Calculate and Draw Gores for a Quassi Semi-sphere or Dome. 2D to 3D Surface

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Introduction: Calculate and Draw Gores for a Quassi Semi-sphere or Dome. 2D to 3D Surface

First you need to know that these instructions are not all exact. Some curves get approximated by straight lines, some angles get approximated to right angles when they actually aren't. But in the end you get an acceptable semi-sphere.


Step 1: Define the Size

Next, we need to define the semi-sphere we want to create.

Size: What is the diameter, or radius, of our sphere going to be ?

e.g. Imagine we want to make a costume, I would go by my shoulder width as the diameter. 50 cm for example, that would make a radius of 25 cm

As you can see in the image the variable a holds the radius.

e.g. a = 25 cm

Now we need to calculate the arc length, b = 2*pi()*a / 4 , where:

  • 2*pi()*a is the circumference of the circle
  • We divide it by 4 because as you can see the arc is a quarter of the circle.

e.g. b = 2*3.14*25cm / 4 = 39.25 cm

Step 2: Define and Calculate the Gores

Define the number of gores you want to use:

  • gores are the number of slices that are cut out and placed together to form the semi-sphere.
    • The more you use the more spherical.
    • Recommended # of gores = 12

The image is a top view of a person and the semi-sphere covering them. Since this is not a 3D image but a 2D projection, you cannot appreciate the perspective. c is at the bottom of the semi-sphere while α is at the top.

Calculate the angle α = 360º / # of gores

e.g. α = 360º / 12 = 30º

Calculate the gore bottom width c = 2*pi()*a / # of gores

e.g. c= 2*pi()*25cm / 12 = 13cm

Step 3: Calculate and Draw the Gore Curves

Here is where the imperfections start. See the note on the bottom for the explanation.

Calculate the circle radius (d) required to obtain the gore curve.

  • d = b / sin(α/2)

e.g. d=39.25cm/sin(15º)=39/0.2588=151.5cm

Now use a piece of string of the length d and position the center at a distance b from the top and draw the arc.

Make a template: I would recommend to make a template to copy paste side by side the curve

Depending on the thickness of the material you use, and the method for connecting the gores to each other, you may want to leave some flaps around 2 cm around the gores to fold inwards and use for joining the gores.

Note:

c which is actually a curve is approximated for a straight line. The error depends on the number of gores you have decided to do. The more gores, the less error.

consequently imposing c as the straight line, also changes b from "perfection". The "gore curve", in a 3 dimensional space would be the same as b, but as you can see that with the approximations we have made this is impossible.

It is possible to calculate the real values, but the added effort is not worth the difference noted in my opinion. None the less I have used this method with fabrics and sponges that have a certain flexibility to them, where imperfections get hidden.

Anyway, you'll get an acceptable semi-sphere.

Step 4: Cut the Gores and Connect All Edges

Cut out the remaining bits and start by gluing the bottom

Then start gluing the gores in pairs.

Use contact glue and use gloves.

Note: Due to imperfections in cutting and gluing you may need to improvise for the top decoration.

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    4 Discussions

    Wow. Thank you! I have been casually searching the Web for such a long time, I'd have to measure in YEARS, for some way to calculate what you refer to as "gores". It has only been in the past few weeks that my occasional web searching revealed that the mathematics I sought was pioneered\ invented millennia ago by Archimedes, who I have during this time been quietly regarding as "my new patron saint". But with only his raw theory, I was searching further for someone who could extract in practical terms, just the part I could use to "gore" a 2-D cylinder, and coax them together into a dome. Hence forth, I offer my greatest gesture of thanks - this glowing comment. "Paucoma", and Archimedes - my two "patron saints".

    Well, what I would do with gores, I don't know, but this is very cool and I learned something new :) Thanks for sharing!

    Nice geometry skills, it looks really stellar! Do you have further plans for this skills? I bet some great finished projects can come out of this technique. I hope we see more from you in the future!

    1 reply

    Hi, Thanks. Actually we made some great costumes of a couple minions. this is where some of the images come from. It was the first steps to some great costumes which I'm attaching as a photo.

    Minions_CafedeFinca.jpg