My Math Forum Polar coordinates trajectory parametrization.

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 June 9th, 2010, 09:04 AM #1 Newbie   Joined: Jun 2010 Posts: 3 Thanks: 0 Polar coordinates trajectory parametrization. Hi everybody, my name is Daniel, i'm a undergraduate of materials science and engineering at UFSC - Brazil. Here I work at the Precision Mechanics Lab. Sorry if I post in the wrong section. I think this is a multidisciplinary problem, evolving calculus, cartesian and polar coordinates. And it's engineering application. Here, I'm working on the development of a software that controls a machine, This machine is a multi-axial scratch tester, for surfaces scratching. (http://www.youtube.com/watch?v=MHMfqiBKpDI) As you can see in the video, the machine has two stages, one that executes a horizontal movement, and the other a rotatory movement. With the machine we scratch surfaces in polar trajectorys (r(x)), were the horizontal stage is the 'r", or radius, and the rotatory stage is the "x", or angle, or theta... Well, we are having success to describe the entered trajectory. But our next challenge it's being pretty hard to solve. We gotta control the tangential velocity of the scratch, and for that, we gotta control the velocity of each stage. The main objective is to make a scratch with a constant tangential velocity. For example: With a trajectory like r(x)=1*cos(x), a cardioid like in the video, how can I get velocity equations for each stage for a constant tangential velocity? Do i have do parametrize in function of the time? I appreciate your help very much.
 June 9th, 2010, 03:05 PM #2 Senior Member   Joined: Feb 2009 Posts: 172 Thanks: 5 Re: Polar coordinates trajectory parametrization. Fala Daniel, aqui quem fala é outro Daniel aqui de Niterói (RJ) e sou mestrando em Matemática da UFF. Não entendi muito bem o que você quer fazer, tem como mandar mais exemplos. Um abraço Daniel
 June 9th, 2010, 04:16 PM #3 Newbie   Joined: Jun 2010 Posts: 3 Thanks: 0 Re: Polar coordinates trajectory parametrization. Ok, i`ll answer in Portuguese and English, Em Português: Cara que massa, algum brasileiro pra me ajudar, é o seguinte: As duas mesas possuem controle de velocidade e de posição, juntando esses dois, eu descrevo uma trajetória. Por exemplo, pra r(x)=1+cos(x) com 0=
 June 9th, 2010, 05:08 PM #4 Senior Member   Joined: Feb 2009 Posts: 172 Thanks: 5 Re: Polar coordinates trajectory parametrization. Se você parametrizar com a parametrização $x(t)=\cos t+\frac{1}{2}\cos(2t)$; $y(t)=\text{sen}t+\frac{1}{2}\text{sen}(2t)$ Não te ajudaria?
 June 10th, 2010, 06:48 AM #5 Newbie   Joined: Jun 2010 Posts: 3 Thanks: 0 Re: Polar coordinates trajectory parametrization. Ok, alright, could be. But how can i get to these results in a generically way, it should fit for every polar coordinates equation. Valeu
 June 10th, 2010, 10:14 PM #6 Senior Member   Joined: Feb 2009 Posts: 172 Thanks: 5 Re: Polar coordinates trajectory parametrization. You must study each case to find a parametrization even in a simple case such as the hyperbole $x^2-y^2=1$ you need to use two parametrizations, one for the part that contains (-1,0) and one for the part that contains (1,0).

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