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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Find the length of the curve $y(x) = \int_1^x\sqrt{t^3 - 1} \, dt$, $1 \leq x \leq 4$

Find the length of the curve $y(x) = \int_1^x\sqrt{t^3 - 1} \, dt$, $1 \leq x \leq 4$. Not sure if I should be using first principle theorem (having some trouble with that) or if there is a simpler/more obvious method.
InARGS
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Is this curve known?

A median of a triangle through mid-point of $(-c,0),(c,0) $ is such so that ratio of cosines of angles between sides/median $$ \cos \phi/\cos \psi =e $$ is a constant. Is the curve known? $$ \frac {((x + c) x + y^2)}{((x - c) x + y^2)} \, \sqrt{…
Narasimham
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Hypocycloid - Direction of circle's rotation and revolution

Ive been trying to derive the equation of a hypocycloid. I am confused with one thing, in the hypocycloid is there a define direction of rotation and revolution of the smaller circle? (by direction I mean anticlockwise and clockwise). Because this…
paul
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Characterizations of cycloid

There are several motions that create a cycloid. I have some examples here. Are there any others? Trace of a fixed point on a rolling circle Evolute of another cycloid (the locus of all its centers of curvature) Involute of another cycloid (trace…
Erfan Salavati
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how to prove plane $ax+by+cz = d$ has normal vector $(a,b,c)$

Given a plane function $ax+by+cz=d$, how can one prove that unit normal vector is $$n = \pm \dfrac{a\boldsymbol{i}+b\boldsymbol{j}+c\boldsymbol{k}}{\sqrt{a^2+b^2+c^2}}$$
mrsir
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What is the f(x) equation for the wave ('jazz stripe') of a Vans Shoes?

I'm curious about how to find an f(x) equation for the "wave" logo of the Vans Shoes. It's a simple 1-dimensional wave. I've been thinking about this for a month and couldn't get the answer. Here's an image for a reference: vans' wave Could…
andirkh
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Radius function from length of curve

I have the following function definition for the length of a curve: $$ l(\theta) = {K_0 \times \sin(\theta) \over \cos(\theta) + K_1} \\ 0 \le \theta \lt \frac \pi 2 \\ K_0, K_1 \ge 0 $$ I would like a function which describes the radius of curve…
ericball
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Can an arbitrary curve in $\Bbb R^2$ be a graph of a certain equation?

Can any curve in $\Bbb R^2$ (which doesn't intersect itself) be a graph of a certain equation? In other words, if given an arbitrary curve in $\Bbb R^2$ (which doesn't intersect itself), is there a equation $f(x,y)=0$ that takes the given curve as…
Minseok Kwon
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How to slice a cone to get a given hyperbola?

Hyperbolas are made whenever a plane is normal to the radius of rotation. Which hyperbola is formed is dependent on the radius and a scaling factor. What radius R and scaling factor would be needed for 1/x? assuming:…
User3910
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if two consecutive copies of a curve do not intersect, can we add an infinite sequence?

Start with a simple non-self intersecting curve on the plane from point $A$ to $B$. Then add a translated copy of that curve next to it, so they from a new curve, and assume that also the new curve does not self-intersect. My question is: if I…
Hendrik Jan
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Real Curves/Circles

Unfortunately I am very ignorant when it comes to mathematics. Please understand and forgive me if this question reflects that. Thank you! I have an observation: Every curve and every circle in the real world is really just a bunch of straight lines…
Dov F
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Rotation of a plane with respect to a straight line

The plane x-2y+3z=0 is rotated through a right angle about its line of intersection with the plane 2x+3y-4z-5=0; Find the equation of the plane in its new position?
user383053
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Lipschitz constant of Polar parametrization

If $p(t)=P(e^{it})=r(t)e^{it}$, where $r(t)>0$ is a mapping between $[0,2π]$ i.e. between the unit circle $T$ and a curve, then it seems that $\mathrm{Lip}(p)=\mathrm{Lip}(P)$ but I have't the proof.
Marijan
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Basic plane question, finding a plane traveling through the heads of 3 given vectors.

Find the equation for the plane passing through the heads of the three given vectors (2, 2, 0) (−1, 2, 1) (1, 1, 4) If I was given 3 points, I know how to do this. Simply find AB x AC and plug one of the points into the equation to get a D…
Luke
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move curve normal to itself

I have a plane curve given by $y = f(x)$. At every point on this curve, I construct a normal direction to this curve and move the point a fixed distance $s$ along the normal to a new point. What is the equation of this new curve ? For example, for…
user265917
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