Dimension of the space; the unit sphere is an (n-1) dimensional manifold
k
Number of subdivisions
method
"dyadic" or "edgewise": the former recursively subdivides the sphere to get
a more uniform grid; the latter uses a faster method using one edgewise subdivision.
p
Power used in the l^p norm; p=2 is the Euclidean norm
positive.only
TRUE means restrict to the positive orthant; FALSE gives the full ball
start
starting shape: "diamond" or "icosahedron"
x
Matrix of points in n-dimensions; each column is a point in R^n
Details
UnitSphere computes a hyperspherical triangle approximation to the unit sphere.
It calls either UnitSphereDyadic or UnitSphereEdgewise
based on 'method'. Both work by subdividing the first octant, and then rotating that
subdivision around to other octants.
Note that 'k' has a different meaning for the different methods. When
method="dyadic", k specifies the number of dyadic subdivisions. When method="edgewise",
k specifies the number of subdivisions as in UnitSimplex, which is then
projected outward to the unit sphere. So when n=2, a dyadic subdivision with k=2 will
result in 16 edges, whereas an edgewise subdivions with k=2 results in 8 edges.
UnitBall computes an approximate simplicial approximation to the unit ball. Specifically,
it generates cones with one vertex at the origin and the other vertices on the surface
of the unit sphere; these later vertices are from UnitSphere. If k is large, these cones
will be very narrow/thin.
Value
an object of class "mvmesh" as described in mvmesh.