# 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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### Prove that this system has no periodic orbits

Question: Show that the system \begin{align} \frac{dx}{dt} & = 10x-0.1xy-0.02x^2+1 \\ \frac{dy}{dt} & = -10y+0.1xy+1 \end{align} has no periodic orbits Attempt: The first thing I tried ...
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### Convex curve has convex interior

Let $c: \mathbb{R} \rightarrow \mathbb{R}^2$ be a simple closed curve with curvature $\kappa \geq 0$. Then the interior of $c$ is convex. I know that in this case  \langle N(t_0), c(t) - c(t_0)...
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### When are cone geodesics planar

I mentioned to my (high school) students today that the intersection of a plane and a cone gives a conic section. One asked whether if you 'unroll the cone' the conic section becomes a straight line ...
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### What is the difference between the following definitions of Vector Functions and Parametric Curves?

The definitions seem exactly the same to me. We have a(t) to be a vector function and x(t) to be a parametric curve where the tip of the position vector traces out a curve in an $n$-dimensional space ...
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### Affine plane curves with constant curvature

Question I want to solve this differential equation for $P : \mathbb{R} \to \mathbb{A}^2$, a plane affine curve. $P'''(t) = \frac{P'(t)}{t^2}$ Someone recognize this equation? Is a famous curve? ...
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### Find the point of tangency between a plane and an ellipsoid

So, it is given that - The tangent plane to the ellipsoid $4x^2 + y^2 + 2z^2 = 16$ is $2x + y + 2z = k$. I’m trying to find k, and the point of tangency between those two. What I did - Assumed that ...
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### When to parameterize to find equation of tangent plane and normal line?

I'm working through some problems asking to find the equation of a tangent plane and the normal line to the surface. I notice that some example questions parameterize the curve before solving and ...
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### Show that $P(x,y)=0$ is a hyperbola if $b^2−4ac>0$ .

The following exercise is from Thomas Garrity's Algebraic Geometry: A Problem Solving Approach. I write the polynomial $P(x, y) = ax^2+bxy+cy^2+dx+ey+h$ in the form $P(x, y) = Ax^2 + Bx + C$ where $A$...