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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398 views

Derivatives of the curvature of a plane curve

I would like to know what is the name of the derivative of the curvature of a plane curve. It should be called "sharpness", but I cannot find a reference. There should also be a connection with the ...
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1answer
162 views

Parametric equation of ellipse given by its foci and sum distance

I am having a problem with converting the equation of a general (tilted) ellipse, from its geometric form to a parametric form: $\sqrt{(x-f_1)^2+(y-f_2)^2} + \sqrt{(x-g_1)^2 + (y-g_2)^2}=S$ where ...
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1answer
281 views

Show that geodesics in a plane are straight lines and vice versa.

I'm trying understand the converse of example $6.2$ of these notes, but I'm stuck when is stated that $x''(t)$ is parallel to $N$, where $N$ is the unit normal to a plane $\Pi$. The author argues that ...
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180 views

A variation of the square peg problem

This question spawned from this recent thread. The notorious square peg problem states that any continuous, simple and closed curve $\gamma$ in the plane contains the vertices of some square. It has ...
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2answers
105 views

Rational curve fitting

Consider $m$ points $(x_i ,y_i )$ in the plane that can be approximated by a curve of the form $y=\frac {c_0+c_1x+c_2x^2}{d_0+d_1x+d_2x^2}$. Suppose that $||(c_0,c_1,c_2,d_0,d_1,d_2)^T||_2=1$. I ...
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1answer
39 views

Stabiliser of a curve under Affine Transformations?

Let $\gamma$ be a curve in the plane, and let $\text{Im}(\gamma) \subset \mathbb{R}^{2}$ be its image in the plane. Is it possible to completely specify the affine transformations of $\mathbb{R}^{2}$ ...
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78 views

The relationship between the curvature of a curve and a circle

Let $c$ be a circle with constant curvature $k_c$, and the green curve $\alpha$ below with curvature at a point $x$ $k_{\alpha}(x)$ I would like to know if this inequality is true: $$k_{\alpha}(q)\...
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47 views

Help with this simple derivative

I'm trying to solve this question from the classical Do Carmo's Differential geometry book. Surprisingly for me, I'm stuck on a very simple question on page 25: MY SOLUTION: Suppose we have already ...
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1answer
48 views

Is the unit tangent with any parametrisation the same as the tangent with arc-length parametrisation?

Suppose we have a plane (or space, I don't think it matters here) curve $\boldsymbol{\alpha}(u) : I \rightarrow \mathbb{R}^2$ with any parametrisation $u$. Is the unit tangent vector $$ \boldsymbol{t}...
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53 views

Prove that the diameter is lower or equal to the half of the perimeter in a closed, convex and planar curve

I'm trying to prove the following statement Prove that the diameter is lower or equal to the half of the perimeter in a closed, convex and planar curve. Is my proof correct? $\textbf{My attempt:...
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285 views

Need help to parametrize the catenary by arc length

The trace of the parametrized curve is $$\alpha(t)=(t,\cosh t),\ t\in\mathbb R$$ is called catenary. I want to show the curvature of the catenary is $$k(t)=\frac{1}{\cosh^2t}$$ Before finding ...
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1answer
101 views

Number of double tangents to an algebraic curve of degree d

Let $C$ be a plane real algebraic curve of degree $d$, i.e., the zero-set of a two-variable polynomial of degree $d$. Q1. Is it the case that the number of double tangents is $O(d^2)$? I believe ...
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1answer
199 views

Prove that the area enclosed by a convex closed regular simple plane curve is lower or equal to width times diameter

I'm try to proof that the area of a convex closed regular simple plane curve is lower or equal to width times the diameter ($A \leq w D$). Intuitively, it's clear that $A \leq w D$, because the convex ...
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1answer
92 views

Connectivity of the space of simple $n$-polylines in the plane

Given $n\geq1$ and points $p\neq q\in\mathbb R^2$, consider the $2(n-1)$-dimensional manifold $\mathcal M_{p,q}^n$ of simple, non-degenerate polylines from $p$ to $q$ having $n$ segments. By "simple" ...
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2answers
137 views

Tangents to a smooth convex curve in the plane

Further to this question Intersections of Tangent Lines to Parabola Different Proof (showing that there are at most two tangents from a point in a plane to a parabola), it struck me that I don't know ...
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1answer
31 views

How to approximate a level curve?

Let $G$ be a $C^\infty$ function $G:\mathbb{R}^2\rightarrow\mathbb{R}$, and let $C:=G^{\leftarrow}(c)$, i.e. $C$ is a level set of $G$. I know that $C$ is bounded (which implies that it's a closed ...
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1answer
90 views

Interpolating a three point curve at any angle using cubic splines

I'm trying to interpolate a curve using cubic splines and three points in the x-y plane. I have some troubles finding the equation for the middle point such that the normal vectors in point P0 is ...
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1answer
232 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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2answers
49 views

Line parallel to another line on a plane

I have the following plane $H= \begin{cases} x=1-3a+b\\ y=a+2b & \\ z=2-a+b\end{cases} $ and line : $l=\begin{cases} x=t+2\\ y=2t-1 & \\ z=t\end{cases} $ on H. How can I find a line lying on ...
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166 views

I found a proof of the Jordan curve theorem for $C^1$ curves. Can anyone check correctness or generalize method to full JCT?

Bear with me, this is a little long. Let $\gamma: S^1 \to \Bbb{R}^2$ be a continuously differentiable simple closed curve. We will often refer to $S^1$ as $[0, 1]$ with the endpoints identified. Let $...
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56 views

Let $J \subset \Bbb{R}^2$ be homeomorphic to a circle. By the JCT, J separates $\Bbb{R}^2$ into two components. Is the bounded one simply connected?

Question in the title. If you separate the plane with a Jordan curve, is the bounded component of its complement simply connected? Intuitively, you would think that the curve might extend to a ...
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162 views

Proof of Schur's Theorem for Convex Plane Curves by Guggenheimer

I'm reading Differential Geometry by Heinrich W. Guggenheimer and I have a doubt about the proof of Schur's Theorem for Convex Plane Curves on page 31. I will put the theorem and the proof here before ...
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3answers
220 views

Finding the equation of a tangent plane given an implicity defined function and obtaining $8$ roots

I want to find the equation of the tangent plane at the point $P(-1,-1,-1)$ given the equation $$x^2 + 10 xyz + y^2 + 8z^2 = 0$$ However, I end up getting eight equations, so I have two questions ...
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2answers
274 views

Convexity of the graph of an implicit function

Let $f(x,y)=0$ be an implicit function and suppose this defines a curve $\gamma$ on $\Bbb{R}^2$. (If it is necessary, assume $f$ is a two variable polynomial.) What are the conditions that guarantees ...
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100 views

Intersection points of a straight line with a closed convex curve

I tried to solve the following problem from Do Carmo. It is very intuitive, but I am having a hard time trying to formalize everything. Show that if a straight line $L$ meets a closed convex curve $C$...
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1answer
61 views

about Euler's $e^{iπ}=−1$

since $e^π= 23.14...$ and this to the power of $i = -1$ , would $23.05...$ equal $-.9995$ ? would $23.27...$ equal $-1.003$ ? in other words, does every positive real number generate a different ...
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1answer
84 views

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 reciprocal of the lemma. What is proper curvature and the Gauss map of a ...
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1answer
42 views

Finding Points of Intersection of the Curve $\mathbf{r}$ with the $yz$-plane

I want to find the point(s) where the curve defined by $\mathbf{r}(t) = \langle 2t^2-t,3t+1,1-2t \rangle$ and the $yz$-plane intersect. The $yz$-plane is defined by the equation $x=0$, so if I set ...
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1answer
48 views

How to distinguish analytically whether the curve touches itself or intersects itself?

I was drawing some curves on a piece of paper and then I realized that if someone else was drawing that curves and asked me to tell whether he drew them so that they intersected themselves at a point ...
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2answers
106 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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1answer
90 views

Plane curve with curvature that tends to zero

Suppose you have immersed plane curve $\gamma:\mathbb{R}_+\rightarrow\mathbb{R}^2$ twice continuously differentiable and parametrized by arc-length such that its curvature $\kappa(t)=\|\gamma''(t)\|$ ...
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2answers
173 views

How to show that the deltoid is a plane algebraic curve of degree $4$?

How can I show that a deltoid is a plane algebraic curve of degree 4? I have searched that the parametric equation for deltoid is given by $$\begin{align} x&= 2 \cos t + \cos 2t \\ y&=2 \sin ...
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3answers
565 views

Brachistochrone problem with floor restriction

An object starts sliding (without friction, under the influence of gravity) from $(0, h)$ along some curve $\gamma(t) = (x(t),\space y(t))$ and just like in the usual Brachistochrone problem it must ...
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2answers
175 views

Locus of vertex of moving parabola

The parabolas $y^2=-12x$ and $y^2=12x$ touch each other at $(0,0)$. The parabola $y^2=-12x$ starts rolling on the parabola $y^2=12x$ (without slipping). Find the locus of the vertex of the moving ...
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1answer
197 views

Find equation of curve equidistant from two ellipses

For visualizing some geometric curve, I have to choose a way between finding sampling points on the curve by numerical processing and finding curve equation of that curve. The curve that I want to ...
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84 views

Intersection of two plane curves in the residue field

I want to propose this problem. Suppose that $C:F(X,Y,Z)=0,\;C':G(X,Y,Z)=0$ are two plane curves defined over a number field $K$. Suppose that they have no common component and that all the ...
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368 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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2answers
286 views

Homogeneous parameterization of a line

In Richard R. Patterson's "Parametric cubics as algebraic curves", it states in 3.1 that a line (2d) in homogeneous coordinates can be parameterized by the homogenous parameter $[t, s]$ as $[s\langle ...
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1answer
34 views

how can I distinguish between these curves

the curve defined as $$f(x,y)=c$$ and $$f(x)=c$$ are plane or space curve i am confuse because $x^2 + y^2=4$ and $x^2=5 $ are both plane curves.But how can i decide about $f(x,y)=c$ and $f(x)=c$ ...
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566 views

The curvature of a plane curve with polar coordinate

let a plane curve with polar coordinates ( $\theta$ , $\rho (\theta )$) and let $k(\theta)$ be it's curvature one can preuve that $k(\theta) = \frac{2({\rho }')^{2}-\rho {\rho }''+ \rho ^{2}}{({\...
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43 views

The first lap of an Archmidean spiral

In polar coordinates, the Archimedean spiral is $r=\theta,\ \theta\ge0.$ Or maybe also $r= c\,\theta,\ \theta\ge0,$ where $c$ does not change as $\theta$ changes. I found myself referring to that ...
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56 views

What does $\partial L$ in $\int _{\partial L}\:xy\:dx\:+\:z\:dy\:+\:\left(x+2y^3\right)dz\:$ mean?

Let the solid L be given by, $L=\left\{\left(x,y,z\right)\in R^3∣\:0\le \:z\le \:1-x^2-y^2,\:y\ge \:-x\right\}$ Observe that this solid is limited by the plane $z=0$, the plane $y=-x$ and the ...
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2answers
275 views

How fast can one move around an ellipse with bounded acceleration?

Given a smooth closed planar curve $\Gamma$, I'm looking for its periodic parametrization $\phi : \mathbb{R}\to\Gamma$ such that the second derivative $\phi''$ is bounded by $1$ in the norm: $|\phi''...
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99 views

Constructing reducible polynomial with two irreducible polynomial

I'm currently trying to prove following statement. Let f,g be homogeneous polynomials of degree n,m respectively, in $k[X,Y,Z]$. Here k is algebraically closed field, and n $\le$ m. Also f does not ...
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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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227 views

Random non-intersecting cubic bezier curves between prescribed anchor points

I am given 4 pairs of red-green points in 2D. each pair corresponds to end points of a cubic bezier curve. My objective is to generate random control points for 4 curves (one going through each pair) ...
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43 views

Derive equation for plane from two parallel curves

I have seven linear regression lines, each with a different value for $z$, i.e. $$\text{eq1}: y = 3x + 2,\, z = 0,\qquad \text{eq2}: y = 2.5x + 3,\, z = 2$$ All of the lines have the same $x$ range (...
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1answer
51 views

Characterizing Conic Section By Eigenvalues

We can bring a function of the form $$ax^2+2bxy+cy^2+dx+ey+f=0$$ To the form of $$\lambda_{1}(x'')^2+\lambda_{2}(y'')^2+k=0$$ In a textbook It is written that if: $\lambda_{1}\lambda_{2}>0$ and ...
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64 views

How do we call this method?

We want to study a parametric curve $f : \mathbb{R} \to \mathbb{R}^2, \ t\mapsto (x(t),y(t))$. To proceed, we use the cartesian equation of a line $y=ax+b$ with $a\in \mathbb{R},b\in \mathbb{R}^{*}$ ...