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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34 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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Is it true that re-parametrization of a curve does not change its length?

I'm trying to understand what is happening with the arc length parametrization, but it seems that something is missing. I considered the curve $a(t)=\left(e^{t}\cdot \sin(t), e^{t}\cdot \cos(t)\...
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14 views

Equilong (Path-Length-Preserving) Map Between Square and Circle?

Does there exist a map between the (half-open) square and the (half-open) circle which preserves path lengths? Such a map would be called an equilong map. Specifically I am searching for a map $f:[0,...
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30 views

Showing that a circle is an osculating circle of a unit-speed curve

Let $\alpha : I\to\mathbb{R}^2$ be a smooth plane curve parametrized by arc length, and assume that $0\in I$. A circle with radius $r$ centred at $p$ is called the osculating circle of $\alpha$ at $0$ ...
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68 views

Is it always possible to draw an Euler diagram with $2^n$ distinct regions?

Assume we have a set $L$ of $n$ Jordan curves which intersect each other and divide a plane into regions. Given a region $r$, we characterize it by $C(r)=\{l \in L: \text{$r$ lies inside $l$}\}$. Let ...
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Stereographic Projection of circle of Complex plane to Sphere

Source : Complex Variables by Ablowitz and Fokas Q : Show that a circle in the z- plane corresponds to a circle on the sphere. Sphere : $X^2+Y^2+(Z-1)^2=1$ Attempt : x,y are in z-plane (complex) ...
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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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Finding the tangent line(s) to a curve in 3D parallel to a plane

Given the curve $$r(t) = (1 − 2t)i + (t^2)j + (t^3/2)k, ~~ t > 0~.$$ Find a point on the curve at which the tangent line is parallel to the plane $$5x + y + z − 3 = 0~.$$ I have done everything ...
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Computing integral when the derivative factors (separates)

I have a specific smooth function $F(x,y,z) = 0$ which implicitly defines a function $\widehat{z}(x,y)$. In my analysis of this function's properties, the term $$M(x,y) \equiv -\frac{\widehat{z}_y}{\...
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29 views

Relationship between affine plane curves and projective plane curves.

Let $\mathbb K$ be a field and $\mathbb A^2_{\mathbb K}:=\mathbb K^2$. An affine plane curve is a set of the form $$C_f(\mathbb K)=\{(a, b)\in\mathbb A_{\mathbb K}^2: f(a, b)=0\}$$ where $f\in \mathbb ...
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$[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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26 views

Help with parametrization of a Surface and finding tangent plane

So i have Surface defined as: $$(x^2+y^2+z^2)^3= (x^2−y^2)^2$$ Where $|x|\leq y$ So I was thinking Spherical Coordinates as base, so something like: $$x=\cos\theta \cos\phi$$ $$y=\cos\theta \sin\...
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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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409 views

A curve is a circle or a line

Let $\gamma(t):\mathbb{R}\to\mathbb{R}^2$ be a continuous curve in the plane such that for every $t_1,t_2\in\mathbb{R}$ the euclidean distance $d(\gamma(t_1),\gamma(t_2))$ depends only on $|t_1-t_2|$. ...
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Are winding number and index of a not smooth closed curve the same?

Let $\gamma:[0,1] \longrightarrow \mathbf{C} \backslash \{0\}$ be a closed curve (continuous and of bounded variation). We call $$\operatorname{Ind}_\gamma(0) \overset{\mathrm{def}}{=} \frac{1}{2 \...
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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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hexagon with plane symmetries with maths and technology. [duplicate]

Find all plane symmetries (rotations and reflections) of a regular pentagon and of the regular hexagon.
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Finding Line tangent to surface and parallel to plane.

I had an exam today and I wanted to know if I solved this question correct. It asked me to, given a surface curve, like $z=x^2+y^2+10$, find the line tangent to it at point $(2,2,1)$ and parallel ...
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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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33 views

Does the system $xy = ab, G(x)+G(y)=G(a)+G(b)$ always have exactly two solutions if $G$ is continuous and injective?

If $f$ and $g$ are commutative operations $$\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R},$$ then for any constants $a,b \in \mathbb{R}$, the system of equations $$f(x,y) = f(a,b), \qquad g(x,y) ...
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1answer
43 views

Total curvature of a parametrized-by-arc-length curve

Suppose we have the following smooth curve $\sigma:]0,2\pi[\leftarrow\mathbb{R}^2, \sigma(t) = (t, \sin t)$. I want to find the total curvature $\kappa := \int_0^{2\pi}||\sigma''(t)||dt$, but first I ...
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Formulations of Cauchy's theorem that don't seem consistent

So I am reading through the book Complex Analysis by Lars Ahlfors, and there is one point that is causing some confusion for me: In chapter 4.2 - Cauchy's integral formula, we first encounter the ...
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74 views

Total curvature of a closed polygonal curve

Let $\tau$ be a closed polygonal curve in the plane, that is, $\tau$ consists of $n$ piecewise-linear segments between the vertices $v_1, \ldots, v_n, v_1$. The total curvature of $\tau$ is defined as ...
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25 views

Is it possible to construct a closed, simple curve in $\mathbb{R}^2$ that has a segment with zero curvature and is differentiable everywhere?

The question title says it all. Presume I am interested in creating a closed, simple curve in $\mathbb{R}^2$ that contains a segment with zero curvature (that is, part of it is a straight line). ...
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44 views

Orthogonal trajectories of unit circles centered on x-axis [closed]

Find ( $c$ is an arbitrary constant ) orthogonal trajectories of circles: $$ (x-c)^2+ y^2= 1 $$
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How does one find a parameter representation for bounded region?

I need help with this question. I have been stuck at it for a few days. My main problem is how I use the curve $K_r$ to find the parametric representation. I have a curve $K_r$ in the $(x,y)$-plane ...
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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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1answer
24 views

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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81 views

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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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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1answer
37 views

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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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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1answer
41 views

$\tilde \phi:\Gamma (W)\rightarrow \Gamma(V)$ 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
25 views

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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89 views

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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1answer
95 views

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
62 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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3answers
25 views

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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1answer
85 views

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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41 views

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 $A$...