INTERFACE HLR2;

(* 
  Oriented projective geometry in three dimensions.
  Created 94-05-04 by J. Stolfi.
  Last edited by stolfi *)
   
IMPORT LR2, LR3, LR3x3;

TYPE
  Point = RECORD c: LR3.T END;  (* c[0..2] are coords [w,x,y]. *)
  Line =  RECORD f: LR3.T END;  (* f[0..2] are coeffs <W,X,Y>. *)
  Sign =  [-1..1];
  
PROCEDURE FromCartesian(READONLY c: LR2.T): Point;
  (*
    Point on "hither" half of the plane (i.e, with positive weight)
    whose Cartesian coordinates are "c". *)
    
PROCEDURE ToCartesian(READONLY p: Point): LR2.T;
  (*
    Cartesian coordinates of point "p" (which must be finite). *)

PROCEDURE Side(READONLY p: Point; READONLY L: Line): Sign; 
  (*
    Returns sign of point "p" relative to line "L": 0 on the line,
    +1 in positive halfplane, -1 in negative halfplane.
    May give inconsistent results for points very close to the line. *)

PROCEDURE Orient(READONLY p, q, r: Point): Sign;
  (*
    Returns the orientation (turning sense) of the triangle "p q r".  
    Specifically, the triangle "[1 0 0] [0 1 0] [0 0 1]" has positive
    orientation. Returns 0 iff the points are collinear.
    May give inconsistent results for points very close to collinear. *)

PROCEDURE Join(READONLY p, q: Point): Line;
  (*
    Return the line through "p" and "q". *)

PROCEDURE Meet(READONLY K, L: Line): Point;
  (*
    Return the point common to "K" and "L". *)

PROCEDURE Dir(READONLY frm, tto: Point): LR2.T;
  (*
    Direction (a unit-length vector) of point "tto" seen from point "frm".
    Works even if one of them is at infinity.  Does not work if both
    are at infinity, or coincident, or antipodal. *)
    
PROCEDURE Normal(READONLY L: Line): LR2.T;
  (*
    The normal direction of line "L": a unit vector which,
    on the hither side of the plane, points from "L" into 
    "L"'s positive halfplane.  Assumes "L" is not at infinity. *)
    
(* PROJECTIVE MAPS *)

TYPE
  PMap =  RECORD dir: LR3x3.T; inv: LR3x3.T END;  
    (* 
      "dir" is the map's matrix, "inv" is its inverse. *)

PROCEDURE MapPoint(READONLY p: Point; READONLY m: PMap): Point;
  (*
    Applies projective map "m" to point "p". *)

PROCEDURE MapLine(READONLY L: Line; READONLY m: PMap): Line;
  (*
    Applies projective map "m" to line "L" *)

PROCEDURE CompMap(READONLY m, n: PMap): PMap;
  (* 
    Returns the composition of "m" and "n", applied in that order *)

END HLR2.