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 [
1]. Here [
2] is the wikipedia reference to the subject.
In the sample chapter available for review [
3] 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/ |
# \ | spread S0
# \ |
# \|
# \
# \
# 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)}]
}
# find spread given coordinates
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)}]
}
#
# find spread given sides
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 :-).
