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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Embedded arcs staying embedded after squaring.

Let $a : [0,1] \to D^2 = \{z \in \mathbb{C} : |z| \leq 1 \}$ be a continuous map with $|a(0)| = |a(1)|$ and further assume that $a$ is an embedded - so $a$ is an embedded arc. Let $f : D \to D$ given ...
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Suppose $g = f \circ u$ with $u$ and $g$ smooth but $f$ not smooth, does this imply $g' = u' = 0.$

Prove or provide a counterexample. Consider $g = f \circ u$ two equivalent continuous curves on an open interval $\mathrm{I}$ with values in $\mathbf{R}^d$ ($d = 1$ is OK). So $u$ is assumed ...
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Find a perpendicular point from a curved line

I have a curved line on a 2d plane. I know its starting point x1,y2 and ending point x2,y2. I need to find a perpendicular point from both ending at a D distance. Here x1,y1 , x2,y2 and D is known i ...
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Showing a curve is not a variety using Bezout's theorem.

I have a question about a step in the following proof: To show that $y-e^x +1 =0$ cannot be written as the solution to a system of polynomial equations $F_1=F_2=...=F_n=0$, first note that any such ...
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Offset a point on a curve in 3D space

I have a curve AB in 3D space in which I know the start A(x,y,z) and end point B (x,y,z). Now, I have a point O (x,y,z) which should be moved along the curve for some distance (D). If it's a straight ...
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Maximum and minimum speed of the ellipse

I have solved the following exercise and I would like to have some feedback on my solution, thanks. Let $a,b>0.$ Find the maximum and minimum speed of the ellipse $\gamma(t)=\left(a\cos(t),b\sin(t)...
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The connected components determined by a path which lies in one of the connected components determined by another path

Let $\alpha$ be a path in the Euclidean plane $\mathbb{R}^2$. Let $A, B$ be distinct connected components of $\mathbb{R}^2\setminus\operatorname{Im}\alpha$. Let $\beta$ be a path in $B$. Let $C$ be a ...
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Is "torsion continuity" a misnomer for G³ geometric continuity?

I've recently been researching parametric vs geometric continuity of splines (piecewise polynomials) in 2D space. The most common terms for each level of geometric continuity are: $G^0$ is positional ...
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Calculate the integral $\int_{\gamma}\omega$ on the curve [closed]

Calculate the integral $\int_{\gamma}\omega$ on the curve $\gamma(t)=(t\cos(2\pi t), t\sin(2\pi t), t^2)$, $t\in[0,1]$ that is contained in the paraboloid S such that: $S=$ {$(x,y,z)$ | $z=x^2 + y^2$} ...
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How can I find the curves whose curvature is $k(s)=\frac{1}{as+b}$? [closed]

I'm trying to find the curves whose curvature is $k(s)=\frac{1}{as+b}$. I guess that they are Logarithmic Spirals. Parametric form of Logarithmic spirals: $x(t)=ae^{(bt)}\cos(t)$ and $y(t)=ae^{(bt)}\...
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A smooth simple closed curve on $\mathbb{R}^2$ is the boundary of a regular domain

Hello I am self studying differential geometry. I am working on the following problem. Let $\sigma:[0,1]\to \mathbb{R}^2$ a smooth simple closed curve on the plane; show that the image of $\sigma$ is ...
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Proving the curvature formula for an arbitrary planar curve using perpendicular bisectors

Given an arbitrary (i.e. not necessarily arc-length parameterised) planar parametric curve $C(t) = \Big(x(t), y(t)\Big)$, I'm looking to prove the formula for its (signed) curvature $$\kappa = \frac{x'...
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A 2-dimensional bounded domain's boundary is connected iff the domain contains the interior of every enclosed closed Jordan curve

I'm looking for a detailed proof, or a reference to a detailed proof, of the following theorem. Let $G$ be a non-empty, bounded domain in the Euclidean plane $\mathbb{R}^2$. Then $G$'s boundary is ...
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Level curves of harmonic functions are analytic curves?

In the paper, of Flatto, Newman, Shapiro about level curves of harmonic functions, namely curves $\Gamma$ for which there exists a harmonic function $u(x,y)$ vanishing on $\Gamma$ but not identically,...
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The boundary of a chain of plane squares

Denote by $\mathbb{Z}$ the set of whole numbers, by $\mathbb{R}$ the set of real numbers, and by $\overline{\mathbb{R}}$ the set extended real numbers $\mathbb{R}\cup\{\pm\infty\}$. We denote ...
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Measure for the number of curves in a shape in $\mathbb{R}^2$

I am looking for a measure of the following form: Say we have some geometric ribbon-like "shape/curve' in $\mathbb{R}^2$. Example: How can we model the number of "curves" (twists, ...
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3 votes
1 answer
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Curves with the same speed and distance to origin

Let $\alpha,\beta:[0,1]\to\mathbb{R}^2$ be two smooth curves satisfying $|\alpha(t)| = |\beta(t)|$ and $|\dot{\alpha}(t)| = |\dot{\beta}(t)|$ for all $t\in[0,1]$. That is, $\alpha$ and $\beta$ have ...
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Extending a rational map $\psi:C\dashrightarrow\Bbb{P}^1$ into a morphism: concrete example

Let all varieties be projective over $\Bbb{C}$. It is well-known that a rational from a smooth curve to another curve extends into a morphism. My question is about a concrete example of this fact. ...
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Proving Jordan curve theorem using simple polygons

I saw a proof for Jordan curve theorem in this post Triangulation of a simple polygon (elementary proof?). However, every path is the limit of polygons. Doesn't it mean that you can proove Jordan ...
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Computing a normal to a plane using cross product

I have a plane $ ax + by +cz = 0 $. I know that the normal to the plan is the 3 dimensional vector $(a,b,c)^T $ because I define $ f(x,y,z) = ax + by +cz $ and the normal vector is $ \nabla f $ Now ...
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Proving that $\int_C \left(\frac {-y}{x^2+y^2}, \frac {x}{x^2+y^2}\right)ds=\varphi(b) - \varphi(a)$

I'm having some trouble with the following exercise: Let $\alpha:[a,b]\to \mathbb R^2\setminus\{(0,0)\}$ be a curve of class $C^1$ and $\varphi:[a,b]\to \mathbb R$ be a $C^1$ function such that: $$\...
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Is there a wave equation which straightens/rounds the lines between troughs and crests?

I'm looking for a periodic wave-shape that can transform from something like a sine wave to a zigzag. I'm particularly interested in: the straightening/rounding of the curve between the crests and ...
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2 votes
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Problem while finding the perimeter of a cardioid.

Pre-Requisite for My Problem There's this cardioid curve and I need to find it's perimeter. The equation given by my teacher is $$R=a(1+\cos\theta)$$ Here, $R$ : the distance to and part of the curve ...
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Geometric inequality involving a circular arc between two vertices of a triangle and lying inside the triangle.

In posing part of an answer to this question, I appealed to geometry in attempting to show that $x<\tan(x)$ for $0<x<\dfrac{\pi}{2}$ Specifically, I assumed that, in the following diagram ...
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Characterization of arclength as unique function on continuous curves that satisfy certain conditions (resolution of "$\pi=4$ paradox")

I was again thinking about the famous $\pi=4$ paradox, and this question in particular: How to convince a layperson that the $\pi = 4$ proof is wrong?, about why the standard sup over polygonal ...
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Can 'f' be called a function in the given problem?

I know that if a variable $z = f(x,y)$, then $z$ or $f$ is a function of $x$ and $y$. Consider $f = xy^2+y=5.$ Clearly, $xy^2+y=5$ is a curve on the x-y plane. $y$ and $x$ are implicitly related, and ...
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A closed regular planar curve of constant width - finding an expression for the opposite point to $\alpha(s)$

A closed regular planar curve $C$ is said to have constant width $μ$ if the distance between any pair of parallel tangent lines to $C$ is always $μ$. If two points on $C$ have parallel tangent lines, ...
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1 answer
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Condition for singular points in $F_{\mu} =X^3+Y^3+Z^3+ \mu XYZ$ in $\Bbb{P}^{2}_{\Bbb{C}}$?

I've been trying to solve the following problem: Given the curve $F_{\mu} =X^3+Y^3+Z^3+ 3\mu XYZ$ in $\Bbb{P}^{2}_{\Bbb{C}}$, show that $F_{\mu}$ has singular points only if and only if $\mu^3=-1$. I ...
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Rotation matrix to construct canonical form of a conic

I want to find the canonical form of the following conic: $$C: 9x^2+4xy+6y^2-10=0.$$ I've found $C$ is a non-degenerate ellipses (computing the cubic and the quadratic invariant), and then I've ...
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What parametrization should I use to evaluate $\int_{\phi}x^{4/3} + y^{4/3}$, where $\phi$ is curve given by $(x^2+y^2)^2 = 9(x^2-y^2)$?

I´ve recently tried calculating this: $$\int_{\phi}x^{4/3} + y^{4/3}$$ where $\phi$ is curve given by $(x^2+y^2)^2 = 9(x^2-y^2)$. And I couldn´t think of any parametrization or substitution that would ...
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Intersection between a plane and a surface in cylindrical coordinates

I want to find the general expression for the intersection line of the following surfaces given in polar coordinates: $$ z=f(r,\theta) $$ $$ \theta=\pi/4 $$ where $f(r,\theta)$ is any real function in ...
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1 answer
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What type of curve is described by $4(\cos{x}+\cos{y})-6(\cos{2x}+\cos{2y})+8\cos{x}\cos{y}=7$?

Does the curve by the function $$4(\cos{x}+\cos{y})-6(\cos{2x}+\cos{2y})+8\cos{x}\cos{y}=7\\x=[-2\pi/3,2\pi/3],\; y=[-2\pi/3,2\pi/3]$$ belong to any known curve family? A collection of curves is found ...
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2 votes
3 answers
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What type of curve is described by $\cos{x}+\cos{x}\cos{y}+\cos{y}=0$?

Does the curve by the function $$\cos{x}+\cos{x}\cos{y}+\cos{y}=0\\x=[-2\pi/3,2\pi/3],\; y=[-2\pi/3,2\pi/3]$$ belong to any known curve family? Examples of curves can be found in Wikipedia (Link1, ...
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Intersection of a projective curve and a hyperplane

Let $\mathcal{C}$ be a projective curve on $\mathbb{P}^2$ over a field $k$ defined by $\Phi(x,y,z)$ such that it is homogenization of an affine curve $\Phi(x,y)$. When we look at the intersection of $\...
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Prove that certain parameterization is one to one

Let $\alpha:I\longrightarrow\mathbb{R}^2$ be a one to one $\mathcal{C}^{\infty}$ curve parameterized by arc length. Im considering the function $$F(s,t)=\alpha(s)+tN(s)$$ where $\{T(s),N(s)\}$ is the ...
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Prove that squares split the plane into regions that have boundary equal to some Jordan curve.

In Tao's proof of the Jordan curve theorem in the appendix of his 246A Notes 3 he covers a simple closed curve $\gamma$ with squares of small sidelength and claims that "the boundaries of these ...
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How can I find $\Delta_1$ for the first of two reversing curves?

This is a problem from the design of roadways. I know someone has the answer, but I haven't been able to work it out myself. The sketch below shows a single curve made up of two asymmetrical ...
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What is the genus of a union of curves? How are they classified?

Consider a cubic plane curve like this: $$y(y - x^2 - 1) = 0$$ which is clearly the "union" of a line ($y = 0$) and a parabola ($y = x^2 + 1$) that have no points in common (not even the ...
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1 answer
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Find a continuous bijective function with the given form and its inverse

I am looking to find a function $g:\mathbb{R}^2\rightarrow\mathbb{R}^2$ and its inverse $g^{-1}$, such that $g$ and $g^{-1}$ are continuous, smooth, and bijective, such that: $$ g(x,0)=\begin{pmatrix}...
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Show that a "smooth" curve intersects boundary of small enough ball at two points

Given a "smooth" (non selfintersecting) curve $f:[0,1] ->$ $\mathbb{R}^2$. $t\in (0,1)$ I want to show that for small enough $r$, there are exactly two points $t_1,t_2$ for which $$\| f(...
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Simple closed curve in $\mathbb{R}^2\backslash \{x_1,...,x_n\}$ as product of generators of fundamental group

Consider $U=\mathbb{R}^2\backslash \{x_1,...,x_n\}$, by Seifert-Van Kampen theorem $\pi_1(U,p)\cong \underbrace{\mathbb{Z}*\mathbb{Z}*...*\mathbb{Z}}_{n-times} \ \forall \ p\in U$ and we can choose as ...
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2 votes
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Differentiability at a point of a cuve

I have to study the differentiability of the following curve $$\begin{array}{cccc} C: & [0, 2\pi ]& \longrightarrow & \mathbb{C} \\ & t & \longmapsto & r(t+i-ie^{-ti}) \end{...
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Is it possible to describe the level sets of cos(x)cos(y) with non-trig functions?

I was looking at $-z = cos(x)cos(y)$ in math3d, and I noticed that if you take a plane -z = T (T is a slider), the intersection (or level set) kind of looks like some $x^n + y^n = r^n$. The math3d ...
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4 votes
2 answers
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How to calculate the area of one of the smaller loops of the curve: $(y^2−x^2)(2x^2−5x+3) =4(x^2−2x+y^2)^2$? [closed]

How to calculate the area of one of the smaller loops of the curve: $$(y^2−x^2)(2x^2−5x+3) =4(x^2−2x+y^2)^2$$?
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General sigmoid function for estimating crop growth at any day during the crop growing season.

These are some of the general Crop Growth Curves, called as Sigmoid Curves in literature, The general equation/function for the sigmoid curve is, as in the function below ...
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Domain (locus) defined by two lines without using arctangent.

In the 2D plane, suppose we have two lines at angles $\alpha$ and $\beta$ (where $\alpha$ > $\beta$, angularly) going through the origin (they are the curves of purely linear functions). What is a ...
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Function notation: meaning of notation $f_k(x)$

I found this in a 2013 MAT question whilst practising past papers and am not sure of the meaning of this notation. Keep in mind my understanding of maths is only at the level of Scottish Highers (...
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Calculating a trochoid-like curve

I want to find the parametric equations of a special curve. I define a large and a small Diameter, $D$ and $d$, respectively, and the number of "lobes", $N$. Then, $$ R = \frac{D \left(N-1\...
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2 votes
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Comparing velocities of two distinct parametrizations of the same curve

I have two positive functions $f(t,a),g(t,a)$ related by an ODE system of form $$\dfrac{df}{dt}=F_1(f(t,a),g(t,a),a),\\ \dfrac{dg}{dt}=F_2(f(t,a),g(t,a),a),$$ where $a$ is a parameter. I know that for ...
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Finding Arc Length of Polar Curve DIRECTLY

I want to find the arc length formula for a polar curve directly without having to use the arc length formula for cartesian coordinates. The integrand $\sqrt{r^2+(r')^2}$ is a bit strange to me ...
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