Expo-Rational B-splines
Transfinite blending
Introduction
Expo-Rational B-splines abbreviated to ERBS is a basis function to be used in a transfinite blending of what we can call “local”functions/curves/surfaces/ In the figure below is this illustrated.
A (global) Expo-Rational B-spline curve (red) with four local curves (green). The global curve is a blending of its local curves, with the Expo-Rational B-splines being the blending functions. The global curve also completely interpolates all existing derivatives of each of the adjacent local curves at the “middle” knot. One important property of an ERBS curve is that at the interpolating points is the global curve interpolating (also with all its derivatives) the local curves.
The ERBS
A graph of an Expo-Rational B-spline; Bk(t) (solid red) and its first derivative (dotted blue). The knots tk-1, tk and tk+1 are also marked on the plot.
ERBS-tensor product surface
This Expo-Rational B-spline tensor product surface is first made by Hermite interpolating (position, first and second derivative) a plane at 3 x 3 points. The positions of the interpolating points are seen as cubes. The interpolation point in the center is moved upwards. Finally the center point is twisted, i.e. rotated approximately 140 degree around the z-axis.
ERBS surface on triangulation
The surface is a connected set of eight triangular Expo-Rational B-spline surfaces. This surface is made by first approximating a sphere. Then the surface is deformed by an editing process.