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Geometry Expressions Newsletter
Geometry Expressions Newsletter
April 2014 
Mechanical Expressions Beta Extended
We have extended the free beta period for our novel symbolic mechanics program Mechanical Expressions for a further 6 months.  

The beta will now extend till the end of September.  If you are a registered beta user, you should have been emailed a new key.

If you did not get a new key, you should just sign up again here.

Don't forget to give us your feedback, we are working to incorporate as many user comments into the final product as we can.

How to... model a bead sliding on a cycloid curve in Mechanical Expressions

An inverted cycloid curve is the solution to the brachistochrone problem: providing the quickest path between two points for a ball rolling under gravity, or a bead sliding on a wire.

Here is how to model this in Mechanical Expressions.

First create the cycloid: a parametric curve with 
X = T+sin(T), Y = -1-cos(T) and with T varying from -3.14 to 3.14.

Now position a point on this curve and use Constrain/Proportional
 to constrain it to lie at parametric location t.

Select the point and use Mechanics Input/Mass to specify its mass. 

Now select the parameter t and use Mechanics Output/Resultant Acceleration to compute its acceleration.

Now create an app using File/Export/HTML5 Javascript App

Specify that the app should contain a simulation, give it a start button, and set t to be draggable.  The result should look like this.

If you add a second bead on a line and a third on a manipulable curve, you can observe that the bead on the cycloid beats both.
Anja Greer Conference
Involute Clock 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
 
Last month's problem came from Archimedes De lineis spiralibus, (in Greek Peri helikon).

This month's problem is from his Liber Assumptorum, whose Greek version is lost, and whose attribution is uncertain.

But the problem could come from a modern test.
  
C lies on the circumference of a circle centered at A. CD cuts the circle at B. BD is the same length as the radius of the circle.  DA extended meets the circle in E.

What is the relationship between angle ADB and angle CAE?

An app which lets you explore this problem is here.
A Solution to February's Problem
 
.

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.


In Geometry Expressions, position point O at the origin, create the polar curve r=T, and constrain a point A at parametric location T=t on the curve.  

Now join OA, draw a line segment OB and constrain it to be perpendicular to OA.

The length of OB as a function of T is the polar equation for the locus of B.

Where the original curve is r=T, the derived curve is r=T^2

You can change the definition of the original curve to find that

r=T^2 yields r=(T^3)/2

r=T^n yields r=(T^(n+1))/n

r=f(T) yields r=(f(T)^2)/|f'(t)|

 
 
eBooks

Our Euclid's Elements now covers books 1-6 and contains over 180 interactive diagrams.
Available on iTunes
 
A breakthrough in the use of interactive diagrams for gaining new insight into mathematics.
Available on iTunescommon core book cover
Common Core Nuggets: apps designed by master teachers to meet specific Common Core Standards.
Available on iTunes

Van Warren's latest iBook for the Kindle, the first of a series on mathematical feedback is available from amazon here

11 Geometry Expressions  eBooks are now available as a bundle for the recession-friendly price of $85.95

Partners
We have a growing network of partners around the world, committed to giving our customers the most convenient access to our products and product support.

Australia/New Zealand

4am Software Australia

4am Software New Zealand

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Scientific Software Solutions

Hong Kong 

ETC Educational Technology Connection (HK) Ltd

Japan

HULINKS

Korea

Creative Learning Corp

Our partner base is continually growing, so check out our partner page here for updates.

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