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Geometry Expressions Newsletter
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Geometry Expressions Newsletter

October 2011 
App competition
Learning calculusAnnouncing our HTML5 app competition.

Create a cool HTML5 app using Geometry Expressions (To create an app, select Export in the File menu and select HTML5/Javascript).

Put it on the web and email a link to


Win a prize:

First prize is a complete set of electronic books ($79 value).

Second prize is a copy of Learning Calculus with Geometry Expressions by L Van Warren ($34.95 value).

Third prize is an electronic copy of the book of your choice ($9.99 value).

All prize winners and honorable mentions will be linked from our web site.

Competition ends November 15th 2011.  Multiple entries are permitted.  Entries will be judged based on:

Utility - Does it solve a useful problem?
Elegance - Is the implementation minimal and beautiful?
Clarity and Usability - Is the application easy to understand and use?

L. Van Warren, University of Arkansas at Little Rock
Irina Lyublinskaya, City University of New York Staten island
Philip Todd, Saltire Software

How to...animate the circle of curvature of an ellipse
osculating circle
Geometry Expressions 3.0 has the ability to export animations as animated gif files.

This is a useful format, as it is quite compact, is well supported by browsers and by presentation software such as PowerPoint, and is the preferred animation format for Wikipedia.

Our GX 3.0 features page has a sample animation showing the circle of curvature of an ellipse, as the point of contact moves around the ellipse.  We will show you how this animation was created. 
  • First draw the ellipse, and use the Point Proportional constraint to specify that C is parametric location t on the circumference.
  • Now create the tangent to the ellipsosculating circle of ellipsee at C and then construct a line (the normal) through C perpendicular to the tangent.
  • Select the normal line and use Construct / Locus.  When a line is selected, this actually creates the envelope of the line.  The envelope of the normal to a curve is the evolute, which is also the locus of the centers of the circles of curvature.  (Exercise:  prove this using Geometry Expressions and an algebra system).
  • Put a point D on the evolute curve and constrain its parametric location (using the Proportion constraint) to be t. (Draw the point away from the normal so you don't pick up an extra incidence constraint.)  
  • Now draw a circle centered at D with C as its radius point.  This is the circle of curvature.
  • Use the Variables Tool Panel to specify initial and final positions for t, animation style (forward and back or just forward) and duration of the animation. 
  • Export the animation using File / Export / Animation. The Animation Export dialog gives you the opportunity to set the resolution of the animation, both spatial and temporal and how many times you wish the animation to repeat.


Problem of the Month
First a simple geometry problem:

C and D lie on the circumference of a circle centered at A. 

F is the intersection of the tangents at C and D.

Show that the circumcircle of CDF passes through A.

Now explore the circumcircle of 3 points on a function and the circle through two points and the intersection of the tangents at those points.

Can you use Geometry Expressions along with your favorite CAS to prove that in the limit as the 3 points coalesce, the circle through two points and the intersection of the tangents has half the radius of the circle through the three points?

0GX Books
Electronic Books
Learning Calculus with Geometry Expressions is a unique electronic resource containing lecture-ready slides and lab-ready Geometry Expressions files.
Available for $6.99 per chapter, or $34.95 for the whole book.

Electronic versions of all our paper Geometry Expressions books are available for $9.99 each.

The complete set of 10 books is available for only $79.95.

You can also buy physical copies of these books

See our YouTube channel for videos exploring the new features in Geometry Expressions.

Quick Links
Geometry Expressions
New features in version 3.0
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