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Geometry Expressions Newsletter
Geometry Expressions Newsletter

February 2011 
Some Examples on the website
The Explore section of the Geometry Expressions website contains hundreds of examples of the software in use.  Some contain a simple picture to give you ideas for creating models, others tell a more involved story.

Here are some of the more substantial ones:

Special Relativity with Geometry Expressions creates models of Minkowski Geometry, time dilation and Lorentz Contraction all with Geometry Expressions.

Newton Implies Kepler follows Feynman's lost lecture showing (without calculus) that Newton's laws imply Kepler's laws.

An assortment of loci is a potpourri of different loci some of which turn out to be conics, some of which don't.

Feynman's and Steiner's Triangles are simply derived triangles whose areas we explore.

Recursive Napoleon Like constructions This article looks at a particular geometric construction performed recursively.  Limiting behavior is obtained by examining a Markov process. 
How to...analyze a locus which is a conic
triangle spline
Sometimes a locus or envelope curve turns out to be a conic, and we are interested in some conic-specific characteristics of the curve, such as location of its center or foci.  However a locus curve is generic and does not have any knowledge of foci.  We describe a trick to get round this problem.

As an example, we take a triangle ABC with vertices coordinates (-a,0), (b,c) and (a,0).  Let D lie proportion t along AB and let E lie proportion t along BC.  We create the locus of the intersection of CD and AE as t varies.

The implicit equation of this locus is quadratic, hence the curve is a conic.  What type of conic is it?

To determine this we sketch an ellipse (or hyperbola or parabola.. the conic will automatically morph to the correct type once an equation is given). We now copy the equation of the locus and paste it in as the equation of the conic.

Once you have the ellipse, its foci are given.  To find the midpoint (below) simply construct the midpoint of the two foci.

triangle spline ellipse

Problem of the Month
bexier spline
ABCD is the control polygon for a cubic Bezier spline. 

     E is proportion t along AB,

     F is proportion t along BC,

     G is proportion t along CD,

     H is proportion t along EF,

     I is proportion t along FG

     J is proportion t along HI.   

The spline curve is the locus of J as t varies between 0 and 1.

Under what conditions on ABCD is the spline curve a parabola?

GX Books
Achilles and the tortoise
Larry Ottman has written a great book of lessons exploring ideas of infinity and limits from a strictly pre-calculus perspective.

Check it out and order 
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
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