# 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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### Constructing a map that accurately mimics 1-point perspective

Problem Provide the set of all possible maps $\Phi(x,y)$ such that the image of the usual square coordinate grid under $\Phi$ is a 1-point perspective grid with vanishing point at $(p,q)$. Note that ...
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### construct series of planar polygonal spirals that approximate planar curve

Let $\gamma:[a,b]\rightarrow \mathbb{R}^2$ a constant speed spiral, that is a) $\gamma$ is locally convex w.r.t. 0, that is $\gamma$ has supporting lines that locally lie above $\gamma$ and $\gamma$ ...
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### For a curve to be smooth, it is necessary that its derivative is never equal to $0$. Why? (Complex Analysis, Curves in the complex plane)

I have a question about curves in the complex plane. A parametrized curve is a function $z(t)$ which maps a closed interval $[a,b]\subset\mathbb{R}$ to the complex plane. We shall impose regularity ...
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### Elliptic curve point "division" by an integer

I am trying to understand whether given a point $Q$ that lies on an elliptic curve $C$ there exists a point $P$ on $C$ such that $n P = Q$ for a given $n\in\mathbb{N}$. In particular I would like to ...
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### Can you identify this curve generated by a rotating Reuleaux triangle?

The image below shows the locus of the centroid of a Reuleaux triangle as it rotates and revolves about a stationary one. The curve is shown as blue dots. I'm trying to determine if the curve has been ...
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### Parameters behind non-symmetric Lissajous loop?

I'm trying to guess what kind of two signals can create this kind of Lissajous curve: However I can't figure out what are the parameters that break the symmetry of the curve. The relative phase ...
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### Sequence involving plane curves is exact

I'm reading through Andreas Gathmann's Algebraic Curves and there is this affirmative on Proposition 2.10 at page 14: Let $P\in\mathbb{A}^2$, and $F, G, H$ curves (or polynomials). If $F$ and $G$ have ...
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### The set of the lines tangent to a simple closed loop on the plane is small.

I have a conjecture. It came to me while studying Jordan Curves. It goes as follows: Given a simple closed loop on the plane, the set of its tangent lines is small (have zero measure). The space of ...
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### Number of lines tangent to multiple points of a non-singular projective plane curve

Let $C:F(x,y,z)=0$ be a projective plane curve, where $F$ is a homogenous polynomial of degree $d$. I want to show that there exists $p\in \mathcal{P}_2$ such that all lines through $p$ are tangent to ...
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### Semantics of the angle between velocity vector and the positive $x$-axis

Let's say a particle moves in plane with curvature equal to $\kappa(t) = 2t$, with constant speed of $\|v(t)\| = 5$, such that $v(0) = 5\textbf{i}$, and the particle never goes to the left of the $y$-...
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### Splitting the barycenter of a curve

Definitions Consider a generic smooth closed curve $c(t):[0,1]\mapsto \mathbb{R}^2$ and define its barycenter $g$ as \begin{equation*} g \triangleq \frac{\int_0^1 c(t)\,\lVert \dot{c}(t)\rVert \text{ ...
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### Compact asymmetric form in the plane?

Not being a mathematician, I may be imprecise in asking my question. For experiments on the visual perception of symmetries in the plane, I'm looking for (a) a closed curve in the plane, that is (b) ...
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### Can closed curves be assumed unit speed?

Andrew Pressley's Elementary Differential Geometry textbook claims on page $21$ that "...we can always assume that a closed curve is unit-speed and that its period is equal to its length." ...
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### About the family of curves orthogonal to curves with equations $|x|^p+|y|^p=1$

This question is a follow-on of this answer of mine. This answer provides a numerical solution ; the initial question was about possible closed form expressions. Q1 : Is there a reference in the ...
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### tangential angle of bifoliate

Consider the curve given by polar equation$$r=f(\theta)={8\cos\theta\sin^2\theta\over3+\cos(4\theta)}$$for $\theta$ in $[0,\pi]$. By Mathworld's equation (9) the tangential angle is given by \phi(θ) ...
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### Orthogonal trajectories to family of curves $\left\Vert x \right\Vert_p=1$ where $x\in\mathbb{R}^2$

I have been trying to find the orthogonal trajectories of the family of $p$-norm curves $\left\Vert x \right\Vert_p=1$, where $x\in\mathbb{R}^2$ and $p>0$. I eventually reached a step where I must ...
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### Image of roulette plane curve being dense in a closed disk

Consider the function $f : \mathbb{R} \to \mathbb{C}$ defined by $f(\theta) = e^{i \theta} + e^{i \xi \theta}$ for all $\theta \in \mathbb{R}$. Let $f[\mathbb{R}] \subset \mathbb{C}$ denote the image ...
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### Reflection of a continuous differentiable curve about a line

Let $g(x)$ be the reflection of the continuous and differentiable curve $f(x)$ about the line $x\cos(\theta)+y\sin(\theta)=r$. Find $g(x)$. Also, examine the case in which a curve $f(x)$ is reflected ...
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### Can we take gradient of a curve?

Consider the case of planar curves in $\mathbb{R}^2$. They can be described by a function $f(x,y) = 0$. For example, a circle can be described by $x^2+y^2=1$. We can take the gradient of this function ...
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### Parallel curve is simple - Differential Geometry

I'm struggling with the injectivity of the parallel curve. Let $\alpha:[0,\ell]\to\mathbb R^2$ with $0<\ell\in\mathbb R$ be a planar closed simple regular curve parametrized by arc length, and its ...
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Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. We say $\Omega$ is a uniform domain with constant $c \geq 1$ if for any $x,y \in \Omega$ there is a rectifiable curve \$\gamma : [0, l_\gamma] \to ...