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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.

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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 ...
2k views

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 $f(r,l)$ ...
474 views

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 ...
770 views

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 ...
789 views

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: ...
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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 ...
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Compute $\int_{\Gamma}\omega$ where $\omega=(y-2z)dx+(x-z)dy+(2x-y)dz$

Compute $\int_{\Gamma}\omega$ where $\omega=(y-2z)dx+(x-z)dy+(2x-y)dz$ and $\Gamma$ is the intersection between: $x^2+y^2+z^2=r^2$ and $x-y+z=0$ My attempt: $\Gamma$ is some kind of ellipse in the ...
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Show that two cardioids $r=a(1+\cos\theta)$ and $r=a(1-\cos\theta)$ are at right angles.

Show that two cardioids $r=a(1+\cos\theta)$ and $r=a(1-\cos\theta)$ are at right angles. $\frac{dr}{d\theta}=-a\sin\theta$ for the first curve and $\frac{dr}{d\theta}=a\sin\theta$ for the second ...
279 views

Parametrization of $a^2+b^2+c^2=2d^2$

Is a complete parametrization of primitive solutions to the equation $$a^2+b^2+c^2=2d^2\qquad a,b,c,d \in \mathbb{Z}$$ known? A reference would be great. Solutions to $a^2+b^2=c^2$ give solutions ...
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Does the 3.3.3 equation $x_1^3+x_2^3+x_3^3=y_1^3+y_2^3+y_3^3$ have a complete integer or rational solution?

Consider the equation $$X_1^3+X_2^3+X_3^3 = Y_1^3+Y_2^3+Y_3^3. \tag{\star}$$ Does this have a complete solution (a.k.a. parameterization) in integers? If not, does it have a complete solution in ...
I need to do a surface integral over the surface bounded by $x + 2y + z = 4$ and the coordinate planes. How do you go about parametrizing this surface? My first thought was with $x = x$, $y = y$, and $... 1answer 192 views Conformal reparametrization We consider $$\sigma (u,v)=(f(u)\cos v, f(u)\sin v, g(u))$$ Picking$u=\theta , v=\phi , f(\theta )=\cos \theta , g(\theta )=\sin \theta$we get that the first fundamental form is$$d\theta^2+\cos^2 \... 1answer 55 views Showing that$f(u,v) = (u,v,u^2-v^2)$is a parametrization. Exercise : Let$f(u,v) = (u,v,u^2-v^2), \; (u,v) \in U$where$U$is a coherent and open subset of$\mathbb R^2$. Show that$f$is a parametrization of a$C^\infty$patch (local surface) of a ... 1answer 68 views Surfaces of revolution perpendicular to the plane xOy Let$S$be a surface of revolution parametrized by$(\varphi(v) \cos u, \varphi(v) \sin u, \delta(v))$. Assume$S$has constant Gaussian curvature equal to 1. Since$K = -\frac{\varphi''}{\varphi}$,$\...
We have cycloid $(x=t-\sin t,y=0,z=1-\cos t)$ and $t\in(0,\pi]$. Now I need do integrate some differential form over manifold $M$ which is created by rotating above cycloid around line $x=\pi,y=0$. ...