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.

964 questions
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 ...
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 ...
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 ...
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 ...
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 ...
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 $\mathbb{R}^{2}$ ...
78 views

53 views

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 ...
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 ...
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 ...
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 ...
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 $C$...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $\kappa(t)=\|\gamma''(t)\|$ ...
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 ... 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 ... 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 ... 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 ... 0answers 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 ... 0answers 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 ... 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 ... 1answer 34 views how can I distinguish between these curves the curve defined asf(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 ... 2answers 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}}{({\... 0answers 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 ... 0answers 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 ... 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''... 0answers 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 ... 0answers 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 ... 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.... 0answers 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) ... 0answers 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$$All of the lines have the same x range (... 1answer 51 views Characterizing Conic Section By Eigenvalues We can bring a function of the form$$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 ...
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 $a\in \mathbb{R},b\in \mathbb{R}^{*}$ ...