LOD switching :in case of Sphere

The projected size of sphere object decrease depends on the distance from camera.
If the distance from camera to sphere object goes to 2 times farther,
The projected size of sphere become $\frac{1}{2}$ smaller.
sphere.png
Below is how to calculate projected sphere's radius.

(1)
\begin{align} p=\frac{nr}{d \cdot (\vec{c}-\vec{v})} \end{align}
p
the radius of projected sphere(this is what I want to know)
d
normalized view vector
c
sphere's center point at loocal coordinates.
v
view point at local coordinates.
r
radius size at loocal coordinates.
n
distance from view point to view frustum's near clipping plane.
en.gif en
ja.gif ja


edit
サポートサイト Wikidot.com