Parametric curves

Before I discuss parametric equations I want to talk about how some textbooks or even teachers may be, unintentionally, misleading. When textbooks introduce functions there is always the discussion that a function of one variable cannot be a circle, because by plotting a circle a function is assuming two different values for the same argument, which is not possible.

Some textbooks present a graph similar to this:

Now, a graph of a piecewise function may have this appearance:

What I'm trying to say is that, unintentionally, a textbook may be making you think that because a trajectory in 2D can have very complicated functions describing each coordinate. Then the trajectory can be a very complex graph. Not quite. In the same way some graphs of single variable functions can only be made with a piecewise function, we'd need piecewise functions to describe a super complicated trajectory with many loops and turns. One equation that describes one trajectory passing through all the points is impractical.

This tangentially relates to numerical methods because sometimes the data is spread in such a way that one single function for all is not enough. We have to look at pieces and different equations for each piece. That or we have to disregard some data to simplify the description.

Parametric equations and graphs

In here I'm assuming that you already know how to plot points and read the Cartesian system of coordinates. The first thing that people learn at school is how to read and use the Cartesian coordinate system to find points. Taking one step further and we can connect the points to form shapes, drawings or trace trajectories. Another step and we can use the concept of functions to plot curves that obey to some rule. This allow us to have more fine grained control over the lines.

A function takes in some input and outputs some value. If position in the plane or 3D space have unique coordinates, can't each coordinate be a function? Yes, they can. [math]\displaystyle{ \overrightarrow{p}(t) = (f(t), \ g(t), \ h(t)) }[/math] is a function that describes position in time. In analytical geometry one should have learned the equation of a line in the parametric form.

[math]\displaystyle{ \begin{cases} x = x_0 & + & va_0 \\ y = y_0 & + & vb_0 \\ z = z_0 & + & vc_0 \end{cases} }[/math]

XYZ are the coordinates. [math]\displaystyle{ (x_0, y_0, z_0) }[/math] a point in the trajectory and [math]\displaystyle{ v(a_0, b_0, c_0) }[/math] a vector and a constant that multiplies the vector. This works well for straight lines. What about curves? Motion in 2D or 3D can have different velocity vectors in each coordinate. Or, at each instant, the velocity vector is really a sum of one vector in X, one vector in Y, a third vector in Z.

[math]\displaystyle{ \begin{cases} x = x(t) \\ y = y(t) \\ z = z(t) \end{cases} \iff \overrightarrow{p}(t) = (x(t), \ y(t), \ z(t)) }[/math]

We can use sine, log, polynomials, squares, anything at each coordinate. They are all linearly independent from each other. What ties everything into one process is that we have one independent variable for all, the time.