I have entitled this chapter implicit functions instead of conic sections because I feel that, for the most part, modern textbooks are emphasizing the wrong ideas here. 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
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.
is our compact conic section with two special cases. Let's
examine the standard form of the ellipse and see how it's related
to the above equation (Note that a and b in the standard form of the
ellipse are not the same a and b from the more general quadratic form
above). The h and k should appear all too familiar from the section on
transformations of functions. They move a special point- in this case,
the center- to a new place in the cartesian plane. The values of a and
b specify the lengths of the major and minor axes and give a way to
determine the locations of the foci. We call 2a the length of the major
axis if a > b and 2b the length of the major axis if b > a.
The implicit functions are given in orange and blue here. I have
suggested a way to parse up the ellipse so each half is can be defined
in terms of familiar functions. It is, of course, possible to parse this up
into more functions. If we are perceptive in analyzing the standard form
of the ellipse and the original equation we will that is necessary, but not
sufficient that both a and c in the original equation are positive. Why?
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
Since the foci are always located along the major axis, they have coordinates . 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 . 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!
A Step Further
Let's explore another relatively obvious question before we continue with noncompact conic sections. Why is there no fxy term in equation 1? Let's explore an ellipse given by the following equation:
This is an example of ellipse that is rotated off axis. In this case, we find the ellipse is at a 45 degree angle to the x-axis but maintains the expected center of (0, 0). It turns out that we need never worry about this situation because of an elementary result that will you see in your linear algebra course. There is always a way to transform the set of points that respects size and shape of the ellipse, but reorients it vertically or horizontally.
Incidentally, you have all the tools you need at your disposal to implicitly solve the equation that gives rise to this ellipse. How would you do it? Including the cross term given by fxy also serves to greatly complicate our understanding of this mathematical object because it allows for the introduction of numerous new degenerate cases (like radius 0 circles!).
Parabolas and Hyperbolas
We'll briefly do an example with a horizontally oriented parabola recalling that we have seen vertically oriented parabolas before. Much like the ellipse or the circle, we have a standard form that is best obtained through completing the square. This has the familiar standard form, but we will add something that will generate a bit more information. Many texts will refer to the following as vertex form, but for our intents and purposes, we will call it the standard form a parabola from this point forward. For our purposes, we will always default to calling the form that allows us to determine what the equation represents with the most information available at first glance the standard form of a geometric object.
The standard form here immediately tells you the vertex of the parabola and the
distance to the focus and the directrix. It also tells you the dilation and the direction
of the parabola.
Hyperbolas have a standard form that appears similar in nature to the ellipse. The single
biggest difference is the presence of a negative coefficient in front of either the x or y term.
Note, that both could not be negative and produce real points that satisfy the equality. Again,
we have a center and a pair of foci. In fact, there are a number of cases where we can obtain hyperbolas that do not fit this standard form. For instance, y = x is a hyperbola. Still, it easier to envision and interpret hyperbolas in the standard form. Note that in standard form, we can readily calculate location of the foci. They will be located a distance of c interior to the vertices which are oriented along the same axis as the positive coordinate and a distance of a or b away from the center (depending on the orientation of the hyperbola).
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 to the right and 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!
Python Programming Project: Implicit Functions
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), )