Classroom Tips and Techniques: Trigonometric
Parametrization of an Ellipse
Robert J. Lopez
Emeritus Professor of Mathematics and Maple Fellow
Maplesoft
Introduction
A circle centered at the origin is easily parametrized trigonometrically as
, where is the radius of the circle. If the center of the
circle is translated to the point
, the trigonometric parametrization readily extends to
.
An ellipse whose standard form in Cartesian coordinates is
can also be parametrized trigonometrically as
since
In this month's article, we discuss a trigonometric parametrization for the ellipse whose Cartesian
equation contains an -term, indicating that the axes of the ellipse are rotated with respect to the
coordinate axes.
Initializations
Trigonometric Parametrization of a Circle
One does not get far into a Calculus course with encountering the trigonometric parametrization
of a circle. Thus, the circle
. Of course, if
is readily parametrized as
, then the circle is given simply by
. In either event, the
trigonometric parametrization is free of the branch issues that makes the explicit representation
so much more awkward to work with.
Simple Trigonometric Parametrization of an Ellipse
The trigonometric parametrization of an ellipse with axes parallel to the coordinate axes is nearly
as familiar as the trigonometric parametrization of a circle. Thus, the ellipse
is readily parametrized as
.
To obtain an equivalent trigonometric parametrization of an ellipse whose axes are not parallel to
the coordinate axes requires application of both a translation and a rotation, calculations that can
be found in a college-level analytic geometry text. For example, I still use the text Analytic
Geometry, C. O. Oakley, Barnes & Noble, 1959, that I acquired years ago for a cost of $1.50.
Unfortunately, I don't think this material appeared in the syllabi of any of the college math
courses I've taught over the years. I did, however, need to make use of the material when
exploring (in the pre-computer days) how penalty functions work in constrained optimization
problems.
Analyzing a Quadratic Equation in Maple
Consider, then, the quadratic equation
whose graph is seen in Figure 1. This figure is obtained as per Table 1 by first loading the
Student Precalculuspackage, then applying the Context Menu to the quadratic equation, selecting
Tutors‫ظ‬Conic Sections.
Tools‫ظ‬Load Package: Student
Precalculus
Figure 1
Loading Student:-Precalculus
Conic Sections Tutor applied to the quadratic equation
Table 1 Application of the Conic Sections Tutor to
.
The option "Constrained Scaling" has been selected in the Plot Options dialog. The standard
form displayed is in the
-plane whose coordinate axes are aligned with the axes of the
ellipse.
Figure 2 shows all the data provided by the Conic Sections Tutor.
class: ellipse
eccentricity: .884
semimajor axis (a): 11.5
semiminor axis (b): 5.37
latus rectum: 5.03
angle: .448
----------------------------In the xy-plane:vertices: [(.573e-1,-2.90), (20.7,7.03)]
foci: [(1.26,-2.32), (19.5,6.46)]
center (h,k): (10.4,2.07)
directrix: y = -2.08*x+53.6
----------------------------In the x'y'-plane:
vertices: [(-1.20,-2.64), (21.7,-2.64)]
foci: [(.131,-2.64), (20.4,-2.64)]
center (h',k'): (10.3,-2.64)
directrix: x' = 23.2
Figure 2
Data provided by the Conic Sections Tutor
Maple states that the quadratic defines an ellipse, and gives its parameters in the original
-plane, and in the
-plane where the coordinate axes are aligned with the axes of the ellipse.
In addition to the eccentricity, the foci and vertices are given in both planes, the equation of the
directrix is given in the
-plane, and the angular offset between the two sets of axes is
provided. Unfortunately, the Tutor does not have enough room to display the exact form of any
of these quantities, which can be large and cumbersome. Hence, the data is reported in
floating-point form with an abbreviated number of digits.
Table 2 shows an alternative analysis of the ellipse, this one provided by the geometry package.
The geometry package has to
be loaded with the syntax
shown to the right.
The ellipse is made known to
the package with the syntax
shown to the right. The
ellipse is named "E" by this
command.
Below, the detail command provides the results (in exact arithmetic) of Maple's analysis of the
ellipse.
The angle of rotation between the sets of axes is not given, and all the data is given just for the
-plane. That this is so can be verified by extracting the coordinates of the foci, writing them in
floating-point form, and comparing the results with those listed in Figure 2. See the calculation
below.
Table 2 Analysis of a quadratic via the ellipse command in the geometry package
A Recipe for the Trigonometric Parametrization of an
Ellipse
and ) can be written in the form
The general quadratic equation in two variables (say,
Compute
, and , where
,
,
The quantities and
are invariant under rotations. If
,
, and and are of
opposite signs, the quadratic equation defines an ellipse. The center of the ellipse, namely,
, is given by
and
Translating the conic to
-coordinates whose origin is at
where
. Rotation of the
removes the
-term. (If
trigonometry gives
puts the ellipse into the form
-axes through an angle , where
, then
so
is defined by
.) Otherwise, basic right-triangle
so that
and
The -coordinate axes are rotated to the
transformation
The result is the equation
-axes, parallel to the
-coordinate axes, by the
, where
and
The standard form
is obtained by defining
and
Consequently, the given ellipse is parametrized by the equations
For convenience, we summarize the requisite calculations in Table 3.
If
If
then
then
,
,
,
Table 3
Calculations leading to a trigonometric parametrization of an ellipse
Example
Let us apply the calculations in Table 3 to the following quadratic. As per Table 3, we block the
calculations in Table 4.
Enter the quadratic.
>
>
Assign the coefficients. The
command used provides the Taylor
coefficient of the requisite power of
the variables. Since the command is
applied to a quadratic, the second
degree Taylor polynomial is the
quadratic, so the coefficients of the
quadratic are precisely the
coefficients of the Taylor
polynomials of the appropriate
degrees.
Compute .
>
Compute .
>
Compute .
>
Compute .
>
Compute .
>
Since
, compute
.
>
Compute
.
Compute
>
.
>
Compute .
>
Compute .
>
Compute .
Compute .
Obtain
.
>
>
>
>
Obtain
.
Table 4
Trigonometric parametrization of the quadratic
To verify that the parametrization obtain in Table 4 actually represents the original ellipse, we
provide Figure 3 in which an implicit plot of the ellipse is superimposed on the parametric plot
determined by the parametrization in Table 4.
Figure 3 Implicit plot of the ellipse determined by the given quadratic superimposed on the
graph of its parametrization. The implicit plot is in red; the parametric curve, in black.
A Sketch of the Derivation
Although Figure 3 suggests that the recipe in Table 3 is correct, it is useful to derive these
results. To this end, we start afresh, with
>
The first step is the translation
, implemented as
>
followed by the rearrangement
>
Extract the coefficients of
and
with
>
then determine the values of
>
and
for which these coefficients vanish. This is done with
Applying these definitions of
and
puts the quadratic into the form
>
The constant term, namely,
>
suggests defining
>
and the determinant
>
If we define
as
via
>
then the constant term becomes
>
, as we see from
At this point, the quadratic can be put into the form
>
where has to be used in place of since an expression has already been assigned to . To
use in this form of the quadratic, the assignment must be removed with
>
so that we can write
>
The
-term is removed with a rotation of the
Using the abbreviations
and
>
or equivalently,
>
The coefficients of
>
are respectively
-system to an
-system via the equations
, the quadratic becomes
The first and third of these expressions are called
and
so that the quadratic becomes
>
It remains to determine the value of for which the coefficient of
by putting the resulting equation into the form
vanishes, a process aided
or
from which it follows that
and hence that
A Maple implementation of these last manipulations is tedious. It begins with the equation
>
and requires moving the terms containing
to the right with
>
This equation is then multiplied by
>
which factors to
>
, obtaining
and
Removing the abbreviations
leads to
>
which can be written as
>
The expression for
appears after a final rearrangement and conversion via
>
The value of is determined by this equation, but since only
expressions for and , it is better to obtain
and
appear in the
from which it follows that
and
The sign convention we have chosen makes an acute angle; the axes of a rotated ellipse can
always be aligned with the horizontal and vertical by rotating through an acute angle.
With the coefficient of the
right side is zero, becomes
-term set to zero, the quadratic, the left side of an equation whose
and hence
The definitions
and
complete the derivation of the formulas.
Legal Notice: The copyright for this application is owned by Maplesoft. The application is
intended to demonstrate the use of Maple to solve a particular problem. It has been made
available for product evaluation purposes only and may not be used in any other context without
the express permission of Maplesoft.