The isosurfaces I'm reading about are defined by a constant value v in a scalar field. The scalar field is defined by placing n vectors in k-space such that
$iso(\vec{x})=\sum_{i=1}^{n}\sum_k(x_k-p_{i_k})^2$
So the field value at each point in space is the sum of all squared distances. I'm interested to find a function a(v) that computes or approximates the area given $iso(\vec{x})=v$ as the isovalue.
Is there maybe a closed form that immediately gives the result? What would be good further reading on that topic? Thanks!
Your isosurface is a sphere, having center at the centroid of the $p_i$ and whose radius $r$ is given by $$ r^2={v\over n}-{1\over n^2}\sum_{i\ne j}(\vec p_i-\vec p_j)^2. $$ That follows from the identity $$ \sum_{i=1}^n(\vec x-\vec p_i)^2= n\left(\vec x -\sum_{i=1}^n{\vec p_i\over n}\right)^2+ {1\over n}\sum_{i\ne j}{(\vec p_i-\vec p_j)^2}. $$ Once $r$ is known it is easy to get the area of the sphere. Notice however that an isosurface exists only if ${v}>{1\over n}\sum_{i\ne j}(\vec p_i-\vec p_j)^2$.