Writing Parametric Equations from Cartesian Equations

[Big-Daddy]

What is the general method for writing Cartesian equations as parametric equations?

For something as simple as y=f(x) we can write x=t and y=f(t) with the same function, but what about something more complicated, generally f(x,y)=0 - how can we make 2 parametric equations to represent a case where, for instance, both x and y have indices (neither 0 nor 1) in the Cartesian equation?

 

[Simon Bridge]

What is the general method for writing Cartesian equations as parametric equations?

I don't think there is one. There are lots of ways to parameterise a relation.

For something as simple as y=f(x) we can write x=t and y=f(t) with the same function, but what about something more complicated, generally f(x,y)=0 - how can we make 2 parametric equations to represent a case where, for instance, both x and y have indices (neither 0 nor 1) in the Cartesian equation?

put: x=g(t), y=h(t) then f(t)=f(g(t),h(t)) is the parameterization.

The details depend on the type of function and what you need the parameterization for.

 

[Big-Daddy]

Ok, firstly, is it always possible to write a couple of parametric equations x(t), y(t) for any Cartesian equation in x and y?

Could you provide some links on how to convert from Cartesian to parametric equations?

 

[Simon Bridge]

You can always parameterize a function but you cannot always do so usefully.
i.e. say that z=f(x,y) represents the height of a terrain above a reference level ... what would be a useful parameterisation?

Depends on what you want to do with it right?

Usually you start out by parameterizing curves.
You can think of the parameter as a time value and the parameterization is the way the position coordinate changes with time. The resulting equation traces out a trajectory.

Treatments are a bit tricky to find since there are no standard ways to go about it.

Paul's notes deal with parametric equations (1st link) and an application to line integrals (2nd link).
http://tutorial.math.lamar.edu/Classes/CalcII/ParametricEqn.aspx
http://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtI.aspx

The arc-length parameterization in some detail:
http://www.math.hmc.edu/math142-01/mellon/Differential_Geometry/Geometry_of_curves/Parametric_Curves_and_arc.html