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- Thread starter nivekious
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[tex]

\gamma:[a,b]\to\mathbb{R}^n

[/tex]

such that [itex]\gamma([a,b])=C[/itex].

This is just the rough idea, does it help?

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HallsofIvy

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The whole point of parameterization is that a path, whether in 2 dimensional space or 3 dimensional space, is a

Another important thing to remember is that there is no

One fairly useful "method" is this: if y= f(x) is a

It is always possible to use "arclength" as parameter: select some point on the path as t= 0 and one direction as t> 0 (both of those can be done arbitrarily). The (x,y) corresponding to t> 0 is the point at distance t from the point at t=0 in that direction, and the (x,y) corresponding to t< 0 is the point at distance t from the point at t=0 in the opposite direction. Of course, determining

You can think "physically": imagine an object moving along the path with a given speed. Then (x(t), y(t)) is the point your object is at at time t. That's often done, of course, in Physics.

Finally, you can use some kind of geometric property. I know that, if I measure angle [itex]\theta[/itex] from the positive x axis, the point on a circle of radius R, center at the origin, at angle [itex]\theta[/itex] is given by (Rcos([itex]\theta[/itex]), Rsin([itex]\theta[/itex]) so I can use x= Rcos([itex]\theta[/itex]), y= Rsin([itex]\theta[/itex]) as parametric equations for that circle.

The important thing to remember is that there is

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