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

 

[CellCoree]

The general method for writing Cartesian equations as parametric equations involves representing each variable, x and y, as a function of a third variable, typically denoted as t. This third variable, known as the parameter, allows us to express the Cartesian equation in terms of two separate equations, one for x and one for y, that can be plotted separately to create a parametric curve.

For more complicated Cartesian equations, such as f(x,y)=0 where both x and y have indices, the process is similar. We can still represent x and y as functions of t, but these functions may involve more complex expressions such as trigonometric functions or logarithms. The key is to manipulate the original Cartesian equation to isolate one variable and then express it in terms of t. This can be repeated for the other variable to create the parametric equations.

It is important to note that there may be multiple ways to represent a Cartesian equation as parametric equations, as the choice of parameter can vary. Additionally, the range of t may also need to be carefully chosen to capture the entire curve. Overall, the process involves breaking down a complex Cartesian equation into simpler components that can be represented parametrically, allowing for a more comprehensive understanding and visualization of the curve.