Mathematics Stack Exchange
Mathematics Stack Exchange

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.

Plane curves are continuous (or smooth) functions $\gamma\colon I\to\mathbb R^2$ from a real interval to the plane. Sometimes the image $\gamma(I)$ is also called a curve.

1306 questions
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Piecewise Smooth Curve

Is the curve defined by: γ(t) = (t,t) for 0≤t≤1 and (2-t,2-t) for 1≤t≤2 piecewise smooth? My logic says yes because one can break it into a finite number of smooth curves (two in this case), but something doesn't add up for me.
Yarduza
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Equation of a vertical plane given $2$ points

A vertical plane passes through points $(1, -1, 1)$ and $(2, 1, 1)$. With three points on a normal plane, I just found two vectors and found the normal by cross-product of the vectors, but I'm not sure what to do here. Help, please.
user3139573
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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?
Vít Tuček
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Maximum area enclosed by a string attached at fixed points

Two fixed points A and B have a string of length L attached between them. Supposing that the string does not intersect the line segment AB, then the string and AB will form a closed figure. What shape will the string form when this closed figure…
DavidButlerUofA
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Lattice Points in x-y plane

What are Lattice Points? Which points in x-y planes are Lattice Points? Is (m,n) a lattice point where m,n are any integers?
user3481652
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Curve intersection criteria

I have two curves, which are given by sets of values: $C = [( x{_1} ,y{_1}),(x{_2},y{_2}),(x{_3},y{_2}),...,(x{_n},y{_n})]$ $C^' = [( x^'{_1} ,y^'{_1}),(x^'{_2},y^'{_2}),(x^'{_3},y^'{_2}),...,(x^'{_m},y^'{_m})]$ where $m \ne n$ Is there any…
Anton Semenov
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Find the point where equations $x=t^2-t$ and $y= t^3 -3t-1$ cross itself.

Find the point where equations $x=t^2-t$ and $y= t^3 -3t-1$ cross itself. This's the first time I meet this kind of problem, can someone give me some idea? Thank you.
JSCB
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Help with parametrization of a Surface and finding tangent plane

So i have Surface defined as: $$(x^2+y^2+z^2)^3= (x^2−y^2)^2$$ Where $|x|\leq y$ So I was thinking Spherical Coordinates as base, so something like: $$x=\cos\theta \cos\phi$$ $$y=\cos\theta \sin\phi$$ $$z=sin\theta $$ And since…
MathIsTheWayOfLife
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Proper parametrization of a closed curve

Let $\gamma:I\to\mathbb{R}^2$ be a closed plane curve, for simplicity, a unit circle. Therefore, we have $$\gamma(\varphi) = (\cos \varphi, \sin \varphi).$$ What is the proper domain of $\varphi$? Wikipedia says it's $\varphi \in [0,2\pi]$ with…
Fizikus
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about Euler's $e^{iπ}=−1$

since $e^π= 23.14...$ and this to the power of $i = -1$ , would $23.05...$ equal $-.9995$ ? would $23.27...$ equal $-1.003$ ? in other words, does every positive real number generate a different value when raised to the $i$ power or are all $n^i$…
Bantokfomoki
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Need to find the equation of a curve having only the direction of it at a given point

Temperature T of a plate lying in xy plane is defined T(x,y)=50-(x^2)-(2y^2). An ant, which is initially at (2,1) moves along a curve ensuring the temperature is decreasing as rapidly as possible. I need to find the equation of this curve. The…
Liucija
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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 $(x, y, z)=(1,2,5)$?
zmyky
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Proof of Netto's theorem

I am trying to show any bijective mapping $f:I \to \mathcal{Q}$, where I is the unit interval in $\mathbb{R}$ and $\mathcal{Q}$ is the unit square, is necessarily discontinuous. How do I go about proving this? The context of this theorem is in the…
Al jabra
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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 sinusoidal functions instead of inverted parabolae.
macro
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The change of the angle of the gradient as moving along the curve

I'm given a curve $g = 0$ in 2D specified by g(x,y) = f(x) - y. The normal to the curve is the gradient of $g$ - $(f', -1)$. Now I want express the change in the angle $\theta$ of the normal as I move along the curve. However, I'm not sure how to…
Alex Botev
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