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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Prove that $z = tx + (1 − t)y$ if $d(x, y)= d(x, z) + d(z, y)$

Let $x,y,z$ be elements of $\mathbb{R}^2$ Prove that $z = tx + (1 − t)y$ if $d(x, y)= d(x, z) + d(z, y)$ d is usual euclidean metric.
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157 views

How to derive Jordan curve theorem for three arcs from the two arc version?

The Jordan curve theorem can be stated as follows: Let $p,q\in\mathbb R^2$, $p\ne q$ and $a,b$ be arcs between $p$ and $q$ intersecting only in the endpoints. Then $a\cup b$ divides $\mathbb R^2$ ...
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164 views

How do I turn a “broken” plot into a smooth curve

I developed and solved a differential equation that predicts fluid temperature along the length of a long pipe with time. Analytical solution is such that it is causing a "discontinuity" in the ...
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188 views

Harris Exercise 5.13 (points in general position)

I have a question about the second part of Exercise 5.13 in Harris' Algebraic Geometry: A First Course: Given $n \leq 2d + 1$ points in $\mathbb{P}^2$, characterize the subset of $(\mathbb{P}^2)^n$ ...
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63 views

$r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ where $a>0$

What does the following equation represent? $r^2\cos\theta+2ar\sin^2{\theta\over2}-a^2$ where $a>0$ My approach: I factorized the equation and it became $(a+r\cos\theta)(a-r)=0$ I feel that ...
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55 views

Real valued function associated with the Diophantine equation $a^2(2^a-a^3)+1=7^b$

The parent question that maybe still remains to be answered at this moment is:Solve the Diophantine equation $a^2(2^a-a^3)+1=7^b$ . As far as the parent question is concerned, when generalizing to ...
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56 views

PDEs along parametrized curves in $\mathbb R^2$

For Riemannian surfaces in $\mathbb R^n,n\geq 3$, the Laplace-Beltrami operator can be expressed in terms of the Riemannian metric and one can consider evolution equations (e.g. heat diffusion, wave ...
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418 views

How to see and proof that the hyperbola as a constant difference of distances holds for $\frac{1}{x}$?

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, which is the distance ...
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587 views

Canonical form of a curve (geometry)

I am bothering with this geometric problem more than half a day and couldn't understand it yet. Here it is: In orthonormal coordinate system K=Oxy we have a curve C: $9x^2 - 4xy + 6y^2 + 6x - 8y + ...
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49 views

Functionals defined on curves

I am looking for classical (real) functionals defined on rectifiable curves (neither necessarily simple nor closed) in either $\mathbb{R}^2$ or $\mathbb{R}^3$. There's length, turning number, total ...
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110 views

Addition law on moduli space of curves

Dislaimer: I know very little about this, so if parts of my question don't make sense, please feel free to edit in a way that does, or ask me to clarify. Let $\mathcal{M}_{1,2}$ be the moduli space ...
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619 views

Plane curve: orhogonal projection and closest points

I am reading 'Differential Geometry of curves and surfaces' by Banchoff and Lovett and I'm confused about the following statement on page 28: "Let two curves $C_1$ and $C_2$ have a regular point $P$ ...
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93 views

Approximation in the Plane of Constant Curvature

In Euclidean space, There is a classic Theorem claims that: The length of every rectifiable curve can be approximated by sufficiently small straight line segment with ends on the curve. Now, The ...
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133 views

Vector equation and curvature

Vector equation $r(t)=2\cos(t)\mathbf i + 3\sin(t)\mathbf j\ \ (0 \le t \le 2\pi)$ represents ellipse. I need to find curvature of this ellipse on endpoints of x and y axis that are given with $(...
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185 views

Is the extra condition in this definition superfluous?

I am learning Differential Geometry and someone told me that the second condition of a definition provided in books follows from the first and is hence superfluous. I cannot dispute it, so convince me ...
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499 views

Detection of self intersection point of curve

What numerical procedure is be adopted to detect self-intersecting parametrized points $ [x(t), y(t) ] $ in $ \mathbb R^2 $ ? Observation : @ roots ( t= 2, t=-1 ) parabola has double value with ...
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36 views

regular curves and ordinary cusps

Let $\Gamma$ be the subset of $\mathbb{R}^{2}$ given by $$ \Gamma = \{(t^2,t^3) : t\in \mathbb{R}\} $$ Does there exist a regular curve of class $C^{3}$ (3−times continuously differentiable), say $\...
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17 views

Moving a point along an ellipse given initial point $P$ and arc-length $d$

Fix a reference frame in $\mathbb{R}^2$. Suppose you have an ellipse $\mathcal{E}$, a point $P \in \mathcal{E}$ and a real number $d$, where it is implicitly understood that $d>0$ means movement ...
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28 views

$[u:v]\mapsto[x(u^2v^3):y(u^2v^3):z(u^2v^3)]$ gives a rational curve

Given the map $\mathbb P^1\to\mathbb P^2$ with $[u:v]\mapsto[x(u^2v^3):y(u^2v^3):z(u^2v^3)]$, s.t. $[x:y:z]$ is on $C$(curve) how shall I deduce that the curve is rational ? I think I must show the ...
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25 views

Space-Filling Jordan Curve

My question is about a simple closed curve that is also a space-filling curve. The figure shows 6 iterations of the formation of a Hilbert curve (limit), whose trace is a solid square. I think we may, ...
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22 views

If there any plane curve whose critical points' curvature are invariant by linear transformation?

I was studying if the curvature of $f$: $$ f(x) = \frac{ax}{b+x} $$ can have the critical points located at the same vertical than this other $g$ curve: $$ g(x) = \frac{ax}{b+x} + cx = f(x) + cx $$ ...
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36 views

Intersection multiplicity and contact order of plane curves

A standard fact about intersection multiplicities of algebraic plane curves is that, for curves $X=V(F(x,y))$ and $Y=V(G(x,y))$, if $p\in X\cap Y$ is a non-singular point of $X$ and $Y$ then $$ I_p(X,...
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14 views

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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35 views

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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70 views

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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66 views

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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33 views

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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29 views

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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73 views

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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38 views

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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87 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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63 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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171 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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164 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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14 views

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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83 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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39 views

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)...
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310 views

If a curve lies in a circle of radius $r$, show there is a point at which the curvature $|k(s)|\geq 1/r$

I am self studying curves and came across this problem: Let $\alpha: I \to R^2$ be a simple smooth closed plane curve. i) If the curve lies inside a circle of radius $r$, show there is a point ...
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83 views

Condition for a plane curve to intersect its osculating circle

The osculating circle of a curve $\alpha$ at the point $p \in \alpha$ is the circle $\mathbb{S}^1$ which is tangent to $\alpha$ at $p$ and has radius $\frac{1}{k(p)}$. Show that, if $k'(p) \neq 0$, ...
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97 views

Proper parametrization of a closed curve

Let $\gamma:I\to\mathbb{R}^2$ be a closed plane curve, for simplicity, a unit circle. Therefore, we have $$\gamma(\varphi) = (\cos \varphi, \sin \varphi).$$ What is the proper domain of $\varphi$? ...
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42 views

The equation of a curve

I need to write the equation of a curve which exactly goes throw point points: $$\left(0,\frac{\pi ^2}{6}-1\right),\left(\frac{1}{2},\frac{2}{3}\right),\left(\frac{3}{4},\frac{2\pi}{3}-\frac{88}{63}\...
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229 views

Simple Proof of the Isoperimetric Theorem in the plane

I am wondering whether there is a ‘simple’ proof of the Isoperimetric Theorem in the plane, i.e. that any simple closed curve in $\mathbb{R}^2$ with length $L$ and enclosed area $A$ fulfils $$4\pi A \...
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59 views

Homotopy of Jordan curves

I'm struggling with the following question: Let $\gamma$ be a (not necessarily smooth) closed Jordan curve and let $\operatorname{int} \gamma, \operatorname{ext}\gamma$ denote the interior and the ...
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52 views

How to calculate the length of the portion of curve from given conditions.

Let $l$ be the length of the portion of the curve $x=x(y)$ between the lines $y=1$ and $y=2$ where $x(y)$ staisfy $\sqrt \frac {1+y^2+y^4}{y} \ , x(1)=0$. Then find $l$ . The main thing I didn't get ...
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252 views

Total Curvature of Space Curves

Question: How can one show that the following proposition is true (Only outline of the proof is needed)? For every closed and regular space curve $c:[a,b] \to \mathbb R^3$ of total length $l$ one ...
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108 views

How to prove that this definition of the circle defines only one curve, the circle?

We may start with this definition of the circle: The set of all points in the plane that are at equal distance (different from $0$) from some fixed point in that plane is called a circle. It is ...
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372 views

Mid Point (Locus) Rectangular Hyperbola

If a rectangular hyperbola have the equation, $xy = c^2$, prove that the locus of the middle points of the chords of constant length 2d is $(x^2 + y^2)(x y - c^2) = {d^2} xy$. I tried to apply ...
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54 views

Intersection of plane wave surface and a curve

How would I calculate the intersection of a plane wave surface and a curve? Note that I am asking about a plane wave surface intersecting a curve in a plane, not a simple sin wave equation ...
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1answer
45 views

Count the number of pair of points from a set of points whose mid points also lie in the same set.

So we have been given a set of points. We have to find the total number of points A and B selected from this set such that mid point of these 2 points also lie in the same set? Let me give an example....
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59 views

Special-Affine invariants for curves in $\mathbb{R}^2$

Working through Ivey and Landsberg's Cartan for Beginners, I am attempting exercise 1.7.3.1, i.e. I am trying to determine special affine invariants for curves ${c}:\mathbb{R} \to \mathbb{R}^2$, that ...