Returns \(I(\)p2 is in \(N_{PE}(p1,r))\)
for points p1 and p2
in the standard basic triangle,
that is, returns 1 if p2 is in \(N_{PE}(p1,r)\),
and returns 0 otherwise,
where \(N_{PE}(x,r)\) is the PE proximity region
for point \(x\) with expansion parameter \(r \ge 1\).
PE proximity region is defined
with respect to the standard basic triangle
\(T_b=T((0,0),(1,0),(c_1,c_2))\)
where \(c_1\) is in \([0,1/2]\), \(c_2>0\)
and \((1-c_1)^2+c_2^2 \le 1\).
Vertex regions are based on the center,
\(M=(m_1,m_2)\) in Cartesian coordinates
or \(M=(\alpha,\beta,\gamma)\) in
barycentric coordinates
in the interior of the standard basic triangle \(T_b\)
or based on circumcenter of \(T_b\);
default is \(M=(1,1,1)\), i.e.,
the center of mass of \(T_b\).
rv is the index of the vertex region p1 resides,
with default=NULL.
If p1 and p2 are distinct and either of them are
outside \(T_b\), it returns 0,
but if they are identical,
then it returns 1 regardless of their locations
(i.e., it allows loops).
Any given triangle can be mapped to the standard basic triangle
by a combination of rigid body motions
(i.e., translation, rotation and reflection) and scaling,
preserving uniformity of the points in the original triangle.
Hence, standard basic triangle is useful for simulation
studies under the uniformity hypothesis.
See also (ceyhan:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:arc-density-PE;textualpcds).