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

To understand parametric equations one should understand the concept of functions first. Functions relate one position in a number line to another position in another number line, which means that we can use multiple functions to relate multiple coordinates on a plane or in 3D space. With functions we have more fine grained control over trajectories for example.

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 (vector valued) function that describes position in time. It's a single variable function that produces vectors for each input. In this case time is the parameter. The graph doesn't have to be interpreted as a trajectory, but it's easier to grasp the concept by doing so. The variable time could by anything else, but for now it's easier to think about time and motion.

(In case you don't know, vector + point = point. This is the origin of the equation below)

[math]\displaystyle{ \begin{cases} x = x_0 & + & ta \\ y = y_0 & + & tb \\ z = z_0 & + & tc \end{cases} }[/math]

XYZ are the coordinates. [math]\displaystyle{ (x_0, y_0, z_0) }[/math] a point in the trajectory and [math]\displaystyle{ t(a, b, c) }[/math] a vector and a constant (the parameter) 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. In this case the equation would have two additional vectors, but the same parameter.

[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. In case you didn't grasp the concept of a function, take a second look. For t = 1, t = 3, ... we have different coordinates. If for each coordinate we have a specific parameter, then there is a function connecting one to another.

Graphs of parametric equations

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.

I think that the first graph presented some confusion which I'm going to clarify now. When we learn to plot graphs of functions each point is of the form [math]\displaystyle{ (x,f(x)) }[/math] where [math]\displaystyle{ f(x) }[/math] is the vertical coordinate. When we plot a vector function in 2D or 3D, there is an independent variable, often time. Where is the axis of time in the graph? Nowhere! With functions of one or two variables, the independent variables lie on the axes themselves, with "height" being the axis for the function's values. With vector value functions, the independent variable or parameter, is "outside" the space. I have a textbook that presents examples of plots with no names for the axes. I think the author did that precisely to avoid the previously mentioned confusion.

Some textbooks use this notation:

[math]\displaystyle{ (x,y) = (f(t),\ g(t)) }[/math]

It's really the same thing as

[math]\displaystyle{ \begin{cases} x = f(t) \\ y = g(t) \end{cases} }[/math]

The first form is an equality between coordinates of points. The second is a parametric equation. There is really no conceptual difference between the two. The first is similar to the equation of a line in vector form, which is [math]\displaystyle{ (x,y) = (x_0,y_0) + t(a,b) }[/math] or, using vector notation, [math]\displaystyle{ r = p_0 + t\overrightarrow{\mathbf{v}} }[/math].

Now comes a question: [math]\displaystyle{ r^2 = x^2 + y^2 }[/math] is the equation of a circle. Is it a parametric equation? No! We don't have functions in this equation, albeit we can isolate one of the variables to get a function. What we have is a non-parametric equation of a circle. There isn't a parameter.

About level curves: one may look at the equation of a circle and see that some functions of two variables have level curves that are circles. That leads to a question of whether level curves can be parametrized. Yes, they can. But some curves can intersect themselves and form loops, which cannot be a level curve because a function can never intersect itself.

About trajectories: with functions of two variables a level curve can partially coincide with a trajectory given by [math]\displaystyle{ x(t) }[/math] and [math]\displaystyle{ y(t) }[/math]. It may be possible to make the relationship between the function and the physics of a trajectory that coincides with the level curve. However, multiple functions can have the same level curve, which makes it complicated to think on such relationship.

Can we have multivariable functions for each coordinate? Yes, why not? But with multivariable functions we are talking about parametric surfaces or volumes in space, which is beyond lines on a plane.