COMMENT ⊗   VALID 00014 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00003 00002	
C00005 00003	1.0	INTRODUCTION TO GEOMES AND GEOMEL.
C00010 00004	DATUM AND LINK ACCESSING.
C00041 00005	3.0	WINGED EDGE PRIMITIVES.
C00043 00006	4.0	EULER MAKE PRIMITIVES.
C00045 00007	5.0	EULER KILL PRIMITIVES.
C00046 00008	6.0	EASY POLYHEDRON ROUTINES.
C00047 00009	7.0	EUCLIDEAN TRANSFORMATIONS.
C00050 00010	8.0	GEOMETRIC MEASURE ROUTINES.
C00051 00011	9.0	BODY INTERSECTION AND CUTTING.
C00052 00012	10.0	IMAGE FORMATION ROUTINES.
C00053 00013	11.0	INPUT/OUTPUT ROUTINES.
C00054 00014	12.0	AUXILLARY ROUTINES: III DISPLAY AND ARITHMETIC.
C00058 ENDMK
C⊗;

PART II - GEOMED AS A SAIL OR LISP ACCESSIBLE GRAPHICS COMMAND LANGUAGE.

1.0 Introduction to GEOMES and GEOMEL.
2.0 Memory, Control, Input and Output Routines.
	GEOMED MKUNIV MKNODE KLNODE
	MKWORLD MKCAMERA MKWINDOW 
	OUTB3D OUTGEM OUTCAM OUTVID
	INB3D  INGEM  INCAM  INCRE
3.0 Datum and Link Names:
	XWC YWC ZWC AA BB CC  XPP YPP ZPP
	IX IY IZ  JX JY JZ  KX KY KZ  
	NFACE PFACE NED PED NVT PVT DAD SON BRO SIS  
	ALT ALT2 CW CCW CAR8 CDR8
4.0 Winged Edged Primitives:
	MKB MKF MKE MKV MKFRAME
	WING INVERT EVERT
	ECW ECCW OTHER VCW VCCW FCW FCCW
	BDET BATT BGET
5.0 Euler Routines:
	MKBFV MKEV ESPLIT MKFE GLUEE
	KLBFEV KLFE KLEV UNGLUE
	GLUE MKCOPY SWEEP ROTCOM PYRAMID FVDUAL
	MKCUBE MKCYLN MKBALL
	BIN BUN BSUB MKCVEX 
	MKBUCK ECUT FCUT BCUT
6.0 Euclidean Routines:
	TRANSL ROTATE SHRINK APTRAN INTRAN DISTANCE
	NORM MKROT1 ORTHO1 ORTHO2 DETERM ANGL3V 
7.0 Image Forming Routines:
	GEODPY IIIDPY SHOW1 SHOW2 SHOW3 SHOW4 
	TAKE1 TAKE2 OCCULT SHADOW CLIPER
	VPROJ UNPROJ FACOEF ECOEF
8.0 Arithmetic and Display Routines:
	PI SQRT LOG SIN COS ATAN ATAN2 ASIN ACOS
	DPYBUF DPYSST DPYSET DPYBIG DPYBRT 
	AVECT AIVECT RVECT RIVECT DPYOUT
1.0	INTRODUCTION TO GEOMES AND GEOMEL.

	GEOMED is implemented in PDP-10 machine  code and is composed
of  geometric modeling  subroutines. These  subroutines are  SAIL and
LISP accessible depending on how they are assembled and  loaded. When
assembled and load for SAIL, the GEOMED subroutines are called GEOMES
for  "Geometric  Modeling  Embedded  in  SAIL";  similarly  when  the
routines are assembled and loaded  for LISP, they are referred  to as
GEOMEL standing for "Geometric Modeling Embedded in LISP".

	Strictly  defined, I  would have  preferred to  use  the name
"GEOMED" only for the  editor itself and to  call the larger  project
"GEM" for Geometric Modeling; however this has  not caught on, and
so  the reader is warned  that there are two  objects named "GEOMED",
one being the  interactive drawing  program by that  name, the  other
being the larger geometric modeling project including GEOMEL, GEOMES,
the data structure, the language, and so on.

	As  a graphics  language,   GEOMED is  all semantics  with no
syntax of its own. There are about one hundred  subroutine which take
from one to four arguments, return one or no values, and usually have
considerible side effects on the GEOMED data structure.
Unless  otherwise  noted,   all  arguments  and   values  are
integers; subroutines  executed for effect return integer value zero.

DATUM AND LINK ACCESSING.

	The GEOMED data  structure  is  implemented  as  twelve  word
blocks  containing pointers and data in the fashion usual to graphics
and simulation; an introduction to this technology can  be  found  in
Knuth. The  twelve  word  blocks are called "nodes". Nodes
are referred to by their actual machine  address  in  the  user  core
image, which is an integer called a "link". Subroutines that
take nodes as arguments or return nodes as values pass  links  rather
than  the  nodes  themselves.   In  SAIL,  the user core image can be
accessed as a special array named MEMORY.
3.0	WINGED EDGE PRIMITIVES.


3.1	MKB,MKF,MKE,MKV,MKFRAME.	Make BFEV Nodes.
3.2 	WING,INVERT,EVERT		Make and change wing pointers.
3.3	LINKED				Find if two entities are linked.
3.4	ECW,ECCW,			Edge fetching around FV perimeter.
3.5 	OTHER,VCW,VCCW,FCW,FCCW		Face-vertex fetching from an edge.
3.6	BDET,BATT,BGET			Body parts linking and body get.

INVERT(EDGE);

	An edge is a  directed vector with a positive  and a negative
vertex; INVERT flip the orientation of an edge vector and returns the
same edge.

EVERT(BODY);

	Evert turns  a  body inside  out,   or  vice  versa.   A  GEM
polyhedral  surface has  an  inside  and an  outside;  the inside  is
defined  by the orientation of the four  wing pointers in edge nodes;
the evert primitives changes the  order of these pointers in  all the
edges of the given body.

4.0	EULER MAKE PRIMITIVES.

4.1 	BNEW ← MKBFV;   	Make vertex polyhedron.
4.2 	VNEW ← MKEV(F,V);	MAKES NEW EDGE AND VERTEX SUCH THAT:
				VNEW = NVT(ENEW); V = PVT(ENEW);
	VNEW ← ESPLIT(E);	MAKES NEW EDGE AND VERTEX...
4.3 	ENEW ← MKFE(V1,F,V2);   MAKES NEW FACE AND EDGE SUCH THAT:
				FNEW = NFACE(ENEW); F = PFACE(ENEW);
				  V1 = PVT(ENEW);  V2 = NVT(ENEW).
4.4	ENEW ← GLUEE(F1,V1,F2,V2);	MAKES NEW EDGE, KILLS F2,
				AND MAKES A HOLE OR KILLS A BODY.
				  V1 = PVT(ENEW);  V2 = NVT(ENEW).

4.2	VNEW ← MKEV(FACE,VERTEX);
	VNEW ← ESPLIT(EDGE);

	Make a new  edge and a new vertex in  the given FACE from the
given VERTEX; the  new vertex is  return, VNEW; the  new edge can  be
accessed by taking PED(VNEW).
ESPLIT,	makes a new edge and a  new vertex, VNEW; the new edge may  be
obtained by taking PED(VNEW); the new  edge is place between VNEW and
PVT(EDGE).

4.3	ENEW ← MKFE(V1,FACE,V2);

	Make a new face and a new edge by  joining V1 and V2 of FACE.
The new  edge is returned, ENEW; the new  face may always be obtained
by taking NFACE(ENEW).
5.0	EULER KILL PRIMITIVES.

5.1 	QNEW ← KLBFEV(Q);	Kill entity.
5.2 	   F ← KLFE(E);		Kill E and NFACE(E), return PFACE(E).
5.3 	   E ← KLEV(V);		Kill V and PED(V), return other E of V.
	   V ← KLEV(E);		Kill E and NVT(E), retirn PVT(E).
5.4 	FNEW ← UNGLUE(E);	Undo an GLUEE.
6.0	EASY POLYHEDRON ROUTINES.

6.1	BODY ←	GLUE(FACE1,FACE2);	Glue face-face.
6.2	QNEW ←	MKCOPY(ENTITY);		Make copy.
6.3	FACE ←	SWEEP(FACE,FLAG);	Sweep cylinder.
6.4	FACE ←	ROTCOM(FACE);		Rotation completion.
6.5	PEAK ←	PYRAMID(FV);		Make pyramid on a face (or vertex).
6.6	BODY ←  FVDUAL(BODY);		Make face/vertex dual of a body.
6.7	BNEW ←  MKCUBE(DX,DY,DZ);	Make right rectangular prism.
6.8	BNEW ←  MKCYLN(RADIUS,N,DZ);	Make right cylinder.
6.9	BNEW ←  MKBALL(RADIUS,M,N);	Make sphere approximation.

7.0	EUCLIDEAN TRANSFORMATIONS.

FRAME ← TRANSLATE(INTEGER ENTITY; REAL DELTAX,DELTAY,DELTAZ);
FRAME ←    ROTATE(INTEGER ENTITY; REAL ABOUTX,ABOUTY,ABOUTZ);
FRAME ←    SHRINK(INTEGER ENTITY; REAL SCALEX,SCALEY,SCALEZ);

	When  the  ENTITY argument  is  non-zero,  these  subroutines
perform   the   indicated   Euclidean  Transformation:   translation,
rotation, dilation and reflection. When the ENTITY argument  is ZERO,
then  a  FRAME  node   representing  the  desired  transformation  is
returned.

FRAMES and EUCLIDEAN TRANSFORMATIONS

	A frame  node  has two  intrepretations: it  may  be used  to
represent  a  frame of  reference  or it  may  be used  to  specify a
Euclidean transformation.

	As a frame of reference the XWC,  YWC, ZWC datums contain the
location  of the origin  of the frame  in world  coordinates; and the
remaining nine datums IX,IY,IZ, JX,JY,JZ, KX,KY,KZ are the components
of three  unit vectors  I, J,   and K that  determine a  right handed
rectangular  Cartesian coordinate system. The  nine components of the
unit vectors form an orthonormal rotation matrix.

	As a Euclidean transformation, the frame is applied to the
3-D world coordinates of an entity Q to make new coordinates.
---------------------------------------------------------------------
|  APTRAN's inner most subroutine.				    |
|  Expects arguments in V and Q. Clobbers 1,2,X,Y,Z.		    |
|								    |
|	X ← XWC(V);						    |
|	Y ← YWC(V);						    |
|	Z ← ZWC(V);					            |
|						 		    |
|	XWC(V) ← X*IX(Q) + Y*JX(Q) + Z*KX(Q) + XWC(Q);		    |
|	YWC(V) ← X*IY(Q) + Y*JY(Q) + Z*KZ(Q) + YWC(Q);		    |
|	ZWC(V) ← X*IZ(Q) + Y*JZ(Q) + Z*KZ(Q) + ZWC(Q);		    |
---------------------------------------------------------------------

HOMOGENEOUS COORDINATES

	The interpretation of frame nodes can be given in the
four by four notation of homogeneous coordinates.
8.0	GEOMETRIC MEASURE ROUTINES.


9.0	BODY INTERSECTION AND CUTTING.


10.0	IMAGE FORMATION ROUTINES.


11.0	INPUT/OUTPUT ROUTINES.


12.0	AUXILLARY ROUTINES: III DISPLAY AND ARITHMETIC.