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 Geometric 3D Animations

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 POVRay.
Note: Here objects in 2D geometry are represented by 3D shapes in the xyplane.
Therefore all coordinates must have the zcomponents zero! ( <?,?,0>) 

We want the external tangents to the tangent points
T_{1} and T_{2} of two circles C_{1}(M_{1},r_{1}) and C_{2}(M_{2},r_{2}) with
the radii r_{1} > r_{2} ,
as shown in the opposite image.
The distance of their centers is d.
The difference of their radii is r_{i} = r_{1}  r_{2}.

The triangle M_{1},S,M_{2} has a right angle at S.
The line(T_{1},T_{2}) is parallel to the line(M_{2},S) and has the same length.
So t = T_{1},T_{2} = sqrt( d^{2}  r_{i}^{2}).
The angle α = atan(r_{i}/t). or α = asin(r_{i}/d).

The calulation of the length of the belt around:
The length of the segment around the circle C_{1}:
l_{1} = 2π·r_{1} ·(180+2·α)/360.
The length of the segment around the circle C_{2}:
l_{2} = 2π·r_{2} ·(1802·α)/360.
The length of the complete belt is:
l = l_{1} + l_{2} + 2·t .


External tangents to two circles rendered with POVRay


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


A round conic torus.

A round conic prism.





