dzach 2005-Sept-29: An interesting article about a new book, DIVINE PROPORTIONS: Rational Trigonometry to Universal Geometry, by N J Wildberger, appeared recently in [L1 ]. Here [L2 ] is the wikipedia reference to the subject.

In the sample chapter available for review [L3 ] the author gives a definition of the terms spread and quadrance. In simple words, the spread is an expression of the separation of two lines ( spread = (sin(angle))**2 ) which "replaces" the angle of classic trigonometry in rational trigonometric calculations, while the quadrance is the square of a distance.

Mathematically inclined minds may add more useful applications in this page. Here is a first take on how to find the spread between two lines, using tcl.

``` # Find the spread based on the coordinates of three points P0, P1, P2, which define lines |P0,P1| and |P0,P2|
#
#                   P1
#                  /
#                 /
#                /|
#               / |
#            P0/  |
#               \ |
#                \|
#                 \
#                  \
#                   P2
#
# use point coordinates to find quadrance
proc Qc {x1 y1 x2 y2} {
return [expr {pow(\$x2-\$x1,2)+pow(\$y2-\$y1,2)}]
}
#
# use side length to find quadrance
proc Qs sd {
return [expr {pow(\$sd,2)}]
}
proc Sc {x0 y0 x1 y1 x2 y2} {
# find the quadrances of each side of triangle P0-P1-P2 formed by the points P0,P1,P2
set q0 [Qc \$x2 \$y2 \$x1 \$y1]
set q1 [Qc \$x0 \$y0 \$x1 \$y1]
set q2 [Qc \$x0 \$y0 \$x2 \$y2]
# use the Cross law to find S0
return [expr {1-pow(\$q1+\$q2-\$q0,2)/(4.0*\$q1*\$q2)}]
}
#
proc Ss {sd1 sd2 sd3} {
set q1 [Qs \$sd1]
set q2 [Qs \$sd2]
set q3 [Qs \$sd3]
# use the Cross law to find S1
set res [expr {1-pow(\$q2+\$q3-\$q1,2)/(4.0*\$q2*\$q3)}]
if {\$res>=0 && \$res <=1.0} {
return \$res
} else {
error "This triangle cannot exist!"
}
}```

Examples:

Assume three points P0(0,0), P1(5,0) and P2(10,10), which define a horizontal line |P0,P1| and a slanted line |P0,P2|. The angle between the two lines is 45deg . The spread S0 will be:

``` % Sc 0 0 5 0 10 10
0.5```

Other examples:

``` % Sc 0 0 50 0 100 75
0.36
% Sc 0 0 5 0 10 20
0.8```

Assuming a triangle with sides 4, 5 and 6 units, find the spread opposite to side measuring 4 units (part of an example appearing in the sample chapter of the book mentioned above):

``` % Ss 4 5 6
0.4375```

Assuming a triangle with sides 3, 4 and 5 units, find the spread opposite to side measuring 4 units:

``` % Ss 3 4 5
0.36```

and for the other sides:

``` % Ss 4 5 3
0.64

% Ss 5 3 4
1.0```

This last one says that the spread opposite to side 5 corresponds to a right angle.

More on this subject as soon as the book arrives and is read :-).

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