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Geometry Expressions Newsletter
April 2013   
Virtual Trammel and Temporal Distortion Apps
Two apps created with Geometry Expressions are available for you to try (for free) on your smart phone or tablet. Involute Clock
  • Virtual Trammel gives you the tactile experience of tracing an ellipse with Archimedes Trammel.  It also lets you explore the question: What happens if you turn the trammel upside down and move the base while keeping the handle fixed?
  • Temporal Distortion provides a collection of 24 whimsical clocks which illustrate various mathematical and engineering concepts, and are just plain fun to watch.

How to... Create Mobile Apps Using Geometry Expressions and PhoneGap Build
Creating an HTML5/JavaScript App with Geometry Expressions is easy. 
  1. Create a model in Geometry Expressions with the behavior you want in your app.
  2. Use File / Export / HTML5 / JavaScript App.
  3. Fill in the dialog box to specify which variables should be accessible to the user and how they should appear.  You can also add a title and some text before and after the diagram.
  4. Geometry Expressions will generate a single compact html page containing your entire application.   
  5. This file can be used in a browser on a computer, tablet or smart phone.


You can create a native Mobile App (again without programming) using the PhoneGap Build service from Adobe.  You simply upload the html file created by Geometry Expressions and within a couple of minutes, native apps will be created for iOS, Android, Blackberry, Win 8...

Problem of the Month
What is the envelope of the family of circles whose centers lie on an ellipse and whose circumferences pass through one of the foci of the ellipse?

An app which lets you explore this problem is here.

How about a hyperbola?

An app for the hyperbola is here.

March Problem Solution
spline curve   
ABCD is a trapezoid with parallel sides AB and CD.  E and F are the midpoints of AB and CD respectively.


A closed curve is composed of two Cubic Bezier Splines: one with control polygon EBCF, the other with control polygon FDAE.


Show that the radius of curvature is continuous at E and F.


An app which lets you explore this problem is here.

To show this using Geometry Expressions, we create points A,B,C,D and constrain the coordinates to be (0,a), (0,b), (x,c), (x,d) respectively.  This makes ABCD a trapezoid with parallel sides AB and CD.

E and F are constructed at the midpoints of AB and CD.  A cubic spline is constructed with control polygon EBCF and the envelope of the normals (the evolute) constructed.  Points M and N are placed at parametric locations 0 and 1 on the evolute. 

The osculating circle of the spline at E has center N and the osculating circle at F has center M.

We note that the radii of these circles depend only on the lengths of AB and CD and the distance between them.  Hence, if we created the spline with control polygon EADF it would have the same radius of curvature at its ends. 

Hence the combined curve has continuous radius of curvature at E and F.


Pythagoras proof
Our Euclid's Elements contains over 120 interactive diagrams.
Available on iTunes

Sample chapters of other iPad eBooks are available here

Interactive math Apps which can be viewed on your iPad are available from Euclid's Muse

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

Watch more videos about creating Apps with Geometry Expressions.

Learn how to Export an Animation with Geometry Expressions.
Quick Links
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