End of investigative tour
Mathematical research continues because the pursuit of each question leads to many more questions. As we end this investigative mini-tour, we list questions that we have raised but haven't answered:
- What other properties of circles have analogies in ellipses?
- What happens to the equation of an ellipse if the figure is translated
or rotated?
- What would a logical arrangement be like if we had taken the dilation
property as a definition of the ellipse?
- Does it make sense to talk about a negative eccentricity? Is the eccentricity
of a parabola 1 and of a hyperbola more than 1?
- Given an ellipse, how can we find its foci and directrices? Can they
be constructed with straightedge and compass?
- How far are the directrices from the foci? Where are the directrices
of a circle?
- How can we draw an ellipse accurately? Can we do it with straightedge
and compass?
- Are there ways of defining ovals other than ellipses?
- If we begin with an elliptical cone that is not necessarily circular,
are all sections ellipses? Is one of the sections necessarily a circle?
If you have wandered off of the tour to explore other properties of the ellipse, you might have raised these questions as well:
- What do the polar coordinate equations for the ellipse mean in terms
of the cylinder or cone?
- What figure has the equation obtained by taking the reciprocal of the
right side of the polar equation for an ellipse?
- Can the area of an ellipse be derived directly from the cylindrical
or conical section property, or from the polar equation of the ellipse?
- How does the area of an ellipse compare with that of a circle that has
one diameter between the foci of the ellipse?
- What if not two dimensions? Are there three-dimensional shapes with properties similar to those of ellipses?
All of these questions can form the basis of tours that extend into other realms of mathematics.
