ARITH3(2)                                               ARITH3(2)

          add3, sub3, neg3, div3, mul3, eqpt3, closept3, dot3, cross3,
          len3, dist3, unit3, midpt3, lerp3, reflect3, nearseg3,
          pldist3, vdiv3, vrem3, pn2f3, ppp2f3, fff2p3, pdiv4, add4,
          sub4 - operations on 3-d points and planes

          #include <draw.h>
          #include <geometry.h>

          Point3 add3(Point3 a, Point3 b)

          Point3 sub3(Point3 a, Point3 b)

          Point3 neg3(Point3 a)

          Point3 div3(Point3 a, double b)

          Point3 mul3(Point3 a, double b)

          int eqpt3(Point3 p, Point3 q)

          int closept3(Point3 p, Point3 q, double eps)

          double dot3(Point3 p, Point3 q)

          Point3 cross3(Point3 p, Point3 q)

          double len3(Point3 p)

          double dist3(Point3 p, Point3 q)

          Point3 unit3(Point3 p)

          Point3 midpt3(Point3 p, Point3 q)

          Point3 lerp3(Point3 p, Point3 q, double alpha)

          Point3 reflect3(Point3 p, Point3 p0, Point3 p1)

          Point3 nearseg3(Point3 p0, Point3 p1, Point3 testp)

          double pldist3(Point3 p, Point3 p0, Point3 p1)

          double vdiv3(Point3 a, Point3 b)

          Point3 vrem3(Point3 a, Point3 b)

          Point3 pn2f3(Point3 p, Point3 n)

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     ARITH3(2)                                               ARITH3(2)

          Point3 ppp2f3(Point3 p0, Point3 p1, Point3 p2)

          Point3 fff2p3(Point3 f0, Point3 f1, Point3 f2)

          Point3 pdiv4(Point3 a)

          Point3 add4(Point3 a, Point3 b)

          Point3 sub4(Point3 a, Point3 b)

          These routines do arithmetic on points and planes in affine
          or projective 3-space.  Type Point3 is

               typedef struct Point3 Point3;
               struct Point3{
                     double x, y, z, w;

          Routines whose names end in 3 operate on vectors or ordinary
          points in affine 3-space, represented by their Euclidean
          (x,y,z) coordinates.  (They assume w=1 in their arguments,
          and set w=1 in their results.)

          Name      Description
          add3      Add the coordinates of two points.
          sub3      Subtract coordinates of two points.
          neg3      Negate the coordinates of a point.
          mul3      Multiply coordinates by a scalar.
          div3      Divide coordinates by a scalar.
          eqpt3     Test two points for exact equality.
          closept3  Is the distance between two points smaller than
          dot3      Dot product.
          cross3    Cross product.
          len3      Distance to the origin.
          dist3     Distance between two points.
          unit3     A unit vector parallel to p.
          midpt3    The midpoint of line segment pq.
          lerp3     Linear interpolation between p and q.
          reflect3  The reflection of point p in the segment joining
                    p0 and p1.
          nearseg3  The closest point to testp on segment p0 p1.
          pldist3   The distance from p to segment p0 p1.
          vdiv3     Vector divide - the length of the component of a
                    parallel to b, in units of the length of b.
          vrem3     Vector remainder - the component of a perpendicu-
                    lar to b. Ignoring roundoff, we have
                    eqpt3(add3(mul3(b, vdiv3(a, b)), vrem3(a, b)), a).

          The following routines convert amongst various representa-
          tions of points and planes.  Planes are represented

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     ARITH3(2)                                               ARITH3(2)

          identically to points, by duality; a point p is on a plane q
          whenever p.x*q.x+p.y*q.y+p.z*q.z+p.w*q.w=0.  Although when
          dealing with affine points we assume p.w=1, we can't make
          the same assumption for planes.  The names of these routines
          are extra-cryptic.  They contain an f (for `face') to indi-
          cate a plane, p for a point and n for a normal vector.  The
          number 2 abbreviates the word `to.'  The number 3 reminds
          us, as before, that we're dealing with affine points.  Thus
          pn2f3 takes a point and a normal vector and returns the cor-
          responding plane.

          Name      Description
          pn2f3     Compute the plane passing through p with normal n.
          ppp2f3    Compute the plane passing through three points.
          fff2p3    Compute the intersection point of three planes.

          The names of the following routines end in 4 because they
          operate on points in projective 4-space, represented by
          their homogeneous coordinates.

               Perspective division.  Divide p.w into p's coordinates,
               converting to affine coordinates.  If p.w is zero, the
               result is the same as the argument.

          add4 Add the coordinates of two points.

          sub4 Subtract the coordinates of two points.



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