A number of the projects in this book require the surface of the turned sphere to be marked out in a regular geometric pattern. Some basic mathematics will be needed, but nothing complicated — you probably learned most of this at school. I have included on the next page a reminder of some useful and simple mathematical formulae which are easily forgotten and may be difficult to find when you need them.
Marking equally spaced points on the surface of a sphere
For the sake of simplicity, all examples assume a sphere of 62mm (2-7/16") diameter.
A To find 12 equally spaced points upon the surface of a sphere
I The vertex separation (the spacing between the points) is given by multiplying the diameter of the sphere by 0.526.
2 Taking the sphere diameter as 62mm (20.526 x 62mm = 32.612mm (which can be rounded to 32.5mm) (0.526 x 2.438in = 1.282in = 1-9/32" Set the compass to 32.5mm (l and use this to mark off the 12 equally spaced points as described under heading C next page. (To find six equally spaced points upon the surface of a sphere, refer to heading E next page.)
This article is excerpted from Woodturning Wizardry by David Springett and © 2005 by Fox Chapel Publishing Company Inc.
B Locating the start points (top and bottom) on a sphere made from two hemispheres Once each hemisphere has been fully turned, its top centre position is no longer apparent. To find the position accurately, use the joint line as a starting point.
I Set a pencil compass to 44mm (1-3/4") radius (the calculation for this measurement is described below, section E, step 5) and place the compass point at any point on the joint line. Draw arcs above and below from this point.
2 Place the compass on any other point on the joint line, preferably on the opposite side, and strike arcs above and below from this point.
3 Where these two arcs intersect will be the top and bottom points of the sphere — the first two primary points (Fig 5.1). Erase all pencil marks except for the top and bottom points.
C Marking the 12 primary points on a sphere with known top and bottom points
I Set a pencil compass to 32.5 mm (1-9/32") radius (see heading A above)
2 Place the compass on the top end- grain point of the sphere and mark a circle about that point, then do the same at the bottom end-grain point.
3 Place the compass point at any point on either of these circles and draw a circle about that point.
4 Now place the compass point at any point where the circles intersect, and draw a new circle.
5 Repeat this procedure until all possible points have been used, when it will be noted that there are (including top and bottom points) 12 intersections. These are the primary points of the sphere (Fig 5.2 on previous page).
Marking the 20 constellation points
Constellation points are a series of 20 points equidistant from each other and from their nearest primary points.
I The first stage involves some trial and error. Choose three adjacent primary points. Place the compass point on one of them, extend the compass to where you judge the equidistant point to be, and strike an arc. Strike arcs of the same radius from the second and third primary points.
The three arcs describe a triangle within which the constellation point lies. Refine the compass setting until all three arcs meet in the same point; this is the precise location of the constellation point (Fig 5.3).
Keeping the compass at the same setting, draw circles around each of the primary points. The intersections of these circles mark the constellation points.
E Marking the six main points This is a practical approach for calculating the positions of six equidistant points on a spherical surface.
I Set a pencil compass to the radius of the sphere — in this case 31mm (1-7/32") — and draw a circle of that size on paper.
2 Draw a line horizontally through the centre of the circle. Where this line cuts the circle on one side mark the letter A, on the other side B.
3 Extend the compass to somewhat more than the radius of the circle — say, around 50mm (2in). With the compass point on A, strike an arc above the centre line; then do the same with the compass point on B.
This article is excerpted from Woodturning Wizardry by David Springett and © 2005 by Fox Chapel Publishing Company Inc.
4 Draw a line from the point where the two arcs intersect to the centre of the circle. Where this line cuts the circle, mark the letter C (Fig 5.4).
5 Set the compass point on B and extend the pencil to touch C. Measure this distance; in this example it will be 44mm (1-3/4")
6 Keeping this setting, place the compass point on one end-grain end of the 62mm (2-7/16") wooden sphere. Mark this position clearly with an E for ‘end grain’. Strike an arc all the way round the sphere. This produces a line which we will call the 'equator'.
7 Place the compass point upon any point on the equator, and strike a second arc around the sphere.
8 Place the compass point at the place where these two circles intersect, and strike a third arc around the sphere (Fig 5.5).
9 The six points where these three circles intersect will all be set at an equal distance from one another.
10 At the point directly opposite the one you marked with an F, mark a second B. This is the other end-grain position — you could think of them as the north and south poles.
F Marking eight clearance points
Clearance points are those positions which are equidistant from three adjacent main points. Drilling or turning holes at these points serves to remove waste material which could not easily be reached through the main openings. This exercise again takes a purely practical approach.
I Select three of the six main’ points, which form a triangle.
2 Place the compass point on any one of these three points. Extend the compass towards what you judge to be the central point between the three main points, and strike an arc. With the same compass setting, strike arcs from the second and third points also. Tithe three arcs do not meet exactly, they will describe a triangle around the central position. Readjust the compass until the precise centre is discovered. In our example the measurement is 28mm (1-1/2").
3 Retaining this setting, place the compass on each of the six main points, striking small arcs which will clearly mark out the positions of the eight clearance holes (Fig 5.6).
This article is excerpted from Woodturning Wizardry by David Springett and © 2005 by Fox Chapel Publishing Company Inc.
