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Thayer Watkins
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The Relationship Between the
Solid Angle of a Conical Point
and Its Angular Deficit

The solid angle and the angular deficit do not measure the same thing, but there is a relationship between them. It is an antithetical relation. When the solid angle is 2π the angular deficit is zero. When the angular deficit is 2π the solid anglle is zero.

Let γ be half the apex angle of a cone. On a sphere of unit radius the cone will subtend an area equal to 2π(1 − cos(γ)). This is the solid angle of the cone, which will be denoted as Θ.

The radius of the circle of intersection of the cone and the unit sphere is equal to sin(γ). The length of the circumference of that circle is then 2πsin(γ) but the circumference of the cicle within which the unrolled cone lies is 2π. Thus the angular deficit α of the cone isgiven by

α = 2π − 2πsin(γ) = 2π(1 − sin(γ))
and hence
sin(γ) = 1 − α/(2π)

On the other hand,

Θ/(2π) = 1 − cos(γ)
so
cos(γ) = 1 − Θ/(2π)

Since sin²(γ)+cos²(γ)=1

[1 − α/(2π)]² + [1 − Θ/(2π)]² = 1
which reduces to
2α/(2π) − [α/(2π)]² + 2Θ/(2π) − [Θ/(2π)]² = 1

This shows the antithetical relationship between the two variables. This can also be seen through the differention the above equation; i.e.,

2[1 − α/(2π)](−1/(2π)) + [1 − Θ/(2π)](−(1/(2π)(dΘ/dα) = 0
which reduces to
(dΘ/dα) = − [1 − α/(2π)]/ [1 − Θ/(2π)]

Solid Angle as a Function of the Angular Deficit

The solid angle can be expressed as a function of the angular deficit.

Θ = 2π(1 − cos(γ))
but
cos(γ) = [1 − sin²(γ)]½

But sin(γ) is given by

sin(γ) = 1 − α/(2π)

Therefore

Θ = 2π(1 − [1−(1 − α/(2π)²)½)

When α is equal to zero, Θ is equal to 2π. When α is equal to 2π, Θ is equal to zero.


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