Lesson I: Definition and Basic Properties of Ellipse
Definition: Ellipse is commonly defined as the locus of points P such that the sum of the distances from P to two fixed points F1, F2 (called foci) are constant. That is, distance[P,F1] + distance[P,F2] = 2 a, where a is a positive constant.
Student Work: At this point allow students to draw their own sketches on the computers and observe some properties of the ellipse.
Before students spend several minutes on observing and playing with the ellipse, introduce two important lines (axes) for an ellipse, major and minor axes, and the vertexes of an ellipse (See Figure 2).
of the ellipse are defined as the intersections of the ellipse
and a line passing through foci. The distance
between the vertexes are called major axis or focal axis. A line passing the center and perpendicular to the major
axis is the minor axis.
Click here to investigate parabolas on an interactive java applet (If you do not have GSP software on your computer, it's an perfect opportunity to take a look at.)
Click here for an interactive java applet game which is about to find the foci of an ellipse.
Click here for an interactive java applet toy to construct ellipses with a tool called "Ellipsograph".
Click here for a java applet to investigate relative properties of two congruent ellipses.
Lesson II: Introduction to the algebraic representation of an ellipse
Analytic Equation of an Ellipse
Before introducing a general equation (i.e. the analytic, polar or parametric form) of an ellipse, it is important to understand where this equation comes from. Thus, before giving a formal equation of an ellipse, deal with the following construction of the equation with referring to the notation on the Figure 3:
First of all notice that
(Why? Let the students think about the picture below!)
Let an ellipse lie along the x-Axis and find the equation of the Figure 3 where F1 and F2 are at (-c,0) and (c,0). In Cartesian Coordinates,
This makes the equation in the simplest form:
If, instead of being centered at (0, 0), the Center of the ellipse is at , equation becomes
In class explorations:
Lesson III: Polar and Parametric Equations of an Ellipse
As can be seen from the Cartesian Equation for the ellipse, the curve can also be given by a simple parametric form analogous to that of a circle, but with the x and y coordinates having different scalings,
In Polar Coordinates, the Anglemeasured from the center of the ellipse is called the Eccentric Angle. Writing for the distance of a point from the ellipse center, the equation in Polar Coordinates is just given by the usual
Here, the coordinates and are written with primes to distinguish them from the more common polar coordinates for an ellipse which are centered on a focus. Plugging the polar equations into the Cartesian equation (9) and solving for gives
Lesson IV: Circumference and Area of an Ellipse
Circumference: The circumference of an ellipse with major axis radius, a, and minor axis radius, b, is given (approximately) by
Area: The area of an ellipse with major axis radius, a, and minor axis radius, b, is given by
The Area of an arbitrary ellipse given by the Quadratic Equation
Lesson V: Reflective (Optical) Property of an Ellipse
In an ellipse, lightrays from one focus will reflect to the ther focus (See Figure 4).
Figure 4. Reflective property of an ellipse
here for an interactive java applet
to investigate reflection (optical) property of an ellipse.
Investigation 1. The locus of points for a fixed point on the coming ray (from F2 to P) is an ellipse (See Figure 5). Why? What kind of evidences do we have? What properties does it have (like where is its foci--hint: one of them is F2)?
Investigation 2. What is the locus of points for a fixed point Q on the reflected ray (from P to F1) is an ellipse? (See Figure 6 for various locus which depends on the location of Q). Describe the properties of this curve?
Investigation 3. What happens if Q is on the normal line at the point Q (See Figure 7).