# 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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### 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 ...