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 00002 00002	Euclidean Primitives.
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Euclidean Primitives.

  1. TRANSLATE(Q,R);	Q argument is a body, face, edge or vertex.
  2. ROTATE(Q,R);	R argument is a transformation array with
  3. DILATE(Q,R);	       respect to world coordinates.
  4. REFLECT(Q,R);

	The four Euclidean transformations are translation, rotation,
reflection  and  dilation; and as first mentioned in Klein's Erlangen
Program, 1872, these four primitives form a group. The primitives may
be  applied  to  bodies,  faces, edges or vertices in order to change
vertex world locii. Thus a body is the set of vertices in its  vertex
ring,  a face is the set of vertices on its perimeter, an edge is the
two vertices which are its ends, and a single vertex is  itself;  but
there  are  special  cases  having  to  do  with faces.  (In GEOMED a
special counter, negative Fcnt, is maintained in wire sweep faces  in
order to make solids of rotation). The second argument R is a pointer
to a transformation array in world coordinates of four rows and three
columns:
			XWC, YWC, ZWC
			IX,  IY,  IZ
			JX,  JY,  JZ
			KX,  KY,  KZ

For translation, only the XWC, YWC and ZWC are involved and  all  the
vertices are translated in the obvious fashion:

	X ← X + XWC;	Y ← Y + YWC;	Z ← Z + ZWC;

Whereas  for  rotation  (dilation  and  reflection)   the   innermost
computation applied to each vertex is:

	X ← X + XWC;	Y ← Y + YWC;	Z ← Z + ZWC;
	XX ← IX*X + IY*Y + IZ*Z;
	YY ← JX*X + JY*Y + JZ*Z;
	ZZ ← KX*X + KY*Y + KZ*Z;
	X ← XX - XWC;	Y ← YY - YWC;	Z ← ZZ - ZWC;