Question #9fb71

1 Answer
Oct 12, 2017

#a(t)=(1 - cos(t), 1- sin(-t))#, where #tin(0,2pi)#



Parametrization is the process by which one can express a curve in a different space, in function of a single paramater for instance. It is useful in some situations, such as line integration.

From circle to line

From the circle below, and some trigonometry, we can find the equations:

  • x(t) = 1 - cos(t)
  • y(t) = 1 - sin(-t) (see that it makes the "clock" goes backwards)

For the initial point, set #t=0#

  • x(t) = 1 - cos(0)=0
  • y(t) = 1 - sin(-0)=1

enter image source here

Computer simulation

By using Matlab, we can plot the parametrized curve.

enter image source here

The red circle is the initial point (0,1) and the blue one final; you can show by running the simulation for a small number of points that it is clockwise; furthermore, the only way for the points on the picture is clockwise.

How do you find a parametrized curve?

In order to find a parametrized curve, you must have information about the pathway you want to parametrize, that is what it is for, transforming a function into a space like function. Given the pathway desired, you must have abilities in general with geometry; in general, it is not trivial and may require a "good eye and sense".