Articles of parametric

How can I re-write an equation (or system of equations) in parametric form?

For the equation $y = 3x$ I need to re-write $x$ and $y$ in terms of a variable $t$. How can I find the value of each variable in terms of $t$?

Parametric equations where sin(t) and cos(t) must be rational

Suppose there are parametric equations $$ x(t) = at – h\sin(t) $$ $$ y(t) = a – h\cos(t) $$ and it is required that both $\sin(t)$ and $\cos(t)$ should be rational. What the values of $t$ should be in that case? Thanks.

Is parametric form of a given function unique?

Can we say that for any given function in single/multivariable, it is always possible to have a parametric form? (Elementary functions, complicated functions?) Given any function, is parametric form uniquely determined?

Parametric Equations for a Hypercone

The n-dimensional cone, with vertex at the origin, central angle, $\alpha$ and central axis in the direction of the unit vector $\xi$ is defined to be all those points, $x\in {R^n}$ whose dot product with $\xi$ is |$x$|$cos(\alpha)$. How would I find parametric equations for this surface?

Relation between $\sin(t)$($\cos(t)$) and $\sin(at)$ ($\cos(at)$) when both are rational

This question relates to Parametric equations where sin(t) and cos(t) must be rational. Suppose it is given that $\cos(t)$ and $\sin(t)$ are both rational and also $\cos(at)$ and $\sin(at)$, where $a$ is some constant, are both rational too. It is known that when $\cos(t)$ and $\sin(t)$ are rational, $$ \cos t = \frac{m^2-n^2}{m^2+n^2} \quad ; […]

How do we prove that two parametric equations are drawing the same thing?

For example, if I have $$\begin {align} x(t) &= r\sin t\cos t\\ y(t) &= r\sin^2 t\\ \end {align}$$ and $$\begin {align} x(t) &= \frac r 2 \cos t\\ y(t) &= \frac r 2 (\sin t + 1) \end {align}$$ How do we show that the two parametric equations draw the same line?

Show that the parameterized curve is a periodic solution to the system of nonlinear equations

First I tried to convert the system to polar coordinates. This only made things worse (unless I made some idiotic mistake). Can I plug in the given ellipse (rectangular coordinates) into the system and solve from there? If so, we get x’ = -4y and y’ = x. The periodic orbits of this system are […]

Parameterising the intersection of a plane and paraboloid

Suppose we have the paraboloid $z=x^2+y^2$ and the plane $z=y$. Their intersection produces a curve $C$, and certain surfaces bounded by it, for example the disc $S$ which directly fills the area of $C$ and the paraboloid $S’$ given by $z=x^2+y^2$ which extends from $C$ downwards and is bounded by $C$. My question is on […]

How should this volume integral be set up?

I would like to find the (four-dimensional) volume of the region given by $$xy>zw \quad\wedge \quad x>-y \quad\wedge \quad x^2+y^2+z^2+w^2<1,$$ for $x,y,z,w\in\mathbb{R}$ and where the last condition means that the whole thing is bounded by the unit $4$-ball. The boundary of the region given by the two first conditions would be (I think) \begin{align} xy=zw […]

Derive parametric equations for sphere

How do you derive the parametric equations for a sphere? \begin{align} x & = r \cos(\theta)\sin(\varphi), \\ y & = r \sin(\theta)\sin(\varphi), \\ z & = r \cos(\varphi), \end{align} where $\theta$ is from $0$ to $2\pi$ and $\varphi$ is from $0$ to $\pi$. There are no good explanations online.