# 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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### Resolution of plane curve singularity

Given a plane curve $C$ over an algebraically closed field of characteristic 0 and $p\in C$ a singular point, if $f:\tilde{C}\to C$ is a resolution of the singularity and $f^{-1}(p)$ consists of one ...
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### Showing that $x(t)=at^2$, $y(t)=vt-at^2$ parameterizes a parabola

Solving a physics problem I obtained following motion equations $$x(t) = at^2$$ $$y(t) = vt - a t^2$$ And I want to determine what type of curve is it on the interval of $\left<0,v\right>$. ...
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### A criterion for determining if a vector points inside a curve

I have to prove that: given a regular closed simple differentiable plane curve $\alpha : \mathbb{R} \rightarrow \mathbb{R}^2$ parametrized in arc length and positively oriented; a non-zero vector $v$ ...
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### Opposite parametrization of closed plane curve

This may be trivial but I cannot fully wrap my head around it: Suppose we have a simple closed regular differentiable plane curve $\alpha : \mathbb{R}\rightarrow \mathbb{R}^2$ parametrized in arc ...
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### Expression of a form as a sum of powers.

I have a small question about one short sentence appearing in page : $376$ of the following electronic textbook : https://scholar.harvard.edu/files/joeharris/files/000-final-3264.pdf The short ...
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### Five conics problem

I have a small question about one short sentence appearing in page : $290$ of the following electronic textbook : https://scholar.harvard.edu/files/joeharris/files/000-final-3264.pdf The short ...
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Here's how far I have gotten: $$\kappa(s) = 1/s \Rightarrow \theta(s) = \int \frac{ds}{s} = \ln(s)$$ Then $$T(s) = e^{i \cdot \ln(s)} = s^i$$ Then the (non-arc-length parametrized) curve $\gamma(... 1answer 65 views ### Why does this method of finding the equation of the tangent line to a second-degree curve work? Given any second degree curve with the general equation $$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$ and a point$(x', y')on it, if I transform the curve equation as follows \begin{align} (1) &&x^2 &\to&... 0answers 47 views ### Transform vectors from cartesian coordinates to curve coordinate system [2d] I have an object moving in a two-dimensional space and its position is given by cartesian coordinates(x_i, y_i)$. This object also has a velocity vector$({v_x}_i,{v_y}_i)$and an acceleration ... 1answer 14 views ### How to show that a parameterization represents a path? I have the following equation that represents a path$C:y^2=x^3+x^2$and a line given by the parameterization$r(t)=(t^2-1,t^3-t)$. I am told that the parameterization represents the path$C$,How ... 1answer 46 views ### Curvature at a point of a curve For a smooth regular curve, there is the notion of curvature at any point. But if this curve is not differentiable at a point, then how to measure the curvature at that point? In the paper Cufí, ... 2answers 107 views ### Curve whose signed curvature is a function Let$f:[a,b]\rightarrow\mathbb{R}$be a function. Is it always possible to find a curve whose signed curvature is the function$f$? I know if$f$is smooth then it can be possible. But I don't know ... 1answer 51 views ### Example of curve in$R^2$which kills driver affirmation Given a (smooth) curve$C$in$\mathbb{R}^2$, there exists at least one parameterization of a curve$f(t)=(x(t),y(t))$. Given also a speed$v(t_0)$(necessarily tangent at the curve a$t=t_0$) at some ... 1answer 73 views ### Rational plane curves as images of$\mathbb{P}^1$A rational projective plane curve is a curve$C\subseteq \mathbb{P}^2$such that there exists a birational map$\mathbb{P}^1\dashrightarrow C$. My question is whether the following statement is true: ... 2answers 67 views ### tangent plane to level set I am confused betweeen tangent plane to the level set(is it the same as level surface?) and to the tangent plane on the surface? I know the formula$z=f(x,y): z-z_0 = f_{x}(x_0)(x-x_0) +f_y(y_0)(y-...
I have a 1-form in the $(x,y)$ plane, and I can write it as: $$\tilde\omega= f \,dA = g \,dB$$ With $f, g,dA,dB \neq 0$. I want to prove that if the following equality holds then the functions ...
### How to construct a second point $Q$ and the third point on a cubic curve?
It is written on Milne's elliptic curve book that Let $C_F$ be a nonsingular cubic Projective curve over $\Bbb Q$. From any point $P \in C_F(\Bbb Q)$ we can construct a second point in \$C_F(\Bbb Q)...