# 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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### Finding points outside two lanes

I have two set of points. One set defines the left lane and other set the right lane. And I also have another set of points(black and purple below) and I need to find points in third set which are ...
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### Bending Energy of regular planar curves is parameterization invariant

I need to show that the Bending Energy of a Planar Curve $$\int_I {\kappa^{2}||\gamma^{'}||}$$ is invariant under a reparameterization of $\gamma$ I'm not really sure how exactly I go about doing ...
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### Elliposid in $\mathbb{R}^3$ is given by $2x^2+2y^2+z^2=338$. Find radius of sphere that touches ellipsoid in points $(x,y,10)$

Elliposid in $\mathbb{R}^3$ is given by $2x^2+2y^2+z^2=338$. Find radius of sphere that touches ellipsoid in points $(x,y,10)$. Sphere has center on $z-axis$ This is what I have so far. Sphere ...
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### How can I find the normalization of a singular hyperelliptic curve?

Given a hyperelliptic plane curve $$X = \text{Spec}\left( \frac{\mathbb{C}[x,y]}{y^2 - x(x-1)(x-2)^4} \right)$$ how can I find its normalization? I know I am suppose to compute the integral ...
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### How to find a plane curve $\gamma$ such that $G\circ\gamma=\Gamma$?

Let $G: \mathbb{R}^2\longrightarrow\mathbb{R}^2$ be a smooth function. Let also $\Gamma:[0,1]\longrightarrow\mathbb{R}^2$ be a smooth curve. I am working on the problem of finding a curve $\gamma$ ...
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### Periodic curves

I am building an Android library with different kinds of progress views represented with curves. As you can see it in the gif below, I have used the following curves: Lemniscate of Bernoulli, ...
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### Rate of increase of the function

How do I find the value of the rate of increase of the function $z=x^2+y^2$ along the direction of the given line $y=3x-1$ and in the $xy$ plane at the point $(x, y, z)=(1,2,5)$?
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### Strictly convex curves and lines

Let $\gamma\colon [0,1] \to \mathbb R^2$ (continuous) be a simple closed plane curve and $\,\mathcal C$ its image. Let $x,y$ be some functions such that $\gamma(t)=(x(t),y(t))$. $\gamma$ is said ...
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### Property of strictly convex curves

Let $\gamma\colon [0,1] \to \mathbb R^2$ (continuous) be a simple closed plane curve and $\,\mathcal C$ its image. Let $x,y$ be some functions such that $\gamma(t)=(x(t),y(t))$. $\gamma$ is said ...
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### closed planar curves invariants

http://www.mathcurve.com/courbes2d/generique/generique.shtml In this french website, he presents an invariant of closed planar curves: number of double points. He says that there is no curves with 1 ...
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### Jordan curve theorem (Maehara's proof)

There is one very simple proof for Jordan curve theorem http://www.maths.ed.ac.uk/~aar/jordan/maehara.pdf But Maehara's proof contains one "trivial" step when we fit Jordan curve $J$ in a ...
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### Techniques for finding functions with known values and derivatives at two points

I want to find a function $\gamma(t) = (x(t),y(t))$ such that for two values of $t$, we have $\gamma'(t)$ and $\gamma(t)$ have some value, and at no point does the curvature ever exceed $r$. What ...
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### Equation for sickle shaped plane curve

Is there a parametric equation for a plane curve with the shape of a sickle cell, e.g. half nephroid and half circle? I couldn't find one so far. Thanks! I'm looking for an equation consisting of ...
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### Plane integral for continuous curves

I'm trying to understand complex path integral $\int_C f(z)dz$ for continuous closed curve $C$. Is it necessary that $C$ is rectifiable and not just generally continuous? Do we get all the ...
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### Describe curves by words

I am trying to describe the general aspect of the following curves My ideas: The curves are continuous, and defined for positive values of x, and giving negative values. Have logarithmic growth (...
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### Angle between tangent vector and point of a cardiod.

Consider the cardioid $\rho=2a\left( 1 - \cos \phi \right)$. Show that the angle between the tangent vector and an arbitrary point (different from the origin) of the curve is half the polar angle. ...
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I have numerically obtained some curves, corresponding with it I have also obtained some roots. I strongly believed these curves can be fitted with some (elliptic) functions taken the roots as ...
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### When working with trochoids what does θ stand for?

These are the formulas with which you can draw trochoids. $x = aθ - b sin(θ)$ $y = a - b cos(θ)$ I'm trying to make trochoids but I got hung up on this symbol $θ$, what is it and how do I use it, I ...
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### How to find a plane that is tangent to 3 spheres?

So there are spheres with radius of 1 centered at (1,2,0), (4,5,0) and (1,3,2). How can one find a plane that is tangent to all 3 spheres? Visually, it looks like as if the spheres are sitting on a ...
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### Curvature in $\mathbb{R}^2$

Let $f(t) = (x(t),y(t))$, not necessarily parametrized by arclength. We define the unit tangent vector, $T(t) = (1/|f'(t)|)(x',y')$. Also the normal vector, $N(t) = (1/|f'(t)|)(-y',x')$, which is ...
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### A 4th grade curve meets a line in one point with multiplicity 4

Suppose a 4th grade curve meets a line in one point with multiplicity 4. Example: the lemniscate $(x^2 + y^2)^2 = y^2 - x^2$ meets the line $x=y$ when the condition $x^4=0$ holds. This shows that ...
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### Deriving tangent plane equation from scalar equation of plane

There's one step in the derivation that I don't understand. I'll explain some of the derivation and then explain which step I don't understand. Begin with scalar equation of plane: A(x-x0) + B(y-yo)...
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### Equation for a Vinyl curve

This video seems to show an explicit map from the torus to $\mathbb{R}^2$. Does it factorize through the projection $\mathbb{R}^3 \to \mathbb{R}^2$? What is the equation of the curve?