Analog gadgets

Come new school year, wide spread of computers, networks, World Wide Web notwithstanding, student are going to use scissors, rulers, compass and other (analog) gadgets for class work, home assignments and ultimately learn and achieve not necessarily academic goals. There is indeed a great variety of analog devices available that often provide an unexpected and edifying experience. Some could be simulated on a computer with added flexibility and ease of access. One is presented below.

Now, I assume that everyone remembers a simple property of right-angled triangles - a median from the right angle to the hypotenuse equals one half of the latter. This fact leads to the way of drawing circles illustrated by the applet below. To see how it works, note that the red segment has rings on both of its ends. By pointing at the rings and dragging the mouse you should be able to move the segment such that its ends will slide along the two axes. The mid point of the segment will then trace a circle. Do you see why?

The gadget takes you one step further. There is actually a third ring in the middle of the segment. This one is also movable. Shift it a little and then slide the segment again. What curve do you obtain this time? It's an ellipse. Using Analytic Geometry, the proof is fairly straightforward. If you follow through you'll be able to compare this gadget with the one that uses a string attached to two points (foci of the ellipse). You have to tighten the string by a tip of a pen and then move the pen keeping the string taut.

You may ponder the difference between analog and digital devices. The string gadget is probably the simpler of the two. However, when it came to programming I had a difficulty to truthfully simulate a loose string. As you may have noticed, no such difficulty arises with the original gadget.

Although simple, this gadget furnishes an opportunity for several exercises:

• Trigonometry: triangle solving
• Conic sections: gathering all ellipse parameters from an equation
• Analytic Geometry: straight line equation r = r₀ + t·(r - r₀) which is useful in locating the drawing point

A reminder

In a circle, an inscribed right angle subtends the circle's diameter. So that the median from the right angle equals the radius of the circle whereas the hypotenuse is equal to its diameter.

Other gadgets

• Abacus
• Angle Trisection
• Broken Calculator
• Cycloids
• Dynamic construction of ellipse and other curves
• Equichordal Points
• Geoboard
• Hart's Inversor
• Paper Folding
• Peaucellier Linkage
• Pythagorean Theorem
• Soroban
• Suan pan

Paul Kunkel from Washington describes a curious device - planimeter - that has been widely used until about 20-30 yeas ago for area estimates.

References

1. V. Gutenmacher, N. Vasilyev, Lines and Curves: A Practical Geometry Handbook , Birkhauser; 1 edition (July 23, 2004)

Ellipse

• Analog device simulation for drawing ellipses
• Brianchon in Ellipse
• Butterflies in Ellipse
• Ellipse in Arbelos
• From Foci to a Tangent in Ellipse
• Gergonne in Ellipse
• Illustration of Pascal's Theorem
• La Hire's Theorem in Ellipse
• Optical Property of Ellipse
• Poncelet Porism in Ellipses
• Reflections in Ellipse
• Three Squares and Two Ellipses