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ARITH3(3)		   Library Functions Manual		     ARITH3(3)

       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

       #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)

       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  projec-
       tive 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.

	      Is the distance between two points smaller than eps?

       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.

	      The reflection of	point p	in the segment joining p0 and p1.

	      The closest point	to testp on segment p0 p1.

	      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 perpendicular to b.   Ig-
	      noring   roundoff,  we  have  eqpt3(add3(mul3(b,	vdiv3(a,  b)),
	      vrem3(a, b)), a).

       The following  routines	convert	 amongst  various  representations  of
       points  and  planes.   Planes are represented 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 indicate 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 corresponding 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  coordi-

       pdiv4  Perspective division.  Divide p.w	into p's coordinates, convert-
	      ing 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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