How to Construct an Hexagonal Parabolic Dome

Spencer Hunter, 1999
This document, unaltered, is in the public domain.

If the parabola x^2 = z is rotated about the origin, the result is the
paraboloid of revolution x^2 + y^2 = z .  The intersection of the
paraboloid with an orthographic plane C=y (where C is a numeric constant)
would be, by substitution, another parabola x^2 + C^2 = z (where C^2 is
also a constant).

It follows, then, that an intersection of the paraboloid with any
orthographic plane is exactly the same parabolic curve that passes through
the origin, only transposed up a bit.  A two-dimensional triangular grid
orthographically projected onto such a paraboloid will form a grid of
identical parabolas.  This fact makes the construction of triangular
templates for a parabolic dome simple.  I will use the "six frequency"
division dome as an example.

First, draw a parabolic segment, using a T-square and some string,
from the origin out to where the slope is about 30 to 45 degrees.  Next,
divide the horizontal axis from the origin out to the end of the segment
into twelve equal parts.  Where these divisions intersect with the
parabola form twelve points.  Connect every other point with a straight
line segment to end up with twelve line segments of different lengths (the
point nearest the origin "connects" to a symmetrical point on the other
side of the origin; simply draw a horizontal line from the point to the
vertical axis and call it half the length of the segment).  After that,
label the segments, starting from the one closest to the origin, a' , a ,
b' , b , c' , c , d' , d , e' , e , f' , and f .

The layout would look something like this:

. a'. b'. c'. d'. e'. f'.
  . a . b . c . d . e . f .
  ^                       ^
  origin of parabola      end of parabolic segment

These are the lengths used to construct the triangular templates.  An
inventory of the templates would look like this (using base, left side,
right side format):

1) a'aa

2) a'b'b'
   abb'

3) a'c'c'
   ac'b
   b'cb

4) a'cc
   ad'c
   b'd'c'
   bdc'

5) a'dd
   add'
   b'e'd'
   be'c
   c'ec

6) a'e'e'
   aee'
   b'ed
   bf'd
   c'f'd'
   cfd'

Paraboloidal hexagons have twelve-way symmetry.  The templates that lie
along the axis of symmetry (the ones that appear at the top of each list
above) need only be duplicated six times.  The others have to be
duplicated twelve times, representing six triangles and their mirror
images.  When the resulting triangles are correctly assembled, the
paraboloidal shape should naturally emerge.  The same holds true for
greater frequency divisions, which require more templates but yield a more
accurate shape.

Pictures of my experimental foil solar reflector are on my web page at
http://www.u.arizona.edu/~shunter/cads.html under "Parabolic domes," or one
level up from this document.



Appendix: How to Draw a Continuous Parabolic Segment

When I was an adolescent, I thought I had originated this method for drawing
a parabola.  Later, I learned that Johannes Kepler had beaten me to it by
some 400 years.  The method depends on knowing how to draw an ellipse
using two pins and a length of string, and understanding the parabola as a
kind of "infinite ellipse."

When drawing an ellipse, one attaches string between two pins pushed into
a board representing the two foci, then uses a pencil to trace the ellipse
by keeping the string taut with the pencil at all times.

One could think of drawing the parabola like a giant ellipse, where one of
the pins has been thrown out to infinity.  The string coming in from the
pin infinitely far away will always be perpendicular to the baseline (and
also perpendicular to the directrix).  Since we can't have infinitely
distant pins connected to infinite lengths of string in the real world, we
use a T-square to effectively simulate the "infinite pin" by keeping the
"incoming string" always perpendicular to the baseline.

First, attach one end of a finite length of string to a pin pushed into a
drawing board (representing the focus of the parabola).  Clamp the other
end of the string to the top of a T-square so that it is lined up with the
edge of the T-square.  Starting with the T-square lined up with the pin,
establish the following conditions:

        1. The pencil keeps the string taut.
        2. The pencil is kept against the edge of the T-square.

Slide the T-square away from the pin while *always* meeting the above two
conditions.  The pencil will trace a continuous parabolic segment.