INTERFACE SPFunction;

(* General functions on the sphere *)

IMPORT Wr, LR3;
 
TYPE T = OBJECT
  METHODS
    eval(READONLY p: Point): LONGREAL;
    grad(READONLY p: Point): Gradient;
    slap(READONLY p: Point): LONGREAL;
    toMaple(wr: Wr.T);
  END;

TYPE
  LONG = LONGREAL;
  LONGS = ARRAY OF LONGREAL;
  Point = LR3.T;     (* A point on the unit sphere *)
  Points = ARRAY OF Point;
  Gradient = LR3.T;  (* A vector tangent to the unit sphere. *)
  
  Basis = ARRAY OF T;
  
PROCEDURE SamplePoints(N: CARDINAL): REF Points;
  (*
    Generates N points, roughly uniformly distributed on the sphere. *)

PROCEDURE Sample(f: T; READONLY p: Points): REF LONGS;
  (*
    Evaluates "f" at each point of "p", returns the results. *)

PROCEDURE DotProduct(
    f, g: T; 
    depth: CARDINAL;
  ): LONGREAL;
  (* 
    Computes the dot product of "f" and "g", that is the 
    integral of "f*g" over the whole sphere, 
    using "(order+1)*(order+2)/2" points over 
    each octant. *)
    
PROCEDURE CustomDotProduct(
    f, g: T; 
    READONLY p: Points;
  ): LONG;
  (* 
    Computes the dot product of "f" and "g", defined as
    "(A/N)*SUM(f(p[k])*g(p[k]))", "N = NUMBER(p)", "A" is the area of
    the sphere, and the "p[k]" is a client-specified set of sample
    points on the sphere. *)
  
END SPFunction.