### 3.3 Critical surface mass density

In the more general case of a three-dimensional mass distribution of an extended lens, the density
can be projected along the line of sight onto the lens plane to obtain the two-dimensional surface mass
density distribution , as
Here is a three-dimensional vector in space, and is a two-dimensional vector in the lens plane. The
two-dimensional deflection angle is then given as the sum over all mass elements in the lens plane:
For a finite circle with constant surface mass density the deflection angle can be written:
With this simplifies to
With the definition of the critical surface mass density as
the deflection angle for a such a mass distribution can be expressed as
The critical surface mass density is given by the lens mass “smeared out” over the area of the Einstein
ring: , where . The value of the critical surface mass density is roughly
for lens and source redshifts of and , respectively. For an
arbitrary mass distribution, the condition at any point is sufficient to produce multiple
images.