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 

 

is readily parametrized as .  Of course, if , 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 Precalculus package, then applying the Context Menu to the quadratic equation, selecting Tutors?Conic Sections. 

Tools?Load Package: Student Precalculus	Loading Student:-Precalculus
Figure 1   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 

The general quadratic equation in two variables (say, and ) can be written in the form 

 

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 puts the ellipse into the form 

 

where .  Rotation of the -axes through an angle , where is defined by 

 

removes the -term.  (If , then so .)  Otherwise, basic right-triangle trigonometry gives 

  

so that 

  and      

The -coordinate axes are rotated to the -axes, parallel to the -coordinate axes, by the transformation 

 

 

The result is the equation , 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 then   If 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 -system to an -system via the equations 

 

 

Using the abbreviations and , the quadratic becomes 

>

 

 

 

or equivalently, 

>

 

 

 

The coefficients of are respectively 

>

 

 

 

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 vanishes, a process aided by putting the resulting equation into the form 

 

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 , obtaining 

>

 

 

 

which factors to 

>

 

 

 

Removing the abbreviations and 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 and appear in the expressions for and , it is better to obtain  

 

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 -term set to zero, the quadratic, the left side of an equation whose right side is zero, becomes 

 

and hence 

 

The definitions and complete the derivation of the formulas.  

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