Intro: 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)
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.
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.