Parametrization of curves
One may think on the graph of each function for each coordinate and try to find a relationship between the graph of each function and the curve. While it may be possible to do that, it's hard to make the association. With functions we can imagine the graph when the do composition, products and sum of functions. However, doing the same for curves doesn't work well because we'd be trying to make an association between graphs which are plotted with an independent and a dependent variable with a graph that is plotted with two independent variables (each coordinate is linearly independent).
The following idea is pure speculation and I have no idea if it works. In physics laboratories there is the classic free fall experiment where a paper strip is burnt with dots created with a device that produces sparks at a regular interval. By measuring the distance between the dots on the paper we draw some conclusions about the motion of a falling object. Now imagine if an object is moving along some curve in 2D given by some equation with two variables. If we could have paper strips for both axis at the same time and mark them at regular intervals, we could have some information about the rate of change in each axis. I was imagining some apparatus where the object moves and some perpendicular light source, one for each axis, projects spot shadows on each axis at regular intervals. It's easier to think on that with some computer simulation than on a real physical apparatus. The fundamental idea here is about seeing the domain of some motion in realtime. Or better say, the rate of change in each axis in realtime.
References for the physics experiment:
- https://learning.physics.iastate.edu/free-fall-spark-tape
- http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/ffallex.html
Converting an equation of two variables into parametric form
The idea is that an equation such as [math]\displaystyle{ 10 = x^2 + y^2 }[/math] can have one variable isolated and expressed as a function of the other. However, if we go straight with algebraic manipulation and write [math]\displaystyle{ x(y) = \ ... }[/math] and [math]\displaystyle{ y(x) = \ ... }[/math] that won't do us much good because we'll end up with two functions with two different variables. What we want to achieve is two functions with the same variable.
