Implicit Functions
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.
1.1 Introduction: Conic Sections vs. 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
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.
1.2 A Historical Note: Conic Sections
We have previously studied lines, parabolas, and absolute value functions. These are all special cases of conic sections, but we will largely omit their descriptions here since we have already seen them. As a fun aside, we will for the briefest of moments examine how the classical Greeks would have thought about these objects as related to two tangent cones. Greek mathematicians imagined a plane intersecting a cone (or possibly a pair of cones). The angle of intersection of that plane determined the object formed. This was not the only way they would have formulated these objects as you will see in other historical notes!
1.3 The Ellipse
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.
Example 1: Construct the associated implicit functions for
Example 2: Graph the result of the implicit functions you constructed for
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
1.4 A Historical Note: Ellipses
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.
1.5 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!).
1.6 A Historical Note of Parabolas
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.
1.7 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.
Example 3: Use the method of completing the square to put
Example 4: Graph your result from putting
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,
Example 5: Use the method of completing the square to put
Example 6: Graph
1.8 Why should we study any of this?
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!
Homework Set:
https://goformative.com/formatives/623d895994e1e4b191da41ae
1.9 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), [0])
plt.show()