Laying out octagons

March 3, 2003

__Octagons in telescope making__. Octagons may appear in secondary cage rings, tube stiffeners, secondary holders,

rocker boxes, and a number of other places in telescopes. In amateur telescope making octagons appear more

frequently than any other shape outside of circles and squares. But unlike these simpler figures, it isn’t immediately

obvious how to layout regular octagons.

__The width is key__. The easiest way to generate an octagon is to make a square the width of the desired octagon, then

clip off the corners by the appropriate amount. So the first thing we need to know is how wide the octagon needs to be.

Sometimes we can make the width whatever we choose, but often the octagon must be inscribed within—or

circumscribed around—a circle of a given diameter (such as the circular shadow of the secondary mirror, or the arc

traced by the three azimuth bearing pads on the ground board). For a circle inscribed within an octagon this is easy—a

circle with a diameter equal to the width of the octagon (or square) will fit exactly within the polygon. Thus, for a circle

inscribed in a regular polygon

W = D

where W = the width of the octagon or square, and D is the diameter of the inscribed circle.

It is just a little more complicated to fit an octagon inside a circle, with the vertices (points) just touching the

circumference. For a circle circumscribing an octagon

D = 1.0824 W

or putting things the other way around

W = 0.9239 D

Once you know the width of the octagon, the length of each side is

L = 0.4142 W

where L is the length of the side.

__Laying out the octagon__. An easy usual way is to first layout an accurate square the same width as the desired octagon,

then make marks 0.2929W and 0.7071W in from one side (the side of the octagon takes up 0.4142 of the width of the

square, leaving 0.2929W on each side; 0.7071 is 0.2929 + 0.4142). Draw two lines perpendicular to this axis passing

through the marks, from one side of the square to the other. Now rotate the square 90 degrees and draw two similar

lines at right angles to the first pair. The intersections between these layout lines and the sides of the square are the

eight points of the octagon. Now just connect the points and you have your octagon.

__Inscribing an octagon within an ellipse__. In a Newtonian the secondary mirror is at a 45 degree angle to the optical axis

of the telescope, and takes the shape of a 45 degree ellipse (that is, the shape of a circle viewed from an angle of 45

degrees rather than square on). It may be desirable to fit an octagonal mirror carrier within the elliptical outlines of the

secondary. To fit an octagon within an ellipse we simply need to know the width of the rectangle within which the

octagon must fit, and adjust our layout lines accordingly. In a 45 degree ellipse the major axis is 1.4142 times the minor

axis (that is, the minor axis times the squareroot of 2). The width of the inscribed octagon will be 0.9239 times the minor

axis of the ellipse. The length of the octagon will be this number times the squareroot of 2, or 1.3066 times the minor

axis (or, if you prefer, 0.9239 times the major axis).

__Layout a rectangle with these dimensions__. Then draw your layout lines as before—make your marks at 0.2929 and

0.7071 times the width of the rectangle, and 0.2929 and 0.7071 times the length of the rectangle. Draw your pairs of

layout lines through these marks, and connect the points as before. You will have a distorted octagon that precisely

fits within the ellipse. You can draw octagons to fit an ellipse of any obliquity once you know the major and minor axes

of the ellipse.

Copyright 2009 Ross Sackett

rocker boxes, and a number of other places in telescopes. In amateur telescope making octagons appear more

frequently than any other shape outside of circles and squares. But unlike these simpler figures, it isn’t immediately

obvious how to layout regular octagons.

clip off the corners by the appropriate amount. So the first thing we need to know is how wide the octagon needs to be.

Sometimes we can make the width whatever we choose, but often the octagon must be inscribed within—or

circumscribed around—a circle of a given diameter (such as the circular shadow of the secondary mirror, or the arc

traced by the three azimuth bearing pads on the ground board). For a circle inscribed within an octagon this is easy—a

circle with a diameter equal to the width of the octagon (or square) will fit exactly within the polygon. Thus, for a circle

inscribed in a regular polygon

W = D

where W = the width of the octagon or square, and D is the diameter of the inscribed circle.

It is just a little more complicated to fit an octagon inside a circle, with the vertices (points) just touching the

circumference. For a circle circumscribing an octagon

D = 1.0824 W

or putting things the other way around

W = 0.9239 D

Once you know the width of the octagon, the length of each side is

L = 0.4142 W

where L is the length of the side.

then make marks 0.2929W and 0.7071W in from one side (the side of the octagon takes up 0.4142 of the width of the

square, leaving 0.2929W on each side; 0.7071 is 0.2929 + 0.4142). Draw two lines perpendicular to this axis passing

through the marks, from one side of the square to the other. Now rotate the square 90 degrees and draw two similar

lines at right angles to the first pair. The intersections between these layout lines and the sides of the square are the

eight points of the octagon. Now just connect the points and you have your octagon.

of the telescope, and takes the shape of a 45 degree ellipse (that is, the shape of a circle viewed from an angle of 45

degrees rather than square on). It may be desirable to fit an octagonal mirror carrier within the elliptical outlines of the

secondary. To fit an octagon within an ellipse we simply need to know the width of the rectangle within which the

octagon must fit, and adjust our layout lines accordingly. In a 45 degree ellipse the major axis is 1.4142 times the minor

axis (that is, the minor axis times the squareroot of 2). The width of the inscribed octagon will be 0.9239 times the minor

axis of the ellipse. The length of the octagon will be this number times the squareroot of 2, or 1.3066 times the minor

axis (or, if you prefer, 0.9239 times the major axis).

0.7071 times the width of the rectangle, and 0.2929 and 0.7071 times the length of the rectangle. Draw your pairs of

layout lines through these marks, and connect the points as before. You will have a distorted octagon that precisely

fits within the ellipse. You can draw octagons to fit an ellipse of any obliquity once you know the major and minor axes

of the ellipse.

Copyright 2009 Ross Sackett