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Geometry Expressions Newsletter
Geometry Expressions Newsletter

November 2010
Student Projects
spherical mirror
For the past few summers, we have sponsored high school students to use Geometry Expressions in their own mathematical research projects.

In the words of one student..

"This is definitely not the traditional math work we  are accustomed to. It is far more interesting."

Alex developed a geometric measure of aberration in a telescope lens.

Ariel compared solar cookers with parabolic and circular cross sections. 

Elmer worked on the problem:  What is the largest circular sector which can lie inside a given triangle?  His results are here and here.

Nick applied Geometry Expressions to model a Savonius vertical axis wind turbine.

Katie looked at some of the mathematics of soccer

Jack researched the catenary as a solar cooker profile.

Walker looked into what becomes of various triangle defined circles when the vertices of the triangle coalesce.  His work is here and here.
How to...create an involute curve
involute1
The involute of a curve is the locus traced out by the end of a piece of string as it wraps around the given curve.

Here's how to create an involute in Geometry Expressions.  (In the picture, our given curve is a logarithmic spiral)
  • First create an arc on the given curve.
  • Now constrain the parametric location of the arc's endpoints. One should be a constant, the other variable (we've used pi and t).
  • Measure the arc length.
  • Create a tangent to the curve at the variable end of the arc.
  • Place a point on the tangent and constrain its distance to be a constant minus the arc length.
The constant represents the length of the string.

The locus of the point on the tangent is the involute curve.

For the logarithmic spiral, the involute is simply a 90 degree rotation of the original curve.

involute 2
Problem of the Month
reflections
Given triangle ABC, parallel lines DC and EA are reflected in BC and BA respectively.

What is the locus of F, the intersection of the reflected lines?

[This problem comes from Alex Turzillo's project on telescope aberration]

GX Books
Farmer and the Mathematician
All Geometry Expressions books are now available as digital downloads at the price of $9.99 each.

They are also available in traditional printed form.

Check them out and order 
here

Online PD
Interested in using Geometry Expressions in class?

Free online Professional Development is now available here

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