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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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28
votes
5answers
4k views

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 ...
12
votes
3answers
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 $f(r,l)$ ...
10
votes
1answer
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 ...
8
votes
1answer
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 ...
7
votes
3answers
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: ...
6
votes
2answers
191 views

Parametrizing the square spiral

Related to this question concerning number spirals I have another one, more specific. While it is rather easy to arrange the natural numbers along an Archimedean spiral by $$x(n) = \sqrt{n}\cos(2\pi\...
6
votes
1answer
58 views

Is any parametrization of a smooth curve smooth? Can we always find a smooth parametrization of a smooth curve?

I assume that this must be true because the parametrization describes the same object, but I cannot recall a theorem that would state this explicitly.
5
votes
4answers
18k views

Parametrization for the ellipsoids

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 ...
5
votes
2answers
134 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 $\Bbb R^2 \to \Bbb R$...
5
votes
1answer
70 views

Finding the area inside an implicitly defined curve $x^2 + (y + \sqrt[3]{|x|})^2=1$

Need help finding the area inside an implicitly defined curve $x^2 + (y + \sqrt[3]{|x|})^2=1$. (I think it is a heart shape). I've been trying to parameterize it with no luck. I also tried to restrict ...
5
votes
1answer
92 views

How might I find, in parametric form, the solution to this (first order, quasilinear) PDE?

Consider the (first order quasilinear) PDE $$ u \frac{\partial u}{\partial x} + (y+1) \frac{\partial u}{\partial y} = u \hspace{10mm} x \in \mathbb{R}, \hspace{4mm} y \in \left( 0, \hspace{2mm} \frac{...
5
votes
1answer
95 views

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 ...
5
votes
1answer
227 views

Finding the volume of a pseudosphere that has been parametrised in $\theta$ and $t$

I've got a problem calculating the volume of the top half of a pseudosphere. The pseudosphere is parametrised by $$\Phi(t,\theta) = \Big ( \frac{\cos(\theta)}{\cosh(t)}, \frac{\sin(\theta)}{\...
5
votes
1answer
84 views

What fields/literature should I explore to learn how to parametrize surfaces, with constraints, that fit data?

This is a somewhat vague question, but I think a good one! I sometimes find myself in the position of having to fit a surface to a point cloud. Because of the nature of the data, I sometimes want to ...
5
votes
0answers
84 views

Genus of trancendental curve

I understand that an algebraic curve is genus 0 iff it can be parameterized using rational functions. I am curious if there is a simple way to know if a transcendental curve is genus 0? Specifically ...
4
votes
1answer
340 views

How many parametrisations are needed to cover a sphere?

I have seen that a sphere can be covered with 6 parametrisations, but is it possible to totally cover a sphere with less parametrisations/charts?
4
votes
3answers
467 views

Evaluate $\ \int_T \vec F \cdot d\vec r $ where $\vec F (x, y, z) = (2xy + 4xz)\vec i + (x^2 + 6yz)\vec j + (2x^2 + 3y^2) \vec k$

I'm a bit confused as to what I've to do for this exam question. $$\vec F (x, y, z) = (2xy + 4xz)\vec i + (x^2 + 6yz)\vec j + (2x^2 + 3y^2) \vec k , \qquad x, y, z ∈ \mathbb{R}. $$ Let $T$ denote ...
4
votes
3answers
168 views

Parameterizing same curve in different directions

$$\textbf{r}_1(t)= \sin t \textbf{i}+ \cos t \textbf{j}$$ and $$\textbf{r}_2(t)= \cos t \textbf{i}+ \sin t \textbf{j}$$ represent the same curve but traversed in different directions. $\textbf{r}_1$ ...
4
votes
1answer
890 views

Parametric Equation for Rectangular Tubing with Corner Radius

I'm working on a problem where I need the parametric equation for a complex shape. Parametric equation of a circle: x = a * cos θ y = a * sin θ Parametric ...
4
votes
1answer
218 views

Why the covariant derivative does not depend of the parametrization?

I'm studying Differential Geometry using the book "Diferential Geometry of Curves and Surfaces - Manfredo P. do Carmo", and he defines covariant derivative as: Let $S \subset \mathbb{R}^3$ a surface, ...
4
votes
3answers
624 views

Find the minimum and maximum distance between the ellipsoid of ecuation $x^2+y^2+2z^2=6$ and the point $P=(4,2,0)$

I've been asked to find, if it exists, the minimum and maximum distance between the ellipsoid of ecuation $x^2+y^2+2z^2=6$ and the point $P=(4,2,0)$ To start, I've tried to find the points of the ...
4
votes
2answers
56 views

Is there a way to parametrise general quadrics?

A general quadric is a surface of the form: $$ Ax^2 + By^2 + Cz^2 + 2Dxy + 2Eyz + 2Fxz + 2Gx + 2Hy + 2Iz + J = 0$$ It can be written as a matrix expression $$ [x, y, z, 1]\begin{bmatrix} A && ...
4
votes
1answer
30 views

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 ...
4
votes
2answers
468 views

Polar radius of a general ellipsoid

Is there a proper parametrization of a general ellipsoid in spherical coordinates? The regular parametrization is this: $$x=a\cdot \cos\phi \cos\theta\\y=b\cdot \cos\phi \sin \theta\\z=c\cdot \sin\...
4
votes
0answers
57 views

Help me see the connection between exponential functions and quadratic curves

We know that the degree 2 equations $x^2 + y^2 =1$ and $x^2 - y^2 =1$ can be parametrized by exponential functions. How come exponential functions show up in this seemingly unrelated area? I think it ...
4
votes
0answers
115 views

Diffeomorphisms preserve tangency between curves

I've been trying to solve the following problem: Prove that if two regular curves $C_1$ and $C_2$ of a regular surface $S$ are tangent at a point $p \in S$, and if $\alpha\colon S \to S$ is a ...
3
votes
4answers
132 views

Smooth parametrization of curve $\cosh x+\cosh y = \operatorname{constant}$

I have a curve (image below) $$ \cosh x+\cosh y = C,\qquad C>2. $$ I would like to get its smooth parametrization of form $$ x = f(t),\qquad y=g(t),\qquad t\in[a,b], $$ so for every point on the ...
3
votes
4answers
220 views

Parametrizing the surface $z=\log(x^2+y^2)$

Let $S$ be the surface given by $$z = \log(x^2+y^2),$$ with $1\leq x^2+y^2\leq5$. Find the surface area of $S$. I'm thinking the approach should be $$A(s) = \iint_D \ |\textbf{T}_u\times \textbf{T}_v|...
3
votes
3answers
1k views

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 ...
3
votes
2answers
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 ...
3
votes
2answers
168 views

Is every path a reparametrization of a path with non zero velocity?

Let $\alpha:[0,1] \to \mathbb{R}^n$ be a non-constant smooth path, and suppose $\alpha'(0)=0$. Do there exist a smooth path $\beta:[a,b]\to \mathbb{R}^n$ and a smooth increasing function $h:[0,1] \to [...
3
votes
1answer
79 views

Hugoniot Locus given by parametric curves

I need to prove that the Hugoniot Locus of a point $\hat{u}$ of the equation $$u_t + f (u) _x = 0,\qquad f\in C^2$$ is the set of $n$ curves $$\begin{cases}\tilde{u}_p(\xi, \hat{u})=\hat{u}+\xi r_p(\...
3
votes
2answers
248 views

How to calculate the surface area of parametric surface?

Suppose you have the surface $z=3xy$ and you want to find the area that lies within the cylinder $x^2+y^2\leq 1$. My homework is forcing me to use the parameterization $$\textbf{r}_1(s,t)= <s\...
3
votes
2answers
1k views

Calculating the surface of revolution of a cardioid.

I have the cardioid $r=1+cos(t)$ for $0\leq{t}\leq{2\pi}$ and I want to calculate the surface of revolution of said curve. How can I calculate it? The parematrization of the cardioid is: $$x(t)=(1+...
3
votes
2answers
1k views

Surface area of sphere within a cylinder

I have to Compute the surface area of that portion of the sphere $x^2+y^2+z^2=a^2$ lying within the cylinder $\Bbb{T}:=\ \ x^2+y^2=by.$ My work: I start with only the $\Bbb{S}:=\ \ z=\sqrt{a^2-x^...
3
votes
1answer
67 views

Parametrizing $(2x+y)^2(x+y)=x$

I need to find $x=x(t)$ and $y=y(t)$ so that the implicitly defined curve on $\mathbb R^2$ $$(2x+y)^2(x+y)=x$$ is converted into an explicit function of the parameter $t$ that can be analysed using ...
3
votes
1answer
356 views

Bézier curves: a detailed explanation of how draw a letter

After I known the background of Bézier curves, I would like to know is you can provide us a detailed example of how to compute the drawing of a letter, using quadratic Bézier curves. As tell us this ...
3
votes
5answers
2k views

Regular curve which normal lines pass through a fixed point

I have the following question: Assume that $α$ is a regular curve in $R^2$ and all the normal lines of the curve pass though the origin. Prove that $α$ is contained in a circle around the ...
3
votes
2answers
51 views

Parametric version of a simple equation

I have a simple relation that I need to plot in a plane. I could do it, but I believe that I don't get the best way. A plane curve is defined implicitely by the following equation : \begin{equation}\...
3
votes
2answers
380 views

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,...
3
votes
2answers
164 views

Solve Diophantine equation with three variables part two

I want to find all solutions of $$x^2+y^2+z^2-xy-yz-zx-x-y-z=0$$ Solutions need not to be primitive. I found several parametric family. For example $(m^2, m^2+m , (m+1)^2)$ $(m^2, m^2+m+2 , (m+1)^2)$ ...
3
votes
2answers
59 views

Parametrically Defined Curves: $f'$ and $g'$ Are Not Simultaneously Zero

I can't find a clear, comprehensive explanation, on this site or elsewhere, for why parametrically defined curves frequently have the condition that the the derivatives of their points $x = f(t)$ and $...
3
votes
3answers
130 views

Parametrizing a set

I'm trying to parameterize the following set $$ \left\lbrace\, r\left(\cos\left(\alpha\right),\sin\left(\alpha\right)\right) \;\middle\rvert \quad 0\leq\alpha\leq2\pi, \quad 0<r\leq 1,\quad \tfrac{...
3
votes
1answer
198 views

Find the intersection multiplicity of $V(F)$ with each branch of $V(G)$ at the following points.

Let $$F = 2X_0^2X_2 - 4X_0X_1^2 +X_0X_1X_2 +X_1^2X_2$$ and $$G = 4X_0^2X_2-4X_0X_1^2+X_0X_1X_2-X_1^2X_2$$ Find the intersection multiplicity of $V(F)$ with each branch of $V(G)$ at $(0:1:0), (0:0:1),...
3
votes
1answer
209 views

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 ...
3
votes
2answers
1k views

Parametrization of a tetrahedron?

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 $...
3
votes
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 \...
3
votes
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 ...
3
votes
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 $K = -\frac{\varphi''}{\varphi}$, $\...
3
votes
1answer
263 views

Rotating cycloid

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 $x=\pi,y=0$. ...