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.

How to write a parametric equation for a circle? Look at a circle on the XY plane. It has upper and lower bounds. Every X is bounded between [math]\displaystyle{ [-a,\ a] }[/math], the same for Y. What function relates to that? Sine and cosine. Therefore [math]\displaystyle{ (x,\ y) = (\sin(t),\ \cos(t)) }[/math] is a parametric equation that plots a circle on the XY plane. Multiply both coordinates by the same constant and we make the radius larger. By different constants each and we have an ellipsis. Multiplying the parameter by any constant changes the velocity of the particle that is running in circles. Why not two sines or two cosines? Because if both coordinates obey to the exact same function we have a straight line, not a circle. Swapping the order of the functions changes the direction, clockwise vs anti-clockwise. Another way to think is to see that the equation of a circle really resembles the identity [math]\displaystyle{ \sin^2(x) + \cos^2(x) = 1 }[/math].

Parametric equation of a parabola: [math]\displaystyle{ (x,\ y) = (t,\ t^2) }[/math]. In this specific case it does overlap with the function [math]\displaystyle{ f(x) = x^2 }[/math], but one is a different object from the other. The function could also be written in the form of an equation [math]\displaystyle{ y = x^2 }[/math]. Careful here! It reads as "every pair where Y is equal to X squared". Don't read it as "every number which is equal to its own square", which translates to [math]\displaystyle{ 1 = 1^2 }[/math] and [math]\displaystyle{ 0 = 0^2 }[/math].