Workshop on Geometry Software

Week 2: Beyond Euclidean Geometry

We constructed a parabola. We start with a line, l and a point not on the line, P. For each point A on l we construct a point Z that is equidistant from P and l , lying on the perpendicular to l at C. You can animate C and have GeoGebra trace Z . The result is a parabola. The line l is called the directrix and the point P is called the focus of the parabola.

We also started the construction of an ellipse from two foci P, Q , and a distance d. A point Z is on the ellipse if the sum of the distance to P and the distance to Q is d.

Dustin helped me figure out the problem about animating the ellipse. You can't animate an arbitrary point; you can only animate a point on a line (or on a circle). So, choose either an arbitrary line, or any circle (I think I did it with the circle with center P of radius d . Choose an arbitrary point on this circle, call it A, construct the ray PA and continue as we did in class.

Assignment:

1. Pepare a Geogebra worksheet with the construction of the parabola. Explain the steps in the construction and give a short justification for the equidistance property.
2. Pepare a Geogebra worksheet with the construction of the ellipse. Explain the steps in the construction and give a short justification for the construction. What happens when d < b?!

We worked with geometric transformations: dilations, translations, rotations and reflections.

1. Pepare a Pretty Picture, a Geogebra worksheet with a construction using the geometric transformation tools discussed in class. Describe precisely the steps you used in the construction. Make it so that varying the freely chosen points maintains the integrity of the figure.

Due: Monday 2/16 at 5:00 am.

Comments: Here are some observations that should be generally useful to all of you for future assignments and revision of this one for the final portfolio.

1. For the parabola and ellipse: Most of you explained the steps in the construction correctly. Only a few explained what property the points on the parabola (or ellipse) satisfy. For the final version, please do that, and *also* explain why the construction creates points that satisfy the property.
2. For the final portfolio include a discussion of what happens in the construction of the ellipse when the distance between the foci is greater than the length of the string.
3. Don't hide the steps in the construction, because we want to see why it works. Use color or opacity to highlight or de-epmphasize.
4. For the pretty picture, use the tools that we played with on Thursday to create the figure; don't just draw a lot freehand. See if you can make your figure work well even after I adjust it by moving points. Explain your construction and think geometrically.
5. For the pretty picture, try using animation, trace, or other tools to make it more interesting.