# 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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### Prove that there are infinitely many triples of integers $(x, y, z)$ s.t. $x^2+y^2=(m^2+n^2)^{z}$ with positive integers $m, n$ and with $z$ even

To prove that there are infinitely many triples of integers $(x, y, z)$ such that $x^2+y^2=(m^2+n^2)^{z}$ with positive integers $m, n$ and with $z$ even, I tried to apply the parametric method, with ...
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### Can an arbitrary path in $\mathbb{R}^3$ be parametrized as $f(t)\hat{u}$ for a basis vector of some arbitrary coordinate system?

This question comes from a recent question in Physics SE I answered, link. This has sparked a question I don't have the background to answer. Consider some arbitrary path in $\mathbb{R}^3$. Some of ...
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### Radius of curvature of parametric equation for a helix

The equations i have for the $x, y, z$ coordinates for a helix are as follows $$x = R\cos(\omega t)$$ $$y = R\sin(\omega t)$$ $$z = vt$$ Where v is the velocity of the particle moving on the path of ...
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### How can i parametrize rationally an hyperbola with the form $-aX^2 + bY^2 =1$?

I know that a rational parametrization of an hyperbola with the form $$-(\frac{x}{c})^2 + (\frac{y}{d})^2 =1$$ is: $$(c·\frac{t^2-1}{2t}, d·\frac{1+t^2}{2t})$$ The problem is that if I transform my ...
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### Parameters behind non-symmetric Lissajous loop?

I'm trying to guess what kind of two signals can create this kind of Lissajous curve: However I can't figure out what are the parameters that break the symmetry of the curve. The relative phase ...
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### Computing a surface integral

I have to verify that the line integral and the surface integral of Stokes' Theorem are equal for the vector field $\boldsymbol{\mathrm{F}}(x,y,z)=(x,y,z)$ and the portion of the surface $S$ defined ...
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### Confusion about orientation of line integral

Consider a continuous curve $\gamma$ in $\mathbb{R}^3$ parameterized by $\mathbf{r}(t)$ as $t:a \rightarrow b$. Now, it is my understanding that line integrals $$\int_\gamma \phi \ ds$$ for a scalar ...
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### Curve is traveled clockwise or anti-clockwise

Given the curve $$\vec{\mathbf{r}}(t) = \bigl(t - \tfrac{1}{3}t^3 \bigr) \, \vec{\mathbf{i}} + (4 - t^2) \, \vec{\mathbf{j}},$$ how can I tell whether it's traveled clockwise or counterclockwise? ...
1 vote
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### How can i find the way to escape robot?

A scout robot is trapped at the center of a square with side length 1km. The scout robot can move at a speed of 3km/h. A guard robot is located at a vertex of the square, and patrols the boundary of ...
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### How can I find a positively oriented parametrization?

I need to find a parametrization of the surface defined by $x^{2}+y^{2}+z^{2}=9$, $x+y+z\geq 1$. The orientation must be compatible with the orientation of its boundary, that is, if the loop lies in ...
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### If $\psi(u,v) = (u, u^2 + v^2, v)$ and $\alpha(t)=(ct - b, 5t^2 - ct + c, t + a)$ is a curve with $\alpha$ in $\text{img}(\psi)$, then $a + b + c$ is
If $\psi(u,v) = (u, u^2 + v^2, v)$ with $(u,v) \in \mathbb{R}^2$ and $\alpha(t) = (ct - b, 5t^2 - ct + c, t + a)$ is a curve with $\alpha \subset \text{img}(\psi)$, then $a + b + c$ is .... Is anyone ...