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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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What kind of Planar Quartic Curve might this be?

I'm trying to smoke out the parameters for a family of curves showing up in a particular optimization problem. I have convinced myself that the solutions always lie on a quartic curve, which is ...
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Strong/weak tangents and limit positions, with rigor

As I'm working from do Carmo's Differential Geometry of Curves and Surfaces, I have found some of his imprecise language regarding strong and weak tangents to be most irksome. I've seen similar posts ...
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prove an inequality between the curvature of these curves

The exercise is: Let $\alpha$ a plane curve such that $|\alpha'(s)|=1$ with curvature $k(s)$. Let $\beta(s)=\alpha(s) + k(s)N(s)$ such that $\beta'(s)\ne 0$ $\forall s$. ($N$ is the normal vector ...
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Confusion about a tangent line approaching an asymptote

I'm working from do Carmo's Differential Geometry of Curves and Surfaces, 2ed. He tends to use language like"the curve $\alpha$ and its tangent line approach [some line] $L$" or "the curve $\alpha$ ...
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How many straight lines and circles can be drawn in the plane so that they are equidistant from all four points?

Four points on a plane are given which are not collinear or all on one circle. How many straight lines and circles can be drawn in the plane so that they are equidistant from all four points? If not ...
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Formula for the osculating conic of a plane curve

A follow up to this question. Presumably similar curves have similar osculating conics, which in turn have identical eccentricities. Thus, the 'local eccentricity' of a plane curve at a point is the ...
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Prove that the curve $\alpha(t)$ is tangent to the $x$ axis.

I have the curve $\alpha(t):(-1,\infty) \rightarrow R^2$ given by $\alpha(t)= ((\frac{nat}{1+t^3}), (\frac{nat^2}{1+t^3}))$ with $n$ a natural an $a$ a constant both of them fixed. I need to prove ...
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Can an arbitrary curve in $\Bbb R^2$ be a graph of a certain equation?

Can any curve in $\Bbb R^2$ (which doesn't intersect itself) be a graph of a certain equation? In other words, if given an arbitrary curve in $\Bbb R^2$ (which doesn't intersect itself), is there a ...
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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 ...
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Parametrization of special family of tori knots

Finding the parametric equations of an (a-c)tori knot knowing that one turn has the following parametric equation: $$\alpha(t)=\begin{pmatrix} x=\Big(R_1+R_2\cos(t)\Big)\cos\biggr(c\arctan\Big(\dfrac{...
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$\tilde \phi:\Gamma (W)\rightarrow \Gamma(V)$ may not extends to all $k(W)$

A problem from Fulton's Algebraic Curves:-- Let $\phi:V\rightarrow W$ be a polynomial map between two affine varieties and $\tilde \phi:\Gamma (W)\rightarrow \Gamma(V)$ be the induced map between co-...
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If $\gamma\colon[a,b]\to\mathbb{C}$ is continuous and $\gamma(b)=-\gamma(a)$, must the curves $\gamma$ and $e^{ic}\gamma$ intersect for all real $c$?

If $[a, b]$ is a compact interval of $\mathbb{R}$ and $\gamma: [a, b] \to \mathbb{C}$ is continuous, denote the connected, compact set $\gamma([a, b])$ by $[\gamma]$. If $h$ is a complex number of ...
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If two closed plane curves are outside each other, can there be a point inside both of them?

I think this recent question (also here) has a quick answer if the conjecture below is true. It looks "obviously" true, but I've learned to distrust my judgement in such matters. It also looks as if ...
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1answer
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Plane clipping by cubic limits

I have a plane equation given by a point and a normal vector, for example. This plane has to lay between $xyz$ limits, $300<x<2700$, $150<y<1350$, $130<z<1370$. I want to know the ...
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Curvature inequality involving a Curve within a disk

If a closed plane curve $C$ is contained inside a disk of radius $r$, prove that there exists a point $p \in C$ such that the curvature k of C at p satisfies $\lvert k\rvert \ge$ $1/r$. I understand ...
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Matrix powers and hyperbola

(We're in $\mathbb{R}^2$) How to find hyperbola equation, that has symmetry axis crossing (0,0) point and for $n=1,2,\ldots$ points ${\begin{pmatrix} 4 & 3 \\ 1 & 1 \end{pmatrix}}^n \begin{...
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Prove that this system has no periodic orbits

Question: Show that the system \begin{align} \frac{dx}{dt} & = 10x-0.1xy-0.02x^2+1 \\ \frac{dy}{dt} & = -10y+0.1xy+1 \end{align} has no periodic orbits Attempt: The first thing I tried ...
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Convex curve has convex interior

Let $c: \mathbb{R} \rightarrow \mathbb{R}^2$ be a simple closed curve with curvature $\kappa \geq 0$. Then the interior of $c$ is convex. I know that in this case $$ \langle N(t_0), c(t) - c(t_0)...
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When are cone geodesics planar

I mentioned to my (high school) students today that the intersection of a plane and a cone gives a conic section. One asked whether if you 'unroll the cone' the conic section becomes a straight line ...
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What is the difference between the following definitions of Vector Functions and Parametric Curves?

The definitions seem exactly the same to me. We have a(t) to be a vector function and x(t) to be a parametric curve where the tip of the position vector traces out a curve in an $n$-dimensional space ...
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1answer
46 views

Affine plane curves with constant curvature

Question I want to solve this differential equation for $P : \mathbb{R} \to \mathbb{A}^2$, a plane affine curve. $ P'''(t) = \frac{P'(t)}{t^2}$ Someone recognize this equation? Is a famous curve? ...
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Find the point of tangency between a plane and an ellipsoid

So, it is given that - The tangent plane to the ellipsoid $4x^2 + y^2 + 2z^2 = 16$ is $2x + y + 2z = k$. I’m trying to find k, and the point of tangency between those two. What I did - Assumed that ...
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When to parameterize to find equation of tangent plane and normal line?

I'm working through some problems asking to find the equation of a tangent plane and the normal line to the surface. I notice that some example questions parameterize the curve before solving and ...
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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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Show that $P(x,y)=0$ is a hyperbola if $b^2−4ac>0$ .

The following exercise is from Thomas Garrity's Algebraic Geometry: A Problem Solving Approach. I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax^2 + Bx + C$ where $A$...
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1answer
201 views

Show that $P(x,y)=0$ is an ellipse if $b^2-4ac<0$.

I tried the following: I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax^2 + Bx + C$ where $A$, $B$, and $C$ are polynomial functions of $y$. This $P(x, y) = Q(x)$ ...
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Show that $P(x,y)=0$ is a parabola if $b^2-4ac=0$.

I tried the following: I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax2 + Bx + C$ where $A$, $B$, and $C$ are polynomial functions of $y$. This $P(x, y) = Q(x)$ ...
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1answer
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Is the role of the boxed condition $z'(t)\neq 0$ to avoid going back?

The role of the boxed condition $z'(t)\neq 0$ is to avoid going back, isn't it?
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48 views

Octahedron Pyramid

So, each octahedron can be inscribed in a cube, so that the corner points of the octahedron are in the midpoints of the side areas of the cube, am I right? From the octahedron $ABCDS_1S_2$, shown in ...
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How can one show that an elliptic curve has a point whose tangent line also meet another point of the curve

I'm coming from a projective setting of a smooth cubic plane curve over a field $K$ and want to show that I can bring it to the Weierstrass long form. The usual method is to start with a point and ...
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How to deduce Jordan Curve Theorem from Schönflies Theorem

Recently I started reading Ethan Bloch's "A First Course in Geometric Topology and Differential Geometry" and I came upon this exercise to deduce the Jordan Curve Theorem from the Schönflies Theorem: ...
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Prove that this class of curves has constant speed and curvature

Let $\gamma: (a,b) \rightarrow \mathbb{R}^2$ be a smooth regular curve such that $\forall s,t \in (a,b)$ , $||\gamma(s)-\gamma(t)||$ is a non-negative real valued function which depends only on $|t-s|$...
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Distance between a line given by planes and origin by Lagrange's multiplier.

Find the minimum distance of the line given by the planes $3x+4y+5z=7$ and $x-z=9$ from the origin, by the method of Lagrange’s multipliers. As- Is it ok to take the Lagrange functions as given ...
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1answer
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Limits of integration for triple integrals

Suppose you have the following problem: Express the mass of a solid tetrahedron T, with vertices $(0,0,0)$, $(1,0,0)$, $(1,4,0)$, and $(1,0,3)$ and density function $p(x,y,z) = 6xyz$. Graphically, I ...
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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 \...
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1answer
56 views

Resolution of plane curve singularity

Given a plane curve $C$ over an algebraically closed field of characteristic 0 and $p\in C$ a singular point, if $f:\tilde{C}\to C$ is a resolution of the singularity and $f^{-1}(p)$ consists of one ...
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Space filling curve's intersections with closed jordan curves [closed]

Does there exist an space_filling curve in $\mathbb{R^2}$ which intersects with each closed jordan curve in plane in an uncountable number of points?
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Showing that $x(t)=at^2$, $y(t)=vt-at^2$ parameterizes a parabola

Solving a physics problem I obtained following motion equations $$ x(t) = at^2 $$ $$ y(t) = vt - a t^2 $$ And I want to determine what type of curve is it on the interval of $\left<0,v\right>$. ...
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A criterion for determining if a vector points inside a curve

I have to prove that: given a regular closed simple differentiable plane curve $\alpha : \mathbb{R} \rightarrow \mathbb{R}^2$ parametrized in arc length and positively oriented; a non-zero vector $v$ ...
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Opposite parametrization of closed plane curve

This may be trivial but I cannot fully wrap my head around it: Suppose we have a simple closed regular differentiable plane curve $\alpha : \mathbb{R}\rightarrow \mathbb{R}^2$ parametrized in arc ...
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1answer
80 views

Derivation of a general tractrix

I would like to derive the differential equation for a general tractrix in parameter form. For me, it is quite obvious that $$\mathbf{A}(t)=\mathbf{P}(t)+\frac{\dot{\mathbf{P}}(t)}{|\dot{\mathbf{P}}(t)...
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A point in every osculating plane of a curve

This question has already been asked here, but it had no answer, so I'm asking it again. Let $I$ be an open interval, and $\alpha: I\rightarrow \mathbb{R}^3$ be a regular curve with curvature $\...
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1answer
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Is torsion $0$ for an osculating curve in Euclidean space?

A curve is called osculating curve if its position vector lies on its osculating plane. Osculating plane for the curve $\alpha(s)$ at some point on it is generated by the tangent vector and normal ...
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1answer
58 views

Writing the equation $r = \theta$ in cartesian cordinates

Trying to write the cartesian version of the equation $r = \theta$ which looks like a spiral when graphed. How is the that going to look? I have: $$ \sqrt{x^2+y^2} = \arctan\left( \frac{y}{x} \...
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If $\sigma: \mathbb{R}\to \mathbb{R}^2$ is a function that spirals,goes to infinity and repeats itself, then is $\sigma$ non-injetive?

Let $\sigma:\mathbb{R}\to \mathbb{R}^2$ be a smooth function such that $$\frac{\text{d}\sigma}{\text{d}t}(s) \neq 0, \quad \forall s \in \mathbb{R},$$ and $$\sigma(t+n) = \sigma(n) +\sigma(t),\ \...
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1answer
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Why does $b(t)=\text{const}.$ follow from $<b(t),\frac{w}{|w|}\ge \text{const}.$

I want to understand the following proof. It is from the book "Differential geometry of curves and surfaces" by C. Tapp. I don't understand why $b(t)=const.$ follows from $<b(t),\frac{w}{|w|}>=...
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2answers
128 views

How to find point in Descartes Folium with slope of -1/3?

I stumbled upon a problem in my calculus book that asked to find the point in $x^3 + y^3 = 3xy$ that had a slope perpendicualr to $y = 3x + 1$ and also was in the first quadrant. I began by getting ...
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1answer
70 views

A fish curve constructed from a circle

In this diagram, there is a circle centred at $A$. $GH$ is the perpendicular bisector of $CD$. Fixing the position of $C$, the path created by point $H$ when $D$ is moving on the circle is plotted in ...
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Finite Unions of Dendrites

I will ask the main question first, and then give the motivation for this one! The question is a bit specific, but seems to be the most general question to ask after handling some obvious ...
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1answer
106 views

Uncountable Collections of Arcs In the Plane with Prescribed Properties

I was wondering, if for each angle $0 \leq \theta < 2\pi$ we have an uncountable collection $A_\theta$ of pairwise-disjoint, closed, straight line segments with length $1$ and slope $\theta$ in the ...