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Geometry Expressions Newsletter
Geometry Expressions Newsletter

January 2011
Classroom Activities
spherical mirror
There are a whole range of different classroom activities on the Explore section of the Geometry Expressions website.

Here are some of our favorites..

String art is a very nice lesson sequence examining the properties of parabolic curves found in simple string art designs.

Light caustics looks at the curves formed by light reflected in a cylinder (such as a coffee cup).

From inductive reasoning to proof by induction is an easy introduction to Geometry Expressions in an activity which uses inductive reasoning and introduces inductive proof.

Sketching an ellipse at the beach uses sticks and string on a beach to draw an ellipse, and some other surprising curves.

Trammels The trammel of Archimedes draws ellipses with given semi-major axes.  This activity maps between this representation and the pin/string representation.

Where to take a rugby kick solves an old standard calculus problem using Geometry Expressions.
How to...do differential geometry
Generic curve
One of Geometry Expressions' unique features is the ability to use generic functions.  When you use generic functions in a parametric curve, you have a starting point to do differential geometry.

For example, the evolute of a given curve is the envelope of its normals.  To derive the equation for the evolute we:
  • First create a parametric curve, and set its x coordinate to be f(T) and y coordinate to be g(T).
  • Now put a point on the curve and constrain its parametric location to be t.
  • Select the point and the curve and construct a tangent.
  • Select the point and the tangent and construct a perpendicular (this is the normal).
  • Select the normal and Construct/Locus to create the envelope.
  • Now select the envelope curve and Calculate/Parametric Equation to display its equation.
     
By creating a general curve in this way, along with the tangent, envelope and locus constructs, you can explore many results in the differential geometry of 2D curves.

Evolute
Problem of the Month
Circle Problem
Point C lies on circle AB.

C' is its image under a rotation about B which maps circle AB onto circle A'B.

Show that the line joining C and C' passes through the intersection between the circles AB and A'B.

GX Books
Farmer and the Mathematician
The latest book from Irina Lyublinskaya and her co-authors Valeriy Ryzhik and Ron Armontrout explores calculus with Geometry Expressions.

Check it out and order 
here
All Geometry Expressions books are now available as digital downloads at the price of $9.99 each.
Order here

They are also available in traditional printed form.
Online PD
Interested in using Geometry Expressions in class?

Free online Professional Development is now available here

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