Prof: 'Rational' geometry simplifies EE toolbox staple

Likewise, "45 degrees is a spread of 1/2, while 30 degrees and 60 degrees are spreads of 1/4 and 3/4, respectively. What could be simpler?" said Wildberger.

The upshot for "engineers, surveyors and scientists," he said, is a set of tools that will "increase accuracy and reduce computation time for geometric problems. Now there are many more opportunities to effectively harness the power of modern mathematical software, such as Mathematica, Matlab, Maple, Magma and Mupad."

The calculations in computer programs involving angles and distances will be simpler using rational trigonometry, since the formulas to perform the trigonometric functions will be arithmetic.

"Simpler is better, and this method is a lot simpler-not just by 10 or 20 percent but by a factor of two to five," said Wildberger. "Some engineers today have forgotten their trigonometry. I hope this new system will provide them with a more natural and easy-to-use alternative, which will then stay in their minds longer."

Classical or Euclidean geometry is taught today using axioms about distances and angles that can only be rigorously defined with calculus. But most people learn geometry and trigonometry before calculus, putting the cart before the horse.

"Divine proportions makes it clear why so many students are turned off by the mathematics of trigonometry," Wildberger said. "The current curriculum attempts to arm you with machine guns to hunt rabbits. Rational trigonometry adds a metrical aspect with the notions of quadrance and spread but stays in the purely algebraic setting, thereby putting Euclidean geometry on a firmer foundation."

Formulas simplified
Wildberger's formulation of geometry does not render sine, cosine and tangent obsolete, but it does make their use unnecessary except for problems that involve angular motion. "The essential roles of sine, cosine and tangent arise when considering angular motion around a circle," he said, "but in rational trigonometry, points and lines are more fundamental than circles."

He argues that rational geometry vastly simplifies many of the formulas EEs use. Even when calculus is used, if quadrance and spread are substituted for distance and angle, then the derivatives and integrals end up being much simpler, Wildberger says.

To demonstrate his point, Wildberger said Snell's law, which is used to determine everything from the direction of magnetic flux and current flow to the angle of reflection and refraction, can be applied with simple arithmetic. "The usual derivation of Snell's law uses the derivative of the square root function. But I give a derivation of Snell's law that shows that only derivatives of linear and quadratic functions are required, so it makes things more elementary."

Wildberger said his book also offers "a fair number of integrals for flux, currents and other quantities that electrical engineers sometimes need to evaluate. The book also introduces rational alternatives to spherical and polar coordinates that I am sure will prove useful to all engineers in solving practical problems. I give a new form of spherical coordinates in two- and three-dimensional space with simpler, more intuitive terms. I also show how to use these for calculations of volumes and surface areas."

Wildberger does not claim to have invented all these concepts, only to have codified them into a usable system. He gives historical credit to many others for the different aspects of rational trigonometry that have been used by various physicists to solve practical problems.

For instance, engineers already solve many trigonometric identities and integrals with various rational substitutions; linear algebra avoids angles by using dot products and cross products; and Einstein's theory of relativity makes use of quadratic intervals.