Surface approximation of point clouds by using multiquadric functions

The following literature provides the easiest way to generate a surface that approximates a given point set.
Only the limitation found in this approach is that the generated surface becomes a height field, i.e. it is always of the form z=f(x,y).
[1] R. L. Hardy, “Multiquadric equations of topography and other irregular surfaces,” J. Geophys. Res., vol. 76, no. 8, pp. 1905–1915, Mar. 1971.

import rhinoscriptsyntax as rs
import Rhino.Geometry as rg
import math
func=lambda xi,xj,yi,yj:math.sqrt(((xj-xi)**2)+((yj-yi)**2)+x)
A=rg.Matrix(len(pnts),len(pnts))
z=rg.Matrix(len(pnts),1)
for i in range(len(pnts)):
    for j in range(len(pnts)):
        pi=pnts[i]
        pj=pnts[j]
        A[i,j]=func(pi.X,pj.X,pi.Y,pj.Y)
        z[i,0]=pi.Z
A.Invert(0.0)  #this parameter should be 0.0
c=A*z
a=list()
for srf in srfs:
    domU=srf.Domain(0)
    domV=srf.Domain(1)
    for i in rs.frange(0,1,0.05):
        for j in rs.frange(0,1,0.05):
            u=domU[0]+i*(domU[1]-domU[0])
            v=domV[0]+j*(domV[1]-domV[0])
            (b,P,A)=srf.Evaluate(u,v,0)
            z=0
            for j in range(len(pnts)):
                z=z+c[j,0]*func(P.X,pnts[j].X,P.Y,pnts[j].Y)
            newP=rg.Point3d(P.X,P.Y,z)
            a.append(newP)
newSrf=rs.AddSrfPtGrid((20,20),a)
newSrf = rg.NurbsSurface.CreateFromPoints(a, 20, 20, 3, 3)
a=list()
a.append(newSrf)

page revision: 5, last edited: 23 Nov 2014 10:05

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Masaaki MIKI