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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2-modulus of curves over a trapezoid
Definition.
For a real number $p \geq 1$, the $p$-modulus of a curve family $\Gamma$ in a metric measure space $(X,d,\mu)$ is $$\operatorname{mod}_p(\Gamma) := \inf\left\{\int_X \rho^p \mathrm{d}\mu : ...
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Are they Cassinian Ovals?
Are these 'Cassinian Ovals' known? Has any physical meaning been found?
For the curves sketched below the foci are at $(2,1),(-2,-3)$ and squared product distances are $ (1.4^2,1.7^2 , 2^2)$. Thanks ...
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How to determinate a toric section?
I just found an a class of curves named toric section.
Essentially like conic sections but instead of a cone intersected by a plane, it is a torus.
They have this equation (except for rototranslations)...
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Covering all Differentiable curves
Let curves be defined as parametric equations of form $R=(y(t),x(t))$ or the set of solutions for the equations $F(x,y)=0$ where $y(t)$, $x(t)$, $F(x,y)$ are a combination of any algebraic functions ...
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Why is nodal cubic the most general singular cubic? [duplicate]
On page 8 (right before Example(I.5.3)) of this notes by Miranda,
it is stated that “nodal cubic is the most general singular cubic”, without further explanation.
Why is this the case?
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Weak form of elastic plane curves
I have a variational problem for plane curves (AKA elastic curves):
$$
\begin{align}
\gamma &: \mathbb{R}/\mathbb{Z} \to \mathbb{R}^2\\
E[\gamma] &= \frac12\int_0^1 ||\gamma''||^2 + p(||\gamma'...
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Can a Jordan curve contain measure-theoretic interior points of the domain it bounds?
Let $I$ be an interval, and $\gamma:I\to \mathbb{R}^2$ a Jordan curve. By the Jordan--Schoenflies Theorem $\gamma(I)$ splits up $\mathbb{R}^2$ in two connected pieces, that is, $\mathbb{R}^2\setminus \...
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Area calculation of Burnside Curve
Burnside curve is given by the equation $y^2=x^5-x.$ I intended to find the Area formula in $(2)$ of the the given link. And I found after some mistakes:
$$A=2\int_{-1}^0 \sqrt{x^5-x}dx=2\int_0^1 x^{1/...
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Jordan theorem : the boundary statement
The Jordan theorem states that if $\Gamma$ is a closed Jordan curve in $C$, then the complement $C\setminus \Gamma$
has two connected components
one is bounded, the other is unbounded
the boundaries ...
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Does any small enough circle around a point of a curve, intersects it at exactly two points?
If we choose any point of a line, a parabola, a circle, a sinusoidal curve etc, as a center of a circle of radius r, then there is a distance d such that, if r<d, the circle intersects the curve at ...
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For a set of data points approximating a closed curve, are there any interpolation methods that minimize SA:V?
I'm researching interpolation methods for a class. If I have the SA:V of the real shape (based on radius), I want to introduce that information into the curve fitting process. I think it would be ...
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Parametric curve resembling a bean.
I am looking for a parametric closed curve that roughly resembles a bean.
I am looking for something with an explicit parametrization of the form $C(t) = (X(t), Y(t))$
I tried searching online but &...
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Can a plane curve have an odd number of vertices?
The celebrated Four-Vertex Theorem says that the number of vertices of a plane curve must be at least 4. Must it be, more specifically, an even number at least 4?
Here's why that seems to me that it ...
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Find the equation of the plane given the double cone and conic section. [closed]
I saw this post recently deriving the equation of a conic section from the equation of a plane and a double cone.
I'm attempting the inverse, solving for the equation of a plane given the double cone ...
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Dividing a topological disk with a straight line segment into two parts of similar area
Given a closed topological disk $D$ in the plane, is it always possible to find a straight line segment $L\subseteq D$ with endpoints on $\partial D$, and whose interior lies inside the interior of $D$...
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Intersection of Curves and boundary
What will be curve of intersection of the cylinder $x^2+y^2=4$ and plane $x+y+z=2$?
I tried to solve by equating both and got $z^2+2xy+2yz+2zx=0$.
But how to change it into parametric form?
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Can a spline be constructed given start and end points, normalized tangent vectors, and curvatures?
I want to create a smooth curve that starts at point $P_0 = (Px_0, Py_0)$ with normalized tangent $T_0 = (Tx_0, Ty_0)$ (where $\sqrt{Tx_0^2 + Ty_0^2} = 1$) and with curvature $κ_0$ (where positive ...
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Can a union of two disjoint simple curves be represented implicitly by $F(x)+G(y)=0$?
Can you find an example of set $S$ with implicit representation $F(x)+G(y)=0$, where $F,G:\mathbb R \to \mathbb R$ are continuous functions such that:
$\ $ i) $S$ is a union of two disjoint endless ...
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A curve that can not be implicitly represented by $\sum_{n=1}^\infty F_n(a_n x + b_n y) = 0$
Give an example of a simple curve $C=\{(x,y)\in \mathbb R^2: f(x,y)=0 \}$ ($f$ is an analytic function) that can not be implicitly characterized by a uniformly convergent series of functions as
$$
\...
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Can a spiral be represented implicitly as $F(x)+G(y)+H(x+y) = 0$?
Can a spiral be represented implicitly as $F(x)+G(y)+H(x+y) = 0$, where $F,G,H$ are continuous differentiable real-valued functions?
Definition: A spiral is a simple curve $C\subset \mathbb R^2$ with ...
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Can an $n$-armed spiral be represented as $F(x)+G(y)=0$?
Can an $n$-armed ($n\in\mathbb N$) spiral in $\mathbb R^2$ be characterised by the equation $F(x)+G(y)=0$, where $F$ and $G$ are some continuous real-valued functions.
Definition: An $n$-armed spiral ...
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Can a spiral be represented as $F(x)+G(y)=0$
Can you characterise a spiral in $\mathbb R^2$ by the equation $F(x)+G(y)=0$, where $F$ and $G$ are some continuous real-valued functions.
A spiral is any continuous parametric curve $(u(t),v(t)),t\...
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Closed curves of the form $F(x)+G(y)=0$
For certain problem in mechanics, it is useful to assume that a simple (smooth probably, but not strictly necessary) closed curve can be expressed in implicit form as
\begin{align}
F\left (x\right)+G\...
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Does $x'(s)y'(s)$ have a geometric meaning?
Consider a simple, closed curve $\left(x(s),y(s)\right)$, being $s$ the arc-length.
The quantity
\begin{align}
\frac{d x(s)}{d s}\frac{d y(s)}{d s}
\end{align}
is found in some problems concerning ...
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Circle-like parametrization of closed curves.
Can every closed, smooth, non-intersecting curve in the plane be parametrized as
\begin{align}
x(t)=a\cos\alpha(t),\:\:\:y(t)=b\sin\alpha(t),
\end{align}
being $a$ and $b$ constants and $\alpha(t)$ a ...
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Finding a plane is tangent to a curve - $H=\left\{\left(6-s^2-t^2,\:s,\:t\right):\:s,t\in \mathbb{R}\:\right\}$
$H=\left\{\left(6-s^2-t^2,\:s,\:t\right):\:s,t\in \mathbb{R}\:\right\}$
At the point $P=(1,1,2)$
so in order to get to solution, my try:
I defined:
$F\left(s,t\right):=\left(6-s^2-t^2,\:s,\:t\right)$
...
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Convert to parametric equation from implicit equation?
Given an implicit equation such as $x^2+y^2=1$ , I know it corresponds to the parametric equations
$
\begin{cases}
x=\cos t\\
y=\sin t
\end{cases}$.
But I don't know how to get from the implicit ...
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Vector valued higher order ODE with scalar product
I'm trying to solve the fourth order ODE characterising elastic curves to compute some elastic curves:
$$ \gamma'''' + 3 \langle \gamma'', \gamma''' \rangle \gamma' + \frac{3}{2} \langle \gamma'',\...
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Two Definitions of Curvature for Plane Curves
Suppose we have a plane curve $C: I \rightarrow \mathbb{R}^2$ that is continuously differentiable ($C^1$) and parameterized by arc length. Tangent lines exists at each point on the curve, so an &...
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Is there a relatively easy way to find whether two plane curves have a "common component"?
I am interested in determining all intersections between two plane curves f(x,y)=0 and g(x,y)=0. (f is degree 4, and g is degree 3.) I would like to use Bezout's Theorem.
I have found 12 total ...
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Unit normal of ellipse
I am reading through a book which states that for an ellipse specified by the points that satisfy:
$f(x_2,x_3)=x_2^2/a^2 + x_3^2/b^2 = 1$
the unit normal is given by
$\mathbf{n}=\frac{\nabla f}{|\...
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Write inner product of two vectors as a function of the angle between them
Suppose $x_i,x_j$ vectors whose starting point is on the original $(0,0)$ and end point are on the unit circle in $\mathbb{R}^2$, then $x_i\cdot x_j$ can be written as
$\cos(\theta_i-\theta_j)$, where ...
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Planar Quartic Curve invariants for number of connex components
I have some affine planar quartic curves over $\mathbb{R}$ (of the general form $a x^4 + b x^3y + c x^2 y^2 + d xy^3 + e y^4 + fx^3 + g x^2y + h xy^2 + i y^3 + j x^2 + k xy + l y^2 + m x + n y + p= 0$ ...
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A geometric problem with families of congruent curves.
Consider the following mapping:
In a square grid on a unit disk we shift the angles between intersecting segments, every shift in general different at different $(x,y)$ points. Take the continuum ...
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Cardiod-like curve traced by bottom-most point of circle rotating around outer circle
Is there a name /any references for the curve traced out by point $C$ in the following gif?
Specifically, let $r$ and $R$ denote the radius of the inner and outer circles. Let $P$ denote the center ...
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Derivative of curve length in terms of angles
Introduction: I am considering the chapter Random walks derived from billiards from the book Dynamics, Ergodic Theory and Geometry, see the link: https://www.math.wustl.edu/~feres/random_billiard.pdf. ...
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Intuitive Explanation that Brachistochrones (Time-Minimizing Curves) have a Constant Change of Angle per Time [closed]
In a 3 Blue 1 Brown video (https://www.youtube.com/watch?v=Cld0p3a43fU), 3b1b gave the following challenge:
Give an intuitive reason why time-minimizing curves going from one point to another under ...
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Bound the length of level sets
Yesterday, I asked a question regarding the relation between the $C^1$ norm of a function $\phi:\Omega\rightarrow \mathbb{R}$, $\Omega\subset \mathbb{R}^2$ and the corresponding length of the zero ...
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Intersection of circle involutes
I'm trying to find the intersection points between two spirals.
If we have one spiral as:
x1 = r*cos(θ) + r*θ*sin(θ)
y1 = r*sin(θ) - r*θ*cos(θ)
and another one as
<...
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How to parameterise a plane? Ie. $2x+3y=z$
I see in the text book, we can equate $x=\cos{}u$, $y=\sin{u}$, $z=v$, but seems to be for cylinders, how do we find a parametric representation of $2x+3y=z$?
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Are there two computable numbers in the unit interval that map to the same point under a space-filling curve?
A space-filling plane curve is a continuous surjective function from the unit interval $[0,1]$ to the unit square. Netto's theorem gives that continuous bijections preserve dimension, so a space-...
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Problem with the definition of the Cissoid of Diocles
As per the definition given on this site,
Given an origin O and a point P on the curve, let B be the point where the extension of the line OP intersects the line $x=2a$ and C be the intersection of ...
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Finding an algebraic equation from a parametric curve
Given a closed parametric curve $c(t)=(c_x(t),c_y(t))$, on some interval $t\in [a,b]$, is it possible to find a closed form function $f(x,y)$, such that the set of points on the curve $C$ is equal to ...
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How to argue that a moving point in 2D crosses a moving line segment?
I have three smooth functions $a, b, z: \mathbb R\to\mathbb R^2$, such that for all $t$, the points $a(t)$, $b(t)$, and $z(t)$ are all different. I want to prove that if $z$ touches the line segment ...
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Can the plane be covered by fewer than continuum many injective curves?
Background: The plane cannot be covered by fewer than continuum many lines. The simplest proof I know goes like this:
Suppose $\mathcal{A}$ is a collection of lines in the plane, $|\mathcal{A}|<\...
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What is the equation for the cup aproximation of the sin function at $x = \frac{\pi}{2}$
What is the equation for the "cup" aproximation of the $\sin$ function at $x = \dfrac{\pi}{2}$?
The cup aproximation $\mathcal{K}$ is defined as follows:
$\mathcal{K}$ is a mapping from $\...
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Are there curves other than circles such that the line through two points on the curve is parallel the line tangent to the curve at midpoint?
In a circle, if we pick any two distinct points $p_1$ and $p_2$ and draw a line passing through $p_1$ and $p_2$ that line is parallel to the line tangent to the circle at the midpoint of the arc ...
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What interpolation of a set of points has a rectification where the angles in the rectification change linearly?
We seek a function from $\mathbb{R}$ to $\mathbb{R}$ such that the function interpolates some points and the rectifications of $f$ have some special properties
DEFINITION OF RECTIFICATION
For any two ...
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Prove that the locus is tangent to the circle
$O,A$ and $B$ are arbitrary points on the plane. Point $C$ moves on the circle with center $O$ and radius $OB$.
Construct a circle with center $C$ and externally tangent to the circle with center $A$ ...
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On "compensation functions"?
The circle $\alpha (t)=r(\cos t,\sin t)$ has the following property:
If we have a halfline from $(0,0)$ pointing to $(\cos t_0,\sin t_0)$, then it crosses the circle at $\alpha (t_0)$.
Which is ...
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