9F. Geometry
We begin with some simple functions (Length, Area, Volume) of various figures (rectangle, box, circle, cone, sphere, pyramid) applied to a length or list of lengths. For example:
m0=: Ar=: */ |
Area of rectangle |
m1=: Ab=: 2: * [:+/ ] *1&|. |
Area of box |
m2=: Vb=: */ |
Volume of box |
m3=: Lci=: 2: * o. |
Length (circumference) of circle (radius) |
m4=: Aci=: [:o. ] ^ 2: |
Area of circle (r) |
d5=: Aco=: o.@* |
Area of cone, excluding base (h r) |
d6=: Vco=: 1r3p1"_ * ] * * |
Volume of cone (h r) |
m7=: As=: 4p1"_ * ] ^ 2: |
Area of sphere (r) |
m8=: Vs=: 4r3p1"_ * ] ^ 3: |
Volume of sphere (r) |
m9=: L=: +/&.(*:"_)"1 |
Length of a vector |
d10=: Lp=: [: L [ , [: L [: -: ] |
Length of edges of pyramid (h w,l) |
d11=: Ap=: [:+/ ]* [:L"1 [,"0-:@] |
Area of pyramid, excluding base (h w,l) |
d12=: Vp=: 1r3"_ * */@, |
Volume of pyramid |
m13=: sp=: -:@(+/) |
Semi-perimeter |
m14=: h=: [: %: [: */ sp - 0: , ] |
Heron's formula for area of triangle |
For example:
h 3 4 5 6 h 51 52 53 1170 h 2 2 2 1.73205
In treating coordinate geometry we will use a list of n elements to represent a point in n-space, and an m by n table to represent a polygon of m vertices. For example:
p=: 3 1 [ q=: 4 1 [ r=: 5 9 NB. Three points T=: p,q,:r NB. A triangle L=: +/&.(*:"_)"1 NB. Length function L p 3.16228 u=: 1&|. Rotate up D=: u-] Displacements ,.&.>(];u;D;L@D)T3 NB. Displacements and lengths (of sides) +---+---+-----+-------+ |3 1|4 1| 1 0| 1| |4 1|5 9| 1 8|8.06226| |5 9|3 1|_2 _8|8.24621| +---+---+-----+-------+ line=: 3 1 4,:1 5 9 NB. A line in 3-space (];-/;L@(-/);L) NB. line Line, disp, length, distances to ends +-----+-------+------+---------------+ |3 1 4|2 _4 _5|6.7082|5.09902 10.3441| |1 5 9| | | | +-----+-------+------+---------------+ T3=: ?.3 3$10 NB. Random triangle in 3-space ,.&.>(];u;D;L@D) T3 +-----+-----+--------+-------+ |1 7 4|5 2 0| 4 _5 _4|7.54983| |5 2 0|6 6 9| 1 4 9|9.89949| |6 6 9|1 7 4|_5 1 _5|7.14143| +-----+-----+--------+-------+
m15 =: L=: +/&.(*:"_)"1 |
Length |
m16=: D=: 1&|.-] |
Displacement |
m17=: LS=: L"1@D |
Lengths of sides |
m18=: S=: 1&o.@(*&1r180p1) |
Sine in degrees |
m19=: C=: 2&o.@(*&1r180p1) |
Cosine in degrees |
m20=: r=: (C,S),:(-@S,C) |
2-dim rotation matrix in degrees |
m21=: b=: <"1@(,"0/~) |
Table of boxed index pairs: do i 0 2 |
d22=: R=: (r@])`(b@[)`(=@i.@3:)} |
3-dim rm: From axis 0 to 2 is 0 2 R a |
d23=: mp=: +/ . * |
Matrix product |
m24=: R3=: (2 0"_ R 0&{)mp(1 2"_ R 1&{)mp(0 1"_ R 2&{) |
R3 p,q,r is p-rotate from axis 2 to 1 on q-r from 1 to 2 on r-r from 0 to 1 |
m25=: Det=: -/ . * |
Determinant |
m26=: Area=: [:Det ] ,. %@!@{:@$ |
Area of triangle |
m27=: Vol=: Area f. |
Volume of simplex in n-space (fixed) |
d28=: dsplitby=: ~:/@:*@:Vol@:(,"1 2) |
Are points pairs (2 by n matrix) x separated by n by n simplex y? |
m29=: Area2=: [: -: [: +/ 2: Det\ ] |
Area of polygon |
Area yields the area of a triangle expressed as a 3 by 2 list of x-y coordinates:
TT 3 3 6 5 2 7 Area TT 7
Area2 also yields the area of a triangle, expressed by a similar table, but with the top row repeated at the bottom:
TT2 3 3 6 5 2 7 3 3 Area2 TT2 7
It is more general, however, and will yield the area of arbitrary polygons:
Polygon 7 2 10 5 6 8 3 6 4 3 7 2 Area2 Polygon 24.5