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18. Geometry
We will illustrate the topic of coordinate geometry by
defining functions for polygons in two dimensions that give
the displacements between adjacent vertices, the length of sides,
the semiperimeter, and the area based on Heron's formula.
We also present a more general definition for area that
not only gives the signed area (positive if the vertices
appear in counter-clockwise order), but also applies to polyhedra
in higher dimensions (in which case the name area might
better be replaced by volume). This definition is based
on the use of the determinant applied to a square matrix obtained
by bordering a table of vertices t by a final row
of the values %!#t . Thus:
(length=: +/&.*:) 5 12
13
disp=: ] - 1&|."1
sides=: length@disp
semiper=: -:@(+/)@sides
HERON=: %:@(*/)@(semiper - 0: , sides)
area=: -/ . * @ (] , %@!@#)
t=: 0 0 4 ,: 3 4 7
(];(1&|."1);disp;sides;semiper;HERON;area) t
+-----+-----+-------+-----+---------+---------------+--+
|0 0 4|0 4 0| 0 _4 4|0 4 4|0.5 3.5 4|2.29129 0j0.5 0|_2|
|3 4 7|4 7 3|_1 _3 4|1 3 4| | | |
+-----+-----+-------+-----+---------+---------------+--+
tet1=:6 0 3 0,3 6 5 8,:7 4 0 5
tet2=: 0,.=i.3 3
tet1;(area tet1);tet2;(area tet2)
+-------+----+-------+---------+
|6 0 3 0|11.5|0 1 0 0|_0.166667|
|3 6 5 8| |0 0 1 0| |
|7 4 0 5| |0 0 0 1| |
+-------+----+-------+---------+
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