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 ...