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- Geometric 3D Animations
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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>) |
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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.
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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).
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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 .
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External tangents to two circles rendered with POV-Ray
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For what can we use this geometry?
Here some examples:
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A round conic torus.
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A round conic prism.
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