For the curve r=r(t), its tangent, normal, and binormal unit vectors, often called T, N, and B, satisfies the Frenet–Serret formulas. So we have:

.

Let n be the unit normal vector of the sphere, and denote as the angle between n and N such that

Differentiate the above equation against s the arc length parameter we get:

For spheres we have n=r the position vector, so

Plugin this one and the above Frenet formula we get

Since B, N, n are all orthogonal to T, their three are in the same plane. And since the angle between N and n is ,

and B, N are orthogonal, the angle between B and n is then .

So we have . Plugin to the above equation we got:

Integrate it we get