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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 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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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Examples of smooth implicit curves and surfaces

I am currently constructing a method to approximate implicitly given plane curves and surfaces, which are smooth and single-sheeted. Now I have finished writing a Matlab function doing all the ...
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Prove the image of $\gamma\colon\left[0,1\right]\to\mathbb{R}^{2}$ has measure zero

Let $\gamma\colon\left[0,1\right]\to\mathbb{R}^{2}$ be a curve such that $\gamma\in\mathcal{C}^{1}$ (Continuously Differentiable). I need to show that $\gamma\left(\left[0,1\right]\right)$ has mesure ...
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Can a spiral have its centroid at the origin?

A spiral is a curve $\gamma$ with the polar equation $r=f(\theta)$ where $f$ is a continuous positive strictly monotone function on some interval $[a, b]$, $-\infty<a<b<\infty$. Best known ...
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Curvature of the curve at point

I have a task which says the following: A planar curve is given by $$x=\cos(t)+t,\\y=t^2+2t+1.$$ I had to calculate for which value of the parameter $t$ does the curve pass through the point $P=(1,1)$...
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Analytic curve through infinitely many points

Given infinitely many (different) points $x_1,x_2,\ldots$ in some bounded domain $U\subset \mathbb{R^n}$ and two arbitrary points $P,Q\in U$ (with $P\neq Q$). I would like to show that there exist an ...
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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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1answer
60 views

Is it always possible to project a smooth projective plane curve to $\mathbb{P}^1$?

In the book Algebraic Curves and Riemann Surfaces by Rick Miranda, the author often makes use of the projection map $\pi \colon X \to \mathbb{P}^1$ given by $[x \colon y \colon z] \mapsto [x \colon z]$...
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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,...
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Deriving the Bending Energy equation for Eulers Elastica

In many (all?) papers regarding elastic curves the bending energy for the elastica is given by $$B[\gamma] = \int_{\gamma} \kappa^2(s)ds$$ where $\gamma$ denotes a planar curve of fixed length and ...
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The distance between 2 points with non zero curvature

Suppose we have 2 points which they are connected by an curve. So it’s not line deferment! How I can find the distance between these 2 points? Is there any general formula? Suppose this problem for ...
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Distinguishing Two Compactifications of $[0,1)$

Pictured below are two subsets of the plane, each a compactification of the closed half-line with remainder a closed arc. I am really frustrated by my inability to prove that the space pictured on ...
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1answer
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Plane curves from interval I to a three dimensional space

I'm studying differential geometry and for the definition of curve we have said that it's the image of a fucntion $r(t)$ , $t\in I$, where $I$ is an interval of the real number. Suppose we have that $...
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Planar Curve $(a\cos(\theta), a\sin(\theta), f(\theta))$

Find $f$ such that following represents a planar Curve $(a\cos(\theta), a\sin(\theta), f(\theta))$ for parameter $\theta$. I have a gut feeling that $f(\theta)= constant$ as otherwise it would become ...
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Caustic curve and “mirror equation”

Suppose that $C$ is a caustic curve with respect to a regular boundary parametrized by $\gamma(t)$. Fix a point $A\in C$, take a tangent ray to $C$ in $A$, intersect it with $\gamma$ (say in $\gamma(T)...
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Generalization of intersection of circles?

Let us consider two circles in the (real) plane: $C_1 : (x-x_1)^2 + (y-y_1)^2 - r_1^2 = 0$ $C_2 : (x-x_2)^2 + (y-y_2)^2 - r_2^2 = 0$ In order to calculate their intersection point we can easily ...
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How to show that $\ell(D)=\deg(D)+1+\frac{(d-1)(d-2)}{2}$ for the divisor $D=\operatorname{div}(z^n)$

I'm reading the proof of Riemann–Roch theorems from those notes. In page $63$, they claims that $\ell(D)=\deg(D)+1+\frac{(d-1)(d-2)}{2}$ for the divisor $D=\operatorname{div}(z^n)$ on a projective ...
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If a line never intersects a differentiable curve then there exists at least one tangent to the curve whose slope is that of the line

It is my conjecture. I have tried proving it by maybe Langranges Mean Value Theorem but to no avail...
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Arc length of the cardioid

Compute the length of the segment of the cardioid $(r, θ) = (1+ \cos(t), t) $ such that $ t \in [0, 2π].$ How do I find the arc length of the cardioid. I did $\mathbf{r}'=\langle -\sin(t),1\rangle$ ...