Mathematics Stack Exchange
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Questions tagged [parametric]

For questions about parametric equations, their application, equivalence to other equation types and definition.

In mathematics, a parametric equation of a curve is a representation of this curve through equations expressing the coordinates of the points of the curve as functions of a variable called a parameter. This contrasts with implicit equations that define a curve as the zero set of some equation in the coordinates.

The parametric forms of curves are well-suited for drawing on a computer, while their corresponding implicit forms are useful for analytic manipulations (intersections, etc.)

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Do "Parabolic Trigonometric Functions" exist?

The parametric equation $$\begin{align*} x(t) &= \cos t\\ y(t) &= \sin t \end{align*}$$ traces the unit circle centered at the origin ($x^2+y^2=1$). Similarly, $$\begin{align*} x(t) &= \cosh t\\ y(t) &= \sinh t \end{align*}$$ draws the right part…
Argon
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Is there an explicit form for cubic Bézier curves?

(See edits at the bottom) I'm trying to use Bézier curves as an animation tool. Here's an image of what I'm talking about: Basically, the value axis can represent anything that can be animated (position, scaling, color, basically any numerical…
subb
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Parametric Equation of a Circle in 3D Space?

So, my dilemma here is... I have an axis. This axis is given to me in the format of the slope of the axis in the x,y and z axes. I need to come up with a parametric equation of a circle. This circle needs to have an axis of rotation at the given…
iAndr0idOs
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Is "imposing" one function onto another ever used in mathematics?

First of all, let me define what I mean by "imposing," and let me clarify that I've only studied this operation in 2D Euclidean space. Now then, to impose one function onto another, you need two things: A function upon which to impose, called the…
Steven
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What is parameterization?

I am struggling with the concept of parameterizing curves. I am not even sure if I know what it means so I tried to look some things up. On Wikipedia it says: Parametrization is... the process of finding parametric equations of a curve, a surface,…
qmd
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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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Seeking proof for the formula relating Pi with its convergents

Could anyone try to prove that the below conjectured formula is valid for relating $\pi$ with ALL of its convergents - those, which are described in OEIS via $\mathrm{A002485}(n)/\mathrm{A002486}(n)$ ? $$ (-1)^n\cdot\left( \pi -…
Alex
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Is it possible to elementarily parametrize a circle without using trigonometric functions?

Just out of curiosity: Is it possible to parametrize a full circle or part of one with elementary functions but without using trigonometric functions? If so, what are advantages/disadvantages compared to the standard parametrizations using $\cos(t)$…
Viktor K.
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Why can't elliptic curves be parameterized with rational functions?

Background: For our abstract algebra class, we were asked to prove that $\mathbb{Q}(t, \sqrt{t^3 - t})$ is not purely transcendental. It clearly has transcendence degree $1$, so if it is purely transcendental, there is a transcendental $u$ and…
Henry Swanson
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Connected unbounded sets $S\subset \Bbb{R}^n$ such that $x\mapsto ||x||^t$ is uniformly continuous on $S$?

Spending the night perusing my old answers, and this question left me wondering about the following. Let's equip $\Bbb{R}^n$ with the usual Euclidean metric, and let us consider the map $N_t:\Bbb{R}^n\to \Bbb{R}$, $N_t(\vec{x})=||\vec{x}||^t$. The…
Jyrki Lahtonen
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Understanding the Equation of a Möbius Strip

I am in HL Math and trying to finish my IA. My topic is the Möbius band. The only problem is, I do not understand the formula that defines it and everywhere I have looked has just given me a math-jargon filled explanation of parametric equations…
Emily
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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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Is there a general way to parameterize all implicit functions?

We all know some curves can be described by $y=f(x)$ and some surfaces can be described by $z=f(x,y)$ However, there exists curves and surfaces which cannot be described by those, such as a circle and a sphere. Therefore, we introduce parameterized…
Alex Vong
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What causes this fractal to curl and unwind?

During my regular recreational late night Desmos foolery, I came across this fractal parametric equation: $$x(t)=\sum_{n=0}^\infty \frac{\cos(2^nt+cn)}{2^n}$$ $$y(t)=\sum_{n=0}^\infty \frac{\sin(2^nt+cn)}{2^n}$$ And decided to animate…
Graviton
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Parametric form of a plane

Can you please explain to me how to get from a nonparametric equation of a plane like this: $$ x_1−2x_2+3x_3=6$$ to a parametric one. In this case the result is supposed to be $$ x_1 = 6-6t-6s$$ $$ x_2 = -3t$$ $$ x_3 = 2s$$ Many thanks.
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