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Geometry Expressions Newsletter
Geometry Expressions Newsletter
February 2014 
How to... export differential equations from Mechanical Expressions
Mechanical Expressions is designed to give you the symbolic mechanics of an instantaneous snapshot in time.

You can ask for the instantaneous acceleration in one or more constraints, and the system assumes that those constraints are in fact free to accelerate.  It works out the appropriate acceleration(s).  

In the picture above, we have asked for the acceleration in the angle constraint.  As we have not specified an angular velocity, it is assumed to be 0.

We can also output these accelerations in the form of differential equations, using Mechanics Output / Differential Equations

Note that rather than a single second order differential equation, we get a system of first order differential equations along with initial conditions.

Also note that we no longer consider the velocity to be 0, except as an initial condition.  So we have picked up extra terms in the acceleration equation.

We would see these terms in the original diagram if we set an angular velocity.
Anja Greer Conference
Learn how to make eBooks for the iPad with interactive mathematical diagrams  at the Anja S. Greer Conference on Mathematics, Science and Technology.

Philip Todd will lead a 5 day workshop at the conference which will teach you how to use Apple's iBook Author software in conjunction with Geometry Expressions to create an interactive mathematical iBook.

Philip will share techniques he developed for his "24 Whimsical Clocks" iBook and for the interactive Euclid's Elements.

The conference takes place at the Phillips Exeter Academy June 22-June 27 2014 


Problem of the Month
 
Another problem with an Archimedes connection.

The purple curve is the spiral whose polar definition is r(θ)=θ.

A is a point on the curve, O is the origin, and B is the intersection of the tangent at A with the perpendicular to OA.

What is the polar equation of the locus of B?  

How about if the original curve is r(θ)=θ^2?

How about if the original curve is r(θ)=θ^n?

   

How about if the original curve is r(θ)=f(θ)?

An app which lets you explore this problem is here.

A Solution to January's Problem
 
This month, a geometry problem which dates back to Archimedes.

CD is the diameter of a circle, and FB is perpendicular to CD.  Tangents to the circle at D and B intersect at E.

Show that CE bisects BF.

Is there an analogous theorem for ellipses?  How about hyperbolas and parabolas?
An app which lets you explore this problem is here.

Solving this problem with Geometry Expressions is straightforward if you are prepared to accept the programs output.  Just enter the drawing, constraining the circle to have radius r and distance AF to be a.  Ensure the center of the circle is constrained to lie on the diameter CD.  Your model should look like this and the distances BG and FG should be identical.

Please don't let this facile computer proof prevent you from looking for a geometric proof.

For conics, an analogous theorem sets BF to be parallel to the tangent at D rather than perpendicular to the circle's diameter (in the circle these conditions are equivalent).

You can explore the analogous conic theorems here.
eBooks

A breakthrough in the use of interactive diagrams for gaining new insight into mathematics.
Available on iTunes

Our Euclid's Elements now covers books 1-6 and contains over 180 interactive diagrams.
Available on iTunes
common core book cover
Common Core Nuggets: apps designed by master teachers to meet specific Common Core Standards.
Available on iTunes
11 Geometry Expressions  eBooks are now available as a bundle for the recession-friendly price of $85.95

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