Mathematica code to calculate weighting functions for bilinear interpolation.
(* Bilinear approximating function within a cell bounded by x1,y1,x2,y2 *) (* values at corners of cell are *) (* (x1,y1) => f1 *) (* (x1,y2) => f2 *) (* (x2,y1) => f3 *) (* (x2,y2) => f4 *) bilin[ x_, t_ ] := f1 ( 1- (x-x1)/(x2-x1))(1-(t-t1)/(t2-t1) ) + f2 ( 1 - (x-x1)/(x2-x1)) (t-t1)/(t2-t1) + f3 (x-x1)/(x2-x1) (1-(t-t1)/(t2-t1) ) + f4 (x-x1)/(x2-x1) (t-t1)/(t2-t1)
(* A linear path across the cell *) (* Starts at xx0,tx0, exits at xx1,tx1 *) tau[ x_ ] := tx0 ( 1-(x-xx0)/(xx1-xx0)) + tx1 (x-xx0)/(xx1-xx0)
(* The bilinear function along the path *) bilin[ x_ ] := bilin[ x, tau[ x ] ]
(* Integral along the path *) res = Integrate [ bilin[x], {x, xx0, xx1 } ]
Print[""] Print["Coeffficient in f1"] cf1 = Simplify[ Coefficient[ Collect[ res, f1],f1]] Print[""] TeXForm[ cf1 ]
Print[""] Print["Coeffficient in f2"] cf2 = Simplify[ Coefficient[ Collect[ res, f2],f2]] Print[""] TeXForm[ cf2 ]
Print[""] Print["Coeffficient in f3"] cf3 = Simplify[ Coefficient[ Collect[ res, f3],f3]] Print[""] TeXForm[ cf3 ]
Print[""] Print["Coeffficient in f4"] cf4 = Simplify[ Coefficient[ Collect[ res, f4],f4]] Print[""] TeXForm[ cf4 ]