Descriptions and Examples for the POV-Ray Raytracer by Friedrich A. Lohmüller
    Elementary Geometry for Raytracing
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External Tangents to two Circles
In the following we write for the square root of a number the expression "sqrt(NUMBER)" and "abs(NUMBER)" for |NUMBER|,
conforming to the syntax used in POV-Ray.

Note: Here objects in 2D geometry are represented by 3D shapes in the xy-plane. Therefore all coordinates must have the z-components zero! ( <?,?,0>)

We want the external tangents to the tangent points    
T1 and T2
of two circles C1(M1,r1)
and C2(M2,r2) with
the radii r1 > r2 , as shown in the opposite image.
The distance of their centers is d.
The difference of their radii is ri = r1 - r2.
The triangle M1,S,M2 has a right angle at S. The line(T1,T2) is parallel to the line(M2,S) and has the same length.
So t = |T1,T2| = sqrt( d2 - ri2).
The angle α = atan(ri/t). or   α = asin(ri/d).
The calulation of the length of the belt around:
The length of the segment around the circle C1:
l1 = 2π·r1 ·(180+2·α)/360.
The length of the segment around the circle C2:
l2 = 2π·r2 ·(180-2·α)/360.
The length of the complete belt is: l = l1 + l2 + 2·t .
External tangents to two circles rendered with POV-Ray

For what can we use this geometry?
Here some examples:

A round conic torus.
A round conic prism.
A roller chain or a conveyor belt.
Animation tutorial:
'Roller Chain'

A rolling bike chain.
Animation tutorial:
'Bike Chain'
See an animation here:
'Animation Bike Chain'
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© Friedrich A. Lohmüller, 2010
http://www.f-lohmueller.de