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C00001 00001
C00002 00002	SUMMARY OF POLYHEDRON PRIMITIVES.
C00005 00003	PRIMITIVES ON POLYHEDRA.
C00009 00004	Euler Primitives.
C00013 ENDMK
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SUMMARY OF POLYHEDRON PRIMITIVES.

A. EULER PRIMITIVES.
	
     1. BNEW ← MKBFV;		make a body, face & vertex.
     2. KLBFEV(Q);		kill a body & all its pieces.
     3.	VNEW ← MKEV(F,V);	make edge & vertex.
     4.	ENEW ← MKFE(V1,F,V2);	make face & edge.
     5.	VNEW ← ESPLIT(E);	split an edge.
     6.	F    ← KLFE(ENEW);	kill face  &  edge leaving a face.
     7.	E    ← KLEV(VNEW);	kill edge & vertex leaving an edge.
     8.	V    ← KLVE(ENEW);	kill vertex & edge leaving a vertex.
     9.	B    ← GLUE(F1,F2);	glue two faces together.
*   10.	BNEW ← UNGLUE(E);	unglue along a seam containing E.

B. SOLID PRIMITIVES.

     1.	VPEAK ← PYRAMID(F); 	form a pyramid on a face.
     2.	F ← PRISM(F);		form a rectangular prism.
     3.	F ← CWPRISMOID(F)	form a clockwise prismoid.
     4.	F ← CCWPRISMOID(F);	form a counter clockwise prismoid.
     5.	ROTCOM(F);		complete a solid of rotation.
     6.	FVDUAL(B);		form face vertex dual of a body.
     7. BNEW ← MKCOPY(B);	make a copy of a body.
     8.	EVERT(B);		turn a body surface inside out.
     9.	B1 ← BUN(B1,B2);	form union of body interiors.
    10.	B1 ← BIN(B1,B2);	form intersection of body interiors.

C. GEOMETRIC PRIMITIVES.

     1.	TRANSLATE(Q,R);
     2.	ROTATE(Q,R);
     3.	DILATE(Q,R);
     4.	REFLECT(Q,R);

D. IMAGE PRIMITIVES.

     1.	PROJECTOR(CAMERA,WORLD);
     2.	ELIST←CLIPER(WINDOW,WORLD);
     3.	OCCULT(WORLD);
*    4.	SHADOW(SUN,WORLD);
*    5.	TV ← MKVID(WINDOW,WORLD);
*    6.	B2D ← MKB2D(WINDOW,WORLD);
*    7.	B2D ← CAREYE(TV);

* under construction, Oct 1972.
PRIMITIVES ON POLYHEDRA.

	In this section a number of primitives for  doing  things  to
polyhedra  are  explained.    Although these primitives are currently
implemented using the winged edge data structure, they do not require
a  particular  polyhedron  representation.     Indeed,  many of these
primitives  were  originally  implemented  in   a   LEAP   polyhedron
representation  very  similar  to  that  of  Falk,  Feldman  and Paul
[reference 5].   Thus, the primitives of this section are on a  level
logically independent from the operations of the previous section.

	Another  aspect  of these primitives is that they can be used
as the basis of a "graphics language" or more accurately as a package
of  subroutines  for geometric modeling. In this vein, the primitives
are currently collected as a  package  called  GEOMES  for  Geometric
Modeling Embedded in SAIL; and as GEOMEL, Geometric Modeling Embedded
in LISP.  A third language, called GEOMED, arises out of the  command
language of a geometric model editor based on the primitives.

	The primitives are shown in four groups in the summary.   The
first  group,  the Euler Primitives, were inspired by Coxeter's proof
of Euler's formula, section 10.3 of [reference 2]. Although the proof
only  required three primitives, additional ones of the same ilk were
developed for convenience.   The second group  is  composed  of  some
polyhedron primitives that were coded using the Euler primitives. The
third group is for primitives that  move  bodies,  faces,  edges  and
vertices; or compute geometric values such as length and volume. This
group is underdeveloped for two reasons:  one, because  I  have  done
these  computations  ad  hoc to date; and two, because they imply the
subject of animation which is large and difficult and not of  central
importance to vision. With the exception of the camera, my worlds are
nearly (but not absolutely) static.  A  less  impoverished  geometric
group  will  be  presented  in the future. The final group, has three
well  developed  primitives  for  making  2D  images;   and   several
primitives  that when finished will realize part of the vision system
that I am trying to build.
Euler Primitives.

	As mention above, the Euler primitives are based on the Euler
Equation F-E+V = 2*B-2*H; where F, E, V, B and H stand for the number
of faces, edges, vertices, bodies and handles that exist.   The  term
"handle"  comes from topology, and is the number of well formed holes
in a surface; a sphere has no handles, a torus has one handle, and an
IBM  flowcharting  template  has  26  handles.     The Euler equation
restricts  the  possible  topologies  of  FEV  graphs  that  can   be
polyhedra;  although  such  Eulerian  polyhedra  do  not  necessarily
correspond to what we normally call  a  solid  classical  polyhedron.
Strict  adherence  to  constructing a polyhedron that satisfies Euler
equation F-E+V = 2*B - 2*H would require only four primitives:

				   	+F -E +V = 2*B - 2*H

   1.	Make Body, Face and Vertex	+1....+1....+1......
   2.	Make Edge and Vertex.		...-1 +1............
   3.	Make Face and Edge.		+1 -1...............
   4.	Glue two faces of one body.	-2 +N -N..........+1
   4.'	Glue two faces of two bodies.   -2 +N -N....-1......

However, the  four  corresponding  destructive  primitives  are  also
possible and desirable:
				   	+F -E +V = 2*B - 2*H

   1.	Kill Body, Face and Vertex	-1....-1....-1......
   2.	Kill Edge and Vertex.		...+1 -1............
   3.	Kill Face and Edge.		-1 +1...............
   4.	Unglue along a seam.	 	+2 -N +N..........-1
   4.'	Unglue along a seam.	 	-2 +N -N....+1......

And finally the operation of splitting an edge at a midpoint into two
edges became so important in  forming  T-joints  during  hidden  line
elimination  that the ESPLIT primitive was introduced in place of the
equivalent KLFE, MKEV, MKFE sequence.

	In  using  the Euler primitives, some non-classical polyhedra
are tolerated as  transitional  states  of  the  construction;  these
transitional states are called:

	Seminal Polyhedron.
	Wire Polyhedron.
	Lamina Polyhedron.
	Shell Polyhedron.
	Face with Wire Spurs on its perimeter.

A seminal polyhedron is like a point; a  wire  polyhedron  is  linear
with two ends like a single piece of wire; lamina and shell polyhedra
are surfaces, and the picturesque phrase about spurs is a restriction
on  how  faces  are  dissected  into more faces.  These terms will be
explained in more detail when they are needed.