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Curves

A curve in the plane or in 3 dimensional space is mathematically the result of a mapping from the real numbers to the plane or space. We call this mapping a path and define it mathematically as follows:

Path

A path in $\mathbb{R}^n$ is a univariate vector-valued function: $\vec{c}:[a,b] \to \mathbb{R}^n$. The curve C is the collection of points $\vec{c}(t)$ with $t \in [a,b]$. We say that $\vec{c}$ parametrizes $C$ and write $\vec{c}(t)=(x_1(t),x_2(t),\dots,x_n(t))$ and call $x_i(t)$ the $i-th$ component scalar function of $\vec{c}$.

Example Curves

 

straight line through $(a,b,c)$:
$\vec{c}(t)=(a,b,c)+t\vec{v}$
parabolic arc:
$\vec{c}(t)=(t,t^2)$
unit circle:
$\vec{c}(t)=(\cos t,\sin t)$ or $\vec{c}(t)=(\cos 2t,\sin 2t)$

cycloid:

A cycloid is a curve a particle on a rolling wheel traces out.

We derive the path $\vec{c}(t)$ that parametrizes this curve. Assume the wheel has a radius $R$, is rolling in the positive $x$-direction along a straight line with constant speed $v$. The position of the center at $t=0$ is at $(0,R)$. The distance between the center and the particle whose path we derive is $r \leq R$

Then the path of the center is described by: \begin{equation} \vec{C}(t)=(vt,R) \end{equation}

The motion of the particle relative to the center is given by: \begin{equation} \vec{d}(t)=(r\cos \theta_t,r\sin \theta_t) \label{eq01} \end{equation} We notice that the particle rotates in clockwise direction started at $\theta_0=-\tfrac{1}{2}\pi$ with an angular velocity: \[ \frac{d\theta_t}{dt}=\frac{2\pi}{2\pi R/v}=\frac{v}{R} \] from which we get: \[ \theta_t=-\tfrac{1}{2}\pi-\frac{v}{R}t \] We use the trignometric relations: $\cos(\phi - \tfrac{1}{2}\pi)=\sin\phi$ , $\sin(\phi - \tfrac{1}{2}\pi)=-\cos\phi$ and $\cos(-\phi)=\cos \phi$ and $\sin(-\phi)=-\sin \phi$ and substitute these relations in $\ref{eq01}$ to get: \[ \vec{d}(t)=(-r \sin(\frac{vt}{R}),-r\cos(\frac{vt}{R})) \] Finally we combine the motion of the center with the relative motion of the particle to get the path for the cycloid curve: \begin{equation} \vec{c}(t)=(vt-r \sin(\frac{vt}{R}),R-r\cos(\frac{vt}{R})) \end{equation}

References



Copyright 2012 Jacq Krol. All rights reserved. Created 2011; last updated September 2012.