Prisms and Antiprisms

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dpietrobon's picture
Submitted by dpietrobon on Sat, 01/03/2025 - 15:36

A prism is a 3D shape with two matching polygonal faces connected by straight edges.

To create these objects in the virtual world we can reuse our code for making polygons and simply connect the edges

to prism :n :r :h
  ; Generates an n-gonal prism of radius r and height h
  make "s 2 * :r * sin( 180 / :n ) ; Side length
  ru 90 fd :h / 2 rd 90 polygon :n :r
  rd 90 fd :h     ru 90 polygon :n :r
  fd :r rt 90 + 180 / :n
  pd ru 90 fd :h rd 90
  repeat :n [ fd :s rd 90 fd :h ru 180 fd :h rd 90 rt 360 / :n ] pu
  rt 90 - 180 / :n fd :r rt 180 rd 90 fd :h / 2 ru 90

An antiprism is like a prism, but its top and bottom faces are rotated and connected with triangles.

 
The following logo procedure defines antiprisms centered at the origin:
 
to antiprism :n :r :h
  ru 90 fd :h / 2 rd 90             polygon :n :r
  rd 90 fd :h     ru 90 rt 180 / :n polygon :n :r pu
  rt 90 fd :r rt 90
  rt 90 / :n pd
  for [ i 1 :n 1 ][
    make "angle1 ( 360 * :i ) / :n
    make "angle2 ( 360 * :i + 180 ) / :n
    make "x1 :r * cos :angle1
    make "x2 :r * cos :angle2   
    make "z1 :r * sin :angle1
    make "z2 :r * sin :angle2
    make "diag sqrt ( :h ^ 2 + ( :x2 - x1 ) ^ 2 + ( :z2 - :z1 ) ^ 2 ) ; Triangle side length
    lookatxyz( :x1 ) (  :h / 2 ) ( :z1 ) fd :diag
    lookatxyz( :x2 ) ( -:h / 2 ) ( :z2 ) fd :diag
  ]
end

Interestingly, the triangular faces of an antiprism form a shape known as a skew regular polygon. That is a polygon which is regular, but not confined to 2-dimensional space alone.