^3+bx^2y+cxy^2+dy^3+ex^2+fxy+gy^2+hx+ky+l=0,$$we are interested in finding rational points on it. ...
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Shortest curve that intersects with all the lines whose distance away from the origin is $1$.
Problem
On Euclidean plane, find the shortest curve that intersects with every straight line whose distance away from the origin is $1$.
background
I have seen this question on zhihu.com which asks to ...
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A subset of a rectangle that 'blocks' every curve that goes from right to left must connect upper and lower sides
I'm trying to find a proof for the following assetion: Given a rectangular region $R$ and a subset $A$ of $R$, if every curve that starts at the left side of $R$ and ends at the right side intersects $...
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Is there is a Jordan curve that approximates a circle, so that any “square inscribed in the curve” must be a tiny square and cannot be a large one?
With regards to the Inscribed Square problem, I would like to know if there is a Jordan curve that visually resembles a circle at a macroscopic scale, but at microscopic scales is extremely jagged, so ...
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Milnor number parity and singularities of parametrized plane curves
I am working with parametrized polynomial plane curves and I have several related questions about Milnor numbers and singularities. I would be grateful for references, precise statements (theorem ...
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When is it possible to continuously vary two parameters $s$ and $t$ such that $f(s)=g(t)$ always holds
While studying a geometry problem I came across this interesting question.
First of all, let $I=[0,1]$. Now, let $f,g:I\to I$ be continuous functions such that $f(0)=g(0)=0$ and $f(1)=g(1)=1$ (they ...
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Properties of a set given by an implicit equation
First of all, english is not my first language so my post may contain some grammatical errors.
I'm a math undergraduate and I'm trying to prove a theorem for which I need to study the zeroes of a ...
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Find the intersection multiplicity when the equations are modified by invertible factors
I am looking at intersection between too plane curves over a field $\mathbb{K}$. I want to determine the multiplicity of the intersection at a point $P$, when the two curves involved are modified in a ...
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A property of spiral closed curves
I have a question related to a simple property of a spiral closed curve, by which I mean figures of the following kind:
I want to somehow prove that this kind of a closed curve satisfies both the ...
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