9/25/2005
Rational Trig and Markets, by Doc Alex Castaldo
Last week the Bloomberg newswire carried the
surprising announcement that Australian
mathematician Norman Wildberger has developed a new
form of trigonometry that does not use sines,
cosines, etc., but can nevertheless solve all
trigonometric problems (and do it better, he claims,
than the classical trigonometry invented 3,000 years
ago). Called Rational Trigonometry, Wildberger's
invention does not even use distance and angle, but
replaces them with two new notions: "quadrance" and
"spread" (see below).
I was skeptical, to say the least, but after
reading the first chapter of his book, I see nothing
obviously wrong or nonsensical in what he is saying.
Here is a brief summary:
Def. The "quadrance" between two points is the
square of the distance, i.e. the quadrance between
A_1 = [x_1, y_1] and A_2 = [x_2, y_2] is defined as
Q(A_1, A_2) = (x_2 - x_1)^2 + (y_2 - y_1)^2
Def. The "spread" betwen two lines L_1 (with
equation a_1*x + b_1*y + c_1 = 0) and L_2 (with
equation a_2*x + b_2*y + c_2 = 0) is defined as
s(L_1, L_2) = (a_1*b_2 - a_2*b_1)^2 / ((a_1^2 +
b_1^2)*(a_2^2 + b_2^2))
The spread is 0 when the lines are parallel, is
equal to 1 when the two lines are perpendicular.
[The spread is like a sine squared, but remember
that we are not supposed to use sines and cosines
anymore].
Wildberger claims that all trigonometric problems
can be solved by applying what he calls The Five
Main Laws of Rational Trigonometry.
We are given three distinct points A_1, A_2 and
A_3. We use the notation Q_1 = Q(A_2, A_3) etc. and
s_1 = s(A_1 A_2, A_1 A_3) etc.
- The Triple Quad Formula
The three points are collinear iff (Q_1 + Q_2 +
Q_3)^2 = 2*(Q_1^2 + Q_2^2 + Q_3^2)
- Pythagoras Theorem
The lines A_1 A_3 and A_2 A_3 are perpendicular
iff Q_1 + Q_2 = Q_3 [this is familiar]
- The Spread Law
For any triangle with non-zero quadrances
s_1/Q_1 = s_2/Q_2 = s_3/Q_3 [This is
recognizable as similar to the Sine Law, but
with all terms squared]
- The Cross Law
For any triangle (Q_1 + Q_2 - Q_3)^2 = 4 * Q_1 *
Q_2 * (1-s_3)
- The Triple Spread formula
For any triangle (s_1 + s_2 + s_3)^2 = 2*(s_1^2
+ s_2^2 + s_3^2) + 4*s_1*s_2*s_3
Does it really work? Does it offer any advantage
over classical trigonometry? Even if it does, will
people want to abandon familiar concepts to learn
the new ones? I doubt it. But Manchester Trading
will try to be the first to apply these concepts to
financial markets. |