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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 ...
Simon M's user avatar
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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 $\gamma$ ...
Mathemann's user avatar
3 votes
2 answers
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The shape of parabola in a 1-point perspective drawing with the vanishing point at the parabola's point at infinity

Edit: After a little bit of careful construction, I've worked out that the map I should have been using instead is $$\phi\left(x,y\right)=\left(\frac{x}{ay+1},\frac{ay}{ay+1}\right)$$ where $a\in(0,\...
Simon M's user avatar
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Volume of an elliptical cylinder cut with a curved plane

I am looking to find the volume that is left when a curved plane defined by $z=-y^3+0.5$ and the xy plane cuts an elliptical cylinder defined from five points ($Ax^2+Bxy+Cy^2+Dx+Ey=F$). I plan to ...
Mr. Mister's user avatar
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25 views

Finiteness of the intersection number [duplicate]

I am taking a course on Algebraic Curves following Gathmann and I am trying to solve exercise 2.7(b) which reads as follows: $F,G$ two curves with no common components through the origin, then every ...
Fernando Rabanillo Novoa's user avatar
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20 views

Proof that the evolute is the geometric locus of points of the centers of curvature

I am trying to prove that the evolute is the geometric locus of points of the centers of curvature. Let r = r(s) be an arc-parametrized curve. The normal line at a point r(s) of the curve is line ...
Grant's user avatar
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2 votes
0 answers
106 views

What is the equation of the evolute of a curve given in implicit equation?

I am trying to find the evolute of a curve given in implicit equation. I tried to find it using the definition, as the envelope of the family of normals, however I didn't come to a conclusion. Then I ...
John's user avatar
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403 views

Why do the roots of $\int_0^x (1-s^2)^n ds$ lie on a lemniscate?

Consider the polynomial $$f_n(x)=\int_0^x (1-s^2)^n ds$$ for some integer $n$. I am interested in these polynomials for reasons unrelated to this question: these are the odd polynomials of which the ...
Wouter's user avatar
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1 answer
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Inclusion of set $A$ in set $B$ or vice versa, where A and B are defined as follows:

Question Consider the sets defined by the real solutions of the inequalities $$ A=\left\{(x, y): x^2+y^4 \leq 1\right\} \quad B=\left\{(x, y): x^4+y^6 \leq 1\right\} . $$ Then (A) $B \subseteq A$ (B) $...
Debu's user avatar
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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 ...
佐武五郎's user avatar
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1 answer
47 views

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 ...
neilps2000's user avatar
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1 answer
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A problem about morphisms from a genus 2 curve to a quartic curve.

I am to answer the following question. Let $X$ be a projective non-singular curve of genus $2$, and let $K$ be an effective canonical divisor on it. Pick any two points $P_1,P_2$ on $X$ with $K \not \...
John Robertson's user avatar
3 votes
0 answers
94 views

A graph with radius of curvature $≥1$ can't have more than 2 distinct real intersection points with a circle of radius 1

Is the following true? If the graph of a continuously twice differentiable function $y(x)$ and the radius of curvature $|\frac1\kappa |$ is $\gt 1$ at all points on the graph (e.g. $y=\sin(x),x\in(0,\...
hbghlyj's user avatar
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1 vote
1 answer
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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 ...
Cye Waldman's user avatar
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2 votes
1 answer
58 views

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 ...
ValientProcess's user avatar
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18 views

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 ...
Mand's user avatar
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0 answers
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Hausdorff dimension of neighbourhood of Jordan domain

Let $D \subseteq \mathbb{R}^2$ be a bounded domain such that $\partial D$ is a Jordan curve, and for $\varepsilon > 0$ let $D^\varepsilon = \{ x \in \mathbb{R}^2 : \text{dist}(x,D) < \varepsilon ...
Julius's user avatar
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2 answers
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Curve is traveled clockwise or anti-clockwise

Given the curve $$ \vec{\mathbf{r}}(t) = \bigl(t - \tfrac{1}{3}t^3 \bigr) \, \vec{\mathbf{i}} + (4 - t^2) \, \vec{\mathbf{j}}, $$ how can I tell whether it's traveled clockwise or counterclockwise? ...
Emmannuelle_Legolas's user avatar
2 votes
1 answer
64 views

What's the distinction between an elliptic curve and its rational points?

I'm confused by this statement: "Be careful that you understand the distinction between the elliptic curve E and the group E(k) of its k-rational points. The group law is defined for the curve E, ...
popstack's user avatar
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2 votes
1 answer
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A centre of a simple closed planar curve

Let $c: [0,1] \rightarrow \mathbb{R}^2$ be a continuous planar curve such that $c(0)=c(1)$. I was wondering if there is a way to find something like a "centre of mass" of planar curve $c$. ...
Anay Jain's user avatar
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Tangent Vector, Principal normal vector, Binormal vector, and Torsion

So I'm trying to fully grasp how all these relate. My current understanding is that the tangent vector describes the direction in which the curve is going/curving. Meanwhile, the principal norm is ...
A Student 's user avatar
1 vote
1 answer
53 views

Locus of 4 colinear points in $\mathbb{P}^2$

I'm trying to solve exercises 2.34 and 2.35 in 3264 and all that (Eisenbud & Harris, 2016, p.80). Exercice 2.34 Let $\varphi\subset(\mathbb{P}^2)^4$ be the locus of 4-tuples of colinear points. ...
Ayoub's user avatar
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Can we find a "s", "t" parametrization of any planar simple curve?

Let $$r: [0,1] \to \mathbb{R}^{2}, $$ be a simple curve. Denote by $\Omega$ the interior of the curve defined by $r(t)$. Without loss of generality, let us assume that $(0,0) \in \Omega$. Does there ...
IdenticallyEulerian's user avatar
4 votes
1 answer
101 views

What geometry is preserved by the translation maps on elliptic curves?

Let $E$ be an elliptic curve (over some field). For any $P \in E$, there is a translation map $T_P: E \to E$ given by $Q \mapsto P+Q$. This map is rational (i.e. the coordinates of $T_P(Q)$ are ...
popstack's user avatar
  • 291
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0 answers
16 views

Law of the areas (2nd of Keplero) proof

I was reading the following proof of the law of the areas (the generalization for central forces): Consider a plane curve $t\mapsto (x(t),y(t))$, that in polar coordinates is given by $t\mapsto \rho(...
Luigi Traino's user avatar
2 votes
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15 views

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 ...
Alma Arjuna's user avatar
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1 vote
1 answer
95 views

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 ...
neilps2000's user avatar
2 votes
1 answer
33 views

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 $y$-...
S11n's user avatar
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0 answers
35 views

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{ ...
matteogost's user avatar
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0 answers
24 views

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) ...
mk9y's user avatar
  • 101
1 vote
2 answers
61 views

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." ...
Bifton Mifts's user avatar
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0 answers
9 views

Possibility of superposition of nonpositive curvature curves produces positive turning

Suppose a finite set of plane curves parameterized by $t$: $p_i(t)=(x_i(t),y_i(t)), i=1...N$, satisfy boundedness: $|p_i(t)|\leq 1\ \ \forall i$ smoothness: $p_i(t)$ differentiable to any order, $\...
George C's user avatar
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1 vote
1 answer
75 views

Integer solutions for a specific genus one, quartic plane curve

As part of a physics project studying renormalization group flows of scalar field theories, I've come across the following quartic plane curve in the variables $m$ and $n$: $$ 36 + 16 m^4 - 108 n + ...
user2309840's user avatar
2 votes
0 answers
28 views

On the shoelace formulas

I've stumbled on this nice formula to compute the barycenter $\bar{c}$ of an arbitrary (but not self-intersecting) polygon \begin{equation} \bar{c} \triangleq \sum_{i=1}^n \frac{A_i}{A} \bar{c}_i \...
matteogost's user avatar
0 votes
0 answers
41 views

Proving the curvature of a plane curve is equal to that of a space curve

Let $\gamma : (a,b) \rightarrow \mathbb{R}^2$ be a regular curve. Let $\iota : \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be the map \begin{equation}\iota\left(\begin{pmatrix}x \\y\end{pmatrix}\right) = \...
spooleey's user avatar
  • 456
1 vote
0 answers
35 views

Injective (closed) piecewise $C^1$ curve not equivalent to the zero chain

This is Exercise 3.1.13 here. Let $\gamma:[a,b] \to U \subset\mathbb{C}$ be a piecewise $C^1$, injective (or can also be closed) curve. Show that there exists a continuous function $f$ such that $\...
ploosu2's user avatar
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0 answers
108 views

parametric reflection of one curve across another

In this other question the user asks for a parametric curve and "imposing" one curve on another. You can find a demonstration here. I have been meaning to use the tangent to answer a similar ...
vallev's user avatar
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9 votes
4 answers
904 views

Motivation for the definition of curvature of a plane curve

I am seeking a motivation for the definition of the curvature of a plane curve. How did people come with the idea of the definition of the curvature? Below I am more specific. The fundamental theorem ...
ghreis's user avatar
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0 answers
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Does a kind of real plane algebraic curve always have a factorization corresponding to connected components?

$f(x,y)$ is a real polynomial such that in the equation $f(x,y)=0$ we can express $x$ in $y$ with composition of polynomial functions and square roots, and can express $y$ in $x$ with composition of ...
hbghlyj's user avatar
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0 votes
0 answers
62 views

Is the relation between all equations and all curves in plane surjective?

Let $O$ be the infinite? set of all operations such as addition, multiplication, trigonometric functions, integrals etc, over the independent variables $x$ and $y$. Let $H_0$ be a subset of $O$. $$H_1 ...
aku jack's user avatar
0 votes
1 answer
54 views

Continuous closed non intersecting curve which lies in bounded region but has an infinite length

In the below picture, author provides an example of a continuous, closed, non-intersecting curve which lies in a bounded region $R$ but which has infinite length. While calculating area bounded by ...
General Mathematics's user avatar
2 votes
1 answer
75 views

Does a connected, locally connected, compact subset $S \subset \mathbb R^2$ which separates the plane contain a Jordan curve?

Let $S \subset \mathbb R^2$ be a connected, locally connected, compact subset of the plane. Suppose there exists $x,y \in \mathbb R^2$ which lie in distinct connected components of $\mathbb R^2 \...
jpmacmanus's user avatar
1 vote
1 answer
68 views

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 ...
Jean Marie's user avatar
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1 answer
37 views

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) ...
hbghlyj's user avatar
  • 3,035
3 votes
2 answers
131 views

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 ...
FabrizzioMuzz's user avatar
2 votes
1 answer
78 views

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 ...
fractal_sounds's user avatar
1 vote
1 answer
57 views

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 ...
Cognoscenti's user avatar
1 vote
0 answers
41 views

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 ...
Sean's user avatar
  • 89
0 votes
0 answers
42 views

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 ...
PinRod3's user avatar
  • 349
3 votes
1 answer
74 views

Connecting two points inside the Koch snowflake without getting too close to the boundary

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 ...
Tobi's user avatar
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