Implicit Functions

Part of the beauty of studying conic sections is that the geometry of objects are captured in cartesian relationships.  Note that I did not use the word functions.  Strictly speaking, they are not functions (though, it is true for the precocious amongst you that we can extend our idea of what a function is and parameterize almost anything!). It turns out, though, that this is generally fine.  They imply a collection of functions hence we group them in a very broad category known as implicit functions.  The point of studying conic sections is to realize that given a relatively innocuous algebraic object like $$ax^2 +bx + cy^2 +dy + e = 0 \quad \text{(equation 1)}$$

We must engage in relatively intensive study to understand all the nuances implied by this equation.  If we do this thoughtfully enough, we may recognize certain patterns that lead us to us towards standard forms of these equations (Mathematicians once much preferred the phrase “canonical forms” because it harkens towards a collection of deep theories at the heart of mathematics known as canonical form theorems).  The importance of these standard forms is that, at a glance, I can glean most of the properties of the mathematical object I am looking at without much thought.  The importance of algebra is that it is a parsimonious language.  It captures infinite objects in all their majesty in tiny, finite collections of symbols.  Broadly speaking there are two very distinct types of conic sections: compact and noncompact.  One way to think about compact objects is that I can draw a representation of their entire graph on a single chart. 

Let’s do a concrete example and then talk about special cases.

The standard form was highlighted in green because it contains more useful information for graphing the ellipse.  Since a > b, we see the major axis is horizontally oriented.  The center can be found at the coordinate (-1, -2).  The vertices are then found by subtracting or adding a and b respectively.  One can then find other useful quantities for graphing such as the coordinates of the foci in the following way $$c = \sqrt{a^2 – b^2} \rightarrow c = \sqrt{\frac{9}{25}-\frac{4}{25}} = \sqrt{\frac{1}{5} = \frac{1}{\sqrt{5}}$$

Since the foci are always located along the major axis, they have coordinates $\right(-1-\frac{1}{\sqrt{5}}, -2\left), \right(-1+\frac{1}{\sqrt{5}}, -2\left)$.  As promised, we’ll discuss the degenerate cases.  If a = b then we have a circle with radius a.  We often write this equation in a slightly modified standard form as $(x-h)^2+(y-k)^2 =r^2$.  If, additionally, r = 0, we say this is a point at (h, k).  Finally, for the time-being, we will only consider r as positive or zero and similarly the major and minor axes lengths of an ellipse will be required to be positive.  Again, this is not strictly necessary, but we cannot graph the result on a cartesian plane!

An alternative way to define an ellipse is as the set of points equidistant from two points.  We call those two points the foci.  Historically, ellipses were discovered many thousands of years ago when ancient peoples such as the classical Greeks realized you could stack to pins into the ground and run a loose string between them.  If you then pulled a string taught with a writing utensil and traced around it you would produce an ellipse. Similarly, the circle was defined to be the set of equidistant from a single point, but was usually drawn with a compass instead of twine.

The parabola was defined as the set of points equidistant from a line and a point not on that line.  They’ve held interest of peoples for a variety of reasons for a long time.  Parabolic arches go back thousands of years as an efficient way to create a load bearing door frame.  It turns out, though, this was probably due to some mistaken understandings.  For much of history, people thought chains hung in a parabolic structure under the force gravity.  This turns out to be incorrect and the curve that a hanging chain traces out is called a catenary.  When people understood this, they were able to make stronger and more efficient load bearing structures called catenary arches.  

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I hope the main point of this section has not been lost on anyone: when we have relations or implicit functions instead of standard cartesian functions then we must carefully study the equations that define them to determine all the possible outcomes.  If we add a cubic term to equation in terms of x, we find the number of possible objects multiplies exponentially and motivates the beginning of the study of a modern field of math known as algebraic geometry.  Some examples are shown below.  If we studied these carefully enough though, we would find analogous quantities and properties along with clever standard forms.  Notice, we also open ourselves up to many more possible cross terms and the same theorem from linear algebra no longer applies!  We have to study those possibilities as well to gain a complete understanding.

One other easy example is the superellipse which has deep links to engineering and to number theory.  A number of famous building including Aztec Stadium in Mexico City, Mexico are superellipses!

Below, you will find the code necessary to begin to explore implicit plots (as tested in python 3.0 on pycharm).  It will produce a rather cute implicit graph.  You mission is to use this code to investigate variations on equation 1 from above that use a cubic term.  Share your results with classmates and begin to classify these functions.  They are part of the very modern field of elliptic functions!

import matplotlib.pyplot as plt
import numpy as np

delta = 0.025
xrange = np.arange(-2, 2, delta)
yrange = np.arange(-2, 2, delta)
X, Y = np.meshgrid(xrange,yrange)
# F is one side of the equation, G is the other
F = X**2
G = 1- (5*Y/4 – np.sqrt(np.abs(X)))**2
plt.contour((F – G), [0])
plt.show()