# 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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### 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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### 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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### 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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### 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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### 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 ...
### 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, ...