# 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.

672 questions
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### Is there a program that receives as input your drawing of a curve and outputs a parametric curve tracing it (reasonably close)?

From what I know, B-Splines is the closest thing that we have to drawing curves and having them defined by the computer. I have some B-Spline code that does this interactively. However, those are a ...
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### Find surface area by calculating surface integrals

Fix a radius $r > 0$ and two angles $ϕ_1$ and $ϕ_2$, with $−π/2 < ϕ_1 < ϕ_2 < π/2$ Find the surface area of the portion of the sphere of radius r with latitudes between $ϕ_1$ and $ϕ_2$. ...
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### Parametrisation of cardioid $r=1-\cos \theta$

I know cardioid $r=1-\cos \theta$ can be parametrized to $$\gamma:[0,2\pi]\rightarrow \mathbb{R^2}, \space \gamma(\theta)=((1-\cos \theta)\cos \theta, (1-\cos \theta)\sin \theta)$$ But how is this ...
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### Circular Arc Prametrization not Using Radius

In an optimization problem I have to parametrize a circular arc. Thus far, I have reduced a more general problem to the figure below: The figure shows a symmetrical circular arc, with chord length L,...
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### Any parametrization $h\colon S^1\to \Gamma$ possible?

Consider the following model, and, in particular, the second picture given in the link, namely the blue line which represents a trajectory which I call $\Gamma$. Is it possible to give any ...
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### Parametric line segment in 3-space

If one wants to parametrize a straight line segment in $\mathbb{R}^3$, which goes from $(1,0,0)$ to $(0,1,\pi/2)$, would this approach be correct? First, we come up with the $xy$-plane equation, ...
If one wants to parametrize the curve of intersection in $\mathbb{R}^3$ of the surfaces $y=x^2$ and $z=x^3$, would it be correct to parametrize this curve as $\bf{g}$$(t)=(t, t^2,t^3)? The reason I'... 2answers 127 views ### I don't understand why we represent functions f:I \subseteq \Bbb R \to \Bbb R^2 the way we do. I don't understand why we represent functions f:I \subseteq \Bbb R \to \Bbb R^2 the way we do: doing an analogy with how we represent functions from \Bbb R to \Bbb R or from \Bbb R^2 \to \Bbb R... 1answer 25 views ### One parametric family that interpolates continuously between identity and natural logarithm on (0,1] I am looking for a family of continuous functions f_p, (0,1]\to\mathbb{R}, and p\in [1,\infty) that fulfill$$ f_1 \equiv \log(x) \\ \lim_{p\to \infty} f_p \to x$$for x\in (0,1]. I ... 1answer 56 views ### Reverse-engineering a parametrization Let's say you have a polynomial depending on complex parameters A,B:$$p(x,y) = (y + Ax)(y + Bx) + (A-B)^2$$One parametrization of zero points of this polynomial is given by$$ x(t) = -(t + t^{-1})... 1answer 50 views ### Analytic formula for parameterizing the below family of curves I'm trying to find an analytic formula for a curve that can look like any of the curves below depending on one or more parameters. My initial thought was to use exponentials, something that might ... 1answer 60 views ### Which is this curve? What curve is given from $$\gamma (\theta)=\left (\sin (\theta +c)\cos \theta , \sin (\theta +c)\sin \theta \right ), \text{ where } c \text{ is a constant }$$ ? How can we find it? 1answer 119 views ### Parametrization of the curve A cycloid is a flat curve that is traced by point of the rim of a circle while the circle rolls without slippage on the line. Show that if the line is the axis$x$and the circle has radius$a>0$, ... 0answers 182 views ### Surface area with double integral - how to parameterize? Problem: Find the surface area of the part of the cylinder$x^2+z^2 = a^2$that is inside the cylinder$x^2+y^2 = 2ay\;$and also in the positive octant$( x \ge 0, y\ge 0, z\ge 0$). Assume$a > 0\$....
Can someone help to describe some possible parametrizations for the ellipsoid: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1?$$ I am thinking polar coordinates, but there may be the ...