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Questions tagged [parametrization]

For questions on parametrization, the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation.

Parametrization is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. "To parameterize" by itself means "to express in terms of parameters".

Parametrization is a mathematical process consisting of expressing the state of a system, process or model as a function of some independent quantities called parameters. The state of the system is generally determined by a finite set of coordinates, and the parametrisation consists thus of one function of several real variables for each coordinate. The number of parameters is the number of degrees of freedom of the system.

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What is the lowest degree polynomial that turns a circle into a nontrivial knot?

The polynomial $$f(x,y) = \begin{bmatrix} (x^2-y^2)(y(4x^2-1)+2) \\ 2xy(y(4x^2-1)+2) \\ x^3-3xy^2 \end{bmatrix} $$ maps the circle $\left\{ (x, y) \mid x^2 + y^2 = 1 \right\}$ to a trefoil knot, and I suspect that this is the lowest…
Zoe Allen
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Is this an ellipse?

Is this parameterisation an ellipse: \begin{align}x(t) &= \frac{2 \cos(t)}{1 + a \sin(t)}\\ y(t) &= \frac{2 \sin(t)}{1 + a \sin(t)}\end{align} where $a$ is a real positive parameter. I tried to do it the naive way but couldn't find a…
Richard Kruel
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Parametric curve resembling a bean.

I am looking for a parametric closed curve that roughly resembles a bean. I am looking for something with an explicit parametrization of the form $C(t) = (X(t), Y(t))$ I tried searching online but "parametric bean" is not yielding much of use.
Makogan
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Idea behind "reparameterization hiding a corner" in single variable calculus

I just solved question #2 on p. 248 from Spivak's Calculus Fourth Edition (2008). Solving it wasn't the issue. I'm trying to understand the idea behind it. This is a screenshot of the question: I'm trying to understand what is meant by…
D.C. the III
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Parametrizing the square spiral

Related to this question concerning number spirals I have another one, more specific. While it is rather easy to arrange the natural numbers along an Archimedean spiral by $$x(n) = \sqrt{n}\cos(2\pi\sqrt{n})$$ $$y(n) =…
Hans-Peter Stricker
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The shape of a candle flame

Has anyone worked out a physically justified equation (perhaps parametrized) for the characteristic (2D outline) shape of a candle flame? Just one half suffices, as it is clearly symmetric about a vertical (in the absence of…
Joseph O'Rourke
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The length of coil winding on cylinder.

Problem A electrical engineer needs a new coil and decides to make one from scratch. He hasn't decided the radius or length of the cylinder on which the coil will be wound. Define a function $f(r,l)$ where $r$=(radius), $l$=(height of the cylinder).…
Tuki
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How to Paramaterize $2\cos(x/2)\cos(y/2)=1$?

$2cos(x/2)cos(y/2)=1$" /> This curve of $2\cos(x/2)\cos(y/2)=1$ looks like a circle squished in from the sides and top and bottom. I know how to parameterize the curve by dividing it into four 90 degree segments, but second derivatives of the…
Roger Wehage
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Area inside a sextic curve. $y^6 + x^4 = 36xy + 4$

I help out high school students do their homework, and one of them said that this was a curve given to them to find the area of, using solely geometric arguments. (Pre-algebra he claims, but the formula clearly has $x$ and $y$.) $$y^6 + x^4 = 36xy…
OmnipotentEntity
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Parametric equation for a space curve

With reference to the following image: the blue curve has trivially a parametrization: $$(x, y, z) = (\cos\theta, \, \sin\theta, \, 0) \; \; \; \text{with} \; \theta \in [0,\,2\pi)$$ I would like to determine the parametric equations of the red…
user247735
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$x^2+y^2+z^2=5(xy+yz+zx)$ -- Is this all solutions?

Problem. Find all integers $x$, $y$, and $z$ that satisfy $$x^2+y^2+z^2=5(xy+yz+zx)\,.$$ Does the following parametrization give all solutions?: $$x=m^2+mn-5n^2\,,$$ $$y=-5m^2+9mn-3n^2\,,$$ $$z=-3m^2-3mn+n^2\,,$$ where $m,n$ are arbitrary…
Is Ne
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How to parametrize a "SpongeBob flower"

I am trying to figure out a way of generating a 5 lobed shaped like the flower sin the sky of spongebob: I.e. a parametric curve that is at least twice differentiable, closed and that transitions from convex to concave in regular lobes.
Makogan
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Which way should you run from the lions?

This is a fun problem that I saw somewhere on the internet a long time ago: Suppose you are at the center of an equilateral triangle with side length $s$. At each of its vertices, there is a lion which is determined to eat you. The lions start at a…
Ovi
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Why do parametrizations to the normal of a sphere sometimes fail?

If I take the upper hemisphere of a sphere, $x^2 + y^2 + z^2 = 1$, to be $\sqrt{1 - x^2 - y^2}$, then the normal is given by $\langle -f_x, - f_y, 1\rangle$ at any point. This leads to an odd result: on the plane $z = 0$, while one might expect all…
Muno
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More $a^3+b^3+c^3=(c+1)^3,$ and $\sqrt[3]{\cos\tfrac{2\pi}7}+\sqrt[3]{\cos\tfrac{4\pi}7}+\sqrt[3]{\cos\tfrac{8\pi}7}=\sqrt[3]{\tfrac{5-3\sqrt[3]7}2}$

I. Solutions In a previous post, On sums of three cubes of form $a^3+b^3+c^3=(c+1)^3$, an example of which is the well-known, $$3^3+4^3+5^3=6^3$$ we asked if there were polynomial parameterizations higher than 3rd degree. The answer was in the…
Tito Piezas III
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