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.
1,360 questions
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Independence of points on $E(\mathbb{Q}(t))$
I have constructed the following elliptic surface over $\mathbb{Q}$ :
$$E_t:y^2=x^3+ (-3t^2-9t+2)x+(2t^3+10t^2+1)$$
where $(t, t+1)$ and $(t+1,t-2)$ are points on $E_t(\mathbb{Q})$.
Using Nagell-Lutz ...
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Finding a parameterization of the intersection between the plane $n_x x+n_y y+n_z z=0$ and the unit sphere
Find a parameterization of the intersection between the plane $n_x x+n_y y+n_z z=0$ and the unit sphere $x^2+y^2+z^2=1$.
Stuck a little on this
Set the equations equal to each other and rearrange:
$$...
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Constant speed vector-valued function
I am trying to find a parabola (or any other shape) as a vector-valued function such that the magnitude of the velocity vector at any point is consistent. The closest I have come up with is:
let $f(t)$...
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locus of centers of deltoids [closed]
It appears to be challenging to find a curve which is a locus of the centers of deltoids passing through $3$ given points.
Maybe someone can count degree of this curve, or find number of deltoids ...
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find a parametrization for a cube
In $\mathbb R^2$, the contour $C:$
\begin{align}
\max\big(|x|,|y|\big) &= c
\\ \Leftrightarrow |x-y|+|x+y|&=2c
\end{align}
of a square can be given with a parametrization
$$c\cdot\left( \...
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Well defined contour integration over a curve
I've got a question about the invariance of a contour integral as defined in complex analysis (for example Marsden: https://ryr2008.wordpress.com/wp-content/uploads/2015/09/marsden-jerrold-michael-j-...
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Makeham's Law Parameter Approximation Using Broyden's Method
I am trying to estimate the parameters for Makeham's Law of Mortality using Broyden's method for my actuarial studies, and I'm having trouble implementing it in R.
Makeham's Law of Mortality
$$
\mu_x =...
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Is this the correct formula for parameterizing a Torus?
I am going over my professor's lecture notes, and I'm pretty sure he made a mistake. He has made errors in the past, so I just want to confirm if this equation is correct or if I am mistaken.
He has ...
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Parametric leaf profile?
I am trying to look for a nice way of parametrizing a leaf (botanical) profile. First a regular leaf, but I also would like to do an oak. The trivial answer is to use a B-Spline, but for multiple ...
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Parametrization $p x^2 + q y^2 = z^3 + r$
Let given $p,q \in \mathbb{Z}^+,r \in \mathbb{Z}$ and equation $p x^2 + q y^2 = z^3 + r$.
If exist solutions of Pell equation $qb^2-3pa^2=r\pm1$, then
$x = a(pa^2 - 3qb^2 + 3r)\\ y = b\\z = pa^2 + qb^...
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Extending to more general conics a property established for chords of circles
This is a follow up of this recent question, now closed.
In order to gather here all the information, let me first recall the question :
Initial (synthetized) question $(Q)$: Being given a circle $(C)$...
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Do geometric properties of a curve depend on its parametrization?
I’m studying how reparametrizations affect the differentiability and smoothness of curves, and I’m trying to understand which properties are intrinsic to the geometric curve and which depend on the ...
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How to obtain a nondegenerate configuration for real parabolas?
Let $P_i=(x_i,y_i)$ be eight distinct points in the plane, expressed in Cartesian coordinates.
Define
$$
m_{ij}=\frac{y_i-y_j}{x_i-x_j}.
$$
A quadruple of points $(P_i,P_j,P_k,P_\ell)$ is said to be ...
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Parametrization of the space curve formed by intersection of two cylinders
My answer includes a 3d visualization of :
Intersecting circular/parabolic cylinders
Please help find 3d curve parametrization w.r.t. a single parameter $t$.
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Geodesic on crescent-shaped 3-d uv-surface
How do you find the parametrization of the shortest curve between two points on the following surface:
I am primarily interested in the parametrization r(t), more so than the length. Although ...
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