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Triangulated Bow Dome

written by Rene K. Mueller, Copyright (c) 2007, last updated Sat, October 6, 2007

Updates

Fri, August 17, 2007: First version, with Bezier curve based model, three main types featured: 0/0 (hemisphere like), 25/0 (cross-tie bow dome like) and 45/0 (peak-tie bow dome like or "oriental").

Introduction

After I studied the Geodesic Dome I thought about triangulating also other non-(hemi)spherical forms, e.g. a general Bow Dome as I covered already using Bezier curve to simulate bending of bows. One of the disadvantages of the dome is that the roof has too litte steep slope in order to have snow slide down, so it would be interesting to have a circular bow dome with a minimum of slope, like a Yurt, yet, triangulated.

Please Note: This is a very preliminary version of the document, and may change dramatically (e.g. such as used notions) or be renamed. In particular I like to optimize the amount of different struts for each version, and later also add an online calculator for each featured option.

Regular Dome


Regular Dome (α = 0°, β = 0°)

Regular Dome
A part of the top area of the dome is where no snow can slide (orange in the illustration), in regions with snow fall this is a disadvantage.

Steep Roof Dome


Steep Roof Dome (α = 25°, β = 0°)

Steep Roof Dome
A steady 25° roof resolves the above mentioned condition.

Dent Roof Dome


Dent Roof Dome (α = 45°, &beta = 0°)

Dent Roof Dome
With a 45° angle from the top, and still remaining vertical start at the bottom, provides an "oriental" shape.

It needs to be said in case of dome, no matter what variant, that the entire surface acts as roof, in other words, all cover requires 100% sealed of rain water.

Stripe Triangulation

The Bezier curve is linearly split into n/2-steps, where n*2 amount of edges of the polygon for the base, that outline is rotated to create a circular space, and triangulated in stripes.

Methods of Stripe Triangulation

There are two ways to stripe triangulate as I focus on (there are many other methods available), keeping the amount of edges constant, or reduce them by half when a certain length is reached:


Method A: Amount of edges kept constant, more narrow triangulation near top

Method B: Amount of edges variable, strut length kept above a minimum

The detail about switching to from 1:1 triangulation to 1:2 when a certain minimum length is reached (aka as "trimming"):


Stripe Triangulation Methods

  • n = amount of sections (preferable 2n, 3*2n and 5*2n so division by 2 is possible without remainder)
    • examples: 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64 ...
    • hint: the w*2n, that w is the amount of edges of the top polygon, but at least 3 (if w = 1 then last polygon is 4)
  • t = 0 .. 1 with 1/(n/2) steps (which is 1/4 of full circumference)
  • smin = 2 &pi rn / n / 4;
  • m0 = 2 n
  • mn+1 = mn/2 if sm < smin
  • sm = 2 π rn / m

References

Overview

Here an overview of a couple of options, for now I feature only Method B ("trimmed lamella") with 1:2 triangles when it narrows:

Triangulated Bow Dome 0° / 0°


0° / 0° / 6

48 struts (6 kinds)

0° / 0° / 8

76 struts (9 kinds)

0° / 0° / 10

125 struts (11 kinds)

0° / 0° / 12

186 struts (13 kinds)

0° / 0° / 16

308 struts (18 kinds)

0° / 0° / 20

475 struts (22 kinds)

Triangulated Bow Dome 25° / 0°


25° / 0° / 6

48 struts (6 kinds)

25° / 0° / 8

76 struts (9 kinds)

25° / 0° / 10

125 struts (11 kinds)

25° / 0° / 12

186 struts (13 kinds)

25° / 0° / 16

308 struts (18 kinds)

25° / 0° / 20

475 struts (21 kinds)

Triangulated Bow Dome 45° / 0°


45° / 0° / 6

48 struts (6 kinds)

45° / 0° / 8

76 struts (9 kinds)

45° / 0° / 10

125 struts (11 kinds)

45° / 0° / 12

168 struts (13 kinds)

45° / 0° / 16

308 struts (18 kinds)

45° / 0° / 20

475 struts (22 kinds)


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