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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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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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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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31 views

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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113 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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72 views

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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79 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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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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51 views

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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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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183 views

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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31 views

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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2answers
79 views

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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155 views

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 $\mathbf{r}'=\langle -\sin(t),1\rangle$ ...
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Parametric equation of a rotated circle in 4 dimensions

I'm attending a differential geometry course, and I'm stuck at one part of a question that we've been asked. The following rotated circle is given in 4 dimensions: $$x_1x_3 + x_2x_4 = \frac{1}{2}$$ ...
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Parametrizing a set

I'm trying to parameterize the following set $$ \left\lbrace\, r\left(\cos\left(\alpha\right),\sin\left(\alpha\right)\right) \;\middle\rvert \quad 0\leq\alpha\leq2\pi, \quad 0<r\leq 1,\quad \tfrac{...
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Expression of a form as a sum of powers.

I have a small question about one short sentence appearing in page : $376$ of the following electronic textbook : https://scholar.harvard.edu/files/joeharris/files/000-final-3264.pdf The short ...
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1answer
27 views

Proving that on a plane curve defined on the reals there exists $t_0$ such that $\vert \alpha(t_0) \vert \leq \vert \alpha(t) \vert \ \forall t$

Let $\alpha: \mathbb{R} \to \mathbb{R}^2$ be a closed plane curve defined on the real line. Suppose that $\alpha$ doesn't go through the origin $\textbf{0} = (0,0)$ and that: $$\displaystyle{\lim_{t \...
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1answer
29 views

Equation for area of closed simple curve

Suppose we have a closed simple curve written as $(x,y) = F(u)$. To find the area enclosed by this curve, we appeal to Green's theorem: $$ A = \frac{1}{2}\int_0^{2\pi} [x\frac{\partial y}{\partial u} ...
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Frenet Formula relating $T$ and $N$.

In the following notes: http://mathematics.stanford.edu/wp-content/uploads/2013/08/Mooney-Honors-Thesis-2011.pdf, the author (on page 5) says that $T$ and $N$ are related by the Frenet formula $\...
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Prove that there are maximum 4 points on every convex closed Jordan curve $C$ which avoids intersection under certain rotaitions

Prove that there are maximum 4 points on an arbitrary given closed convex Jordan curve of $C$ on a plane which for each one like $p$ we have the following property: If we rotate $C$ around $p$ by $\...
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160 views

Five conics problem

I have a small question about one short sentence appearing in page : $290$ of the following electronic textbook : https://scholar.harvard.edu/files/joeharris/files/000-final-3264.pdf The short ...
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2answers
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Which plane curve parametrized by arc-length has curvature $\kappa(s) = \frac{1}{s}$

Here's how far I have gotten: $$\kappa(s) = 1/s \Rightarrow \theta(s) = \int \frac{ds}{s} = \ln(s)$$ Then $$T(s) = e^{i \cdot \ln(s)} = s^i$$ Then the (non-arc-length parametrized) curve $\gamma(...
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Why does this method of finding the equation of the tangent line to a second-degree curve work?

Given any second degree curve with the general equation $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$ and a point $(x', y')$ on it, if I transform the curve equation as follows \begin{align} (1) &&x^2 &\to&...
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Transform vectors from cartesian coordinates to curve coordinate system [2d]

I have an object moving in a two-dimensional space and its position is given by cartesian coordinates $(x_i, y_i)$. This object also has a velocity vector $({v_x}_i,{v_y}_i)$ and an acceleration ...
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1answer
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How to show that a parameterization represents a path?

I have the following equation that represents a path $C:y^2=x^3+x^2$ and a line given by the parameterization $r(t)=(t^2-1,t^3-t)$. I am told that the parameterization represents the path $C$ ,How ...
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1answer
46 views

Curvature at a point of a curve

For a smooth regular curve, there is the notion of curvature at any point. But if this curve is not differentiable at a point, then how to measure the curvature at that point? In the paper Cufí, ...
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2answers
107 views

Curve whose signed curvature is a function

Let $f:[a,b]\rightarrow\mathbb{R}$ be a function. Is it always possible to find a curve whose signed curvature is the function $f$? I know if $f$ is smooth then it can be possible. But I don't know ...
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Example of curve in $R^2$ which kills driver affirmation

Given a (smooth) curve $C$ in $\mathbb{R}^2$, there exists at least one parameterization of a curve $f(t)=(x(t),y(t))$. Given also a speed $v(t_0)$ (necessarily tangent at the curve a $t=t_0$) at some ...
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Rational plane curves as images of $\mathbb{P}^1$

A rational projective plane curve is a curve $C\subseteq \mathbb{P}^2$ such that there exists a birational map $\mathbb{P}^1\dashrightarrow C$. My question is whether the following statement is true: ...
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67 views

tangent plane to level set

I am confused betweeen tangent plane to the level set(is it the same as level surface?) and to the tangent plane on the surface? I know the formula $z=f(x,y): z-z_0 = f_{x}(x_0)(x-x_0) +f_y(y_0)(y-...
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1answer
27 views

What can be said about two expressions of a 1-form?

I have a 1-form in the $(x,y)$ plane, and I can write it as: $$ \tilde\omega= f \,dA = g \,dB $$ With $f, g,dA,dB \neq 0$. I want to prove that if the following equality holds then the functions ...
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How to construct a second point $Q$ and the third point on a cubic curve?

It is written on Milne's elliptic curve book that Let $C_F$ be a nonsingular cubic Projective curve over $\Bbb Q$. From any point $P \in C_F(\Bbb Q)$ we can construct a second point in $C_F(\Bbb Q)...