Maple Application Center Highlights

New Maple applications are submitted to us from users on a daily basis, bringing our total number of applications on the site to over 400. Among the new submissions this summer, here are three that best showcase the power of Maple.

Differential Geometry Visualised with Maple
Submitted by Dr. John Oprea, Cleveland State University

Is a donut more ñroundî than a sphere? Can you shape a ramp so that a ball rolling down it reaches the bottom in the same amount of time, no matter how high you release it? These are questions of differential geometry, the study of curves and surfaces in 3-space. John Oprea submitted six Maple applications showing how to visualise several concepts of differential geometry. We showcase one of them here.

A path in 3-space is characterised by its curvature, which is a mathematical function describing how sharply the path bends at any given time. A basic theorem of differential geometry states that a path can be recreated solely from its curvature function. Although the formula involves integrals which can be solved only rarely, you can still compute and plot the path numerically. Oprea's Maple application implements this procedure. Given curvature k, the formula for the path b as a function of times is

> beta(s)=[Int(cos(theta(u)),u=0..s),Int(sin(theta(u)),u=0..s)];



where



You can transform these integrals into a system of differential equations to be solved numerically and plotted, resulting in the path with the specified curvature. Oprea writes a Maple procedure called recreate() that does just that. He gives the following example:

> k:=t->sin(t)*t:
> #the curvature of the unknown path at time t is sin(t)t)
> recreate(k, -8, 8, -2, 2, 0, 3);



Maple Essentials
An online Maple Tutorial by Mike Pepe, Seattle Central Community College This collection of Maple worksheets guides new users through a hands-on, self-paced introduction to basic Maple commands. Through six sessions, the user learns tools for arithmetic, algebraic manipulations, graphing, solving equations, and defining functions. Each session contains numerous examples, as well as space for the student to experiment and work through exercises. There is also a handy reference page summarising the commands presented in the tutorial.

Taking the Agony out of Trig
Submitted by Wayne Matthews, Camosun College, B.C. Canada

The scene: high school trig class

Teacher:
"Sine and cosine are the vertical and horizontal projections, respectively, of a line segment from the center to the rim of the unit circle."
Student A:
"Huh?"
Teacher:
"Well, letÍs graph it. Sin(x) forms a wave ranging from -1 to 1 over the angles 0 to 2p. See?"
Student B:
"No. What do waves have to do with sine?"
Student C:
"Yeah, you told us last week that sine and cosine were ratios of a triangleÍs sides. Opposite-over-hypotenuse, and all that. Now itÍs a wave?"
Students D-Z:

The above travesty is a problem of visualization. How, indeed, does one convey to students the equivalence of sin(x) as the opposite-over-hypotenuse ratio, as a projection inside the unit circle, and as a wave?