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in G4. By duality, a sum of 3-blades in G4 can always be reduced to a 3-blade. However, a
bivector generated by adding 2-blades in G4 cannot generally be reduced to a 2-blade. This
is the fundamental fact underlying classical line geometry.
A point p or a line A lies on a plane if and only if p("=p or A("=A, respectively.
The line of intersection of two planes = a '" b '" c and = a '" b '" c is given by their
meet (" which can be expanded according to (2.16) and (2.22):
(" = = (a '" b '" c )
=[a ]b '"c +[b ]c '"a +[c ]a '"b . (5.1)
25
Similarly, the intersection point of a plane = a '" b '" c and a line A = p '" q is given by
(" A = A = (p '" q) =[p]q -[q]p (5.2)
or, equivalently,
(" A = A = A (a '"b '"c) =[Aab]c +[Aca]b +[Abc]a. (5.3)
Hence, we have the identity
[abcp]q - [abcq]p =[pqab]c +[pqca]b +[pqbc]a. (5.4)
5.2. PAIRS OF TETRAHEDRONS
The three-dimensional analogue of Desargues theorem says that if the joins of correspond-
ing vertices of two tetrahedra concur, then the lines of intersection of corresponding face
planes are coplanar. To prove this (see [5, p. 108f.]), take a, b, c, d and a , b , c , d as the
vertices of the two tetrahedra. If p is the center of perspectivity, we can write
a = p + a, b = p + b, c = p + c, d = p + d .
Hence
a '" b '" c =(p +a) '" (p + b) '" (p + c)
= p '" (b '" c - a '" c + a '" b) +a '" b '" c.
Therefore, if a point x lies on the line
L = b '" c - a '" c + a '" b
and on the plane a '" b '" c, then x lies on the plane a '" b '" c . In other words, the line L
is the line of intersection of a '" b '" c and a '" b '" c . The remaining lines of intersection of
corresponding face planes are
c '" d - b '" d + b '" c, d '" a - c '" a + c '" d,
a '" b - d '" b + d '" a,
and they all lie in the plane
b '" c '" d - a '" c '" d + a '" b '" d - a '" b '" c.
This concludes the proof.
The following identity was first derived by Forder [5, p. 170f.] who called it the Mbius
identity:
(b '" c '" a ) (" (c '" a '" b ) (" (a '" b '" c ) (" (a '" b '" c )+
26
+(b '"c '"a) ("(c '"a '"b) ("(a '"b '"c) ("(a'"b'"c) =0. (5.5)
The first term on the left side can be calculated by using (5.1) and (5.2) successively to get
(a '" b '" c ) (" (a '" b '" c ) =[abc a ]b '" c +[abc b ]c '" a
and
(c '" a '" b ) (" (a '" b '" c ) (" (a '" b '" c )
= -[abc a ][cab c ]b +[abc b ][cab c ]a - [abc b ][cab a ]c .
Finally, since
(x '" y '" z) (" w =[xyzw] for any x, y, z, w,
we have
(b '" c '" a ) (" (c '" a '" b ) (" (a '" b '" c ) (" (a '" b '" c )
= -[abc a ][cab c ][bca b ] +[a b ca][c a bc][b c ab] .
Exchanging primed and unprimed points, the left side of this equation becomes equal to
the second term on the left side of (5.5), while the right side changes its sign. Thus, (5.5)
is proved.
The Mbius identity may be interpreted as follows: If the planes b '" c '" a , c '" a '" h ,
a '" b '" c meet at a point d lying on the plane a '" b '" c , then the intersection point,
say d , of the planes b '" c '" a, c '" a '" b, a '" b '" c lies in the plane a '" b '" c. Cyclical
permutations of the points in this theorem leads to another theorem, namely: If for two
tetrahedra {a, b, c, d} and {a , b , c , d } the vertices a, b, c, d lie in the face-planes b '"c '"d ,
c '" d '" a , d '" a '" b , a '" b '" c , respectively, and a , b , c lie in b '" c '" d, c '" d '" a,
d '" a '" b, respectively, then d lies in a '" b '" c. This is to say that each tetrahedron has its
vertices on the face-planes of the other. In other words, the two tetrahedra are inscribed
and circumscribed to each other. Such tetrahedra are called Mbius tetrahedra.
5.3. REGULI
The set of all lines joining corresponding points in two projective ranges of points on skew
lines is called a regulus. Let a, a b, b c, c be any pairs of corresponding points such that
c = a + b and c = a + b . Then a + b and a + b correspond to each other for every
" *" {- ", + "}, and their join is
a '" a + (a '" b + b '" a ) +2b '"b .
If we set A = a '" a , B = b '" b , C = c '" c , then
C =(a +b) '"(a +b ) =A+B+(a'"b +b '"a ) ,
and the join of corresponding points is given by the linear combination
(1 - )A +(2 -)B+C .
27
Therefore, the lines of a regulus, called generators, are linear combinations of any three lines
in the system. Every point p on a generator of a regulus is given by a linear combination
of the form
p = a + b + (a + b ) . (5.6)
Any line L that cuts three generators of a regulus cuts all of them. For if L intersects
A, B, C, that is, if L '" A = L '" B = L '" C = 0, then L intersects all linear combinations
of A, B, C. Such a line L is called a directrix of the regulus. If L, M, N are directrices of
a regulus, then any linear combination of them is also a directrix. Thus L, M, N generate
a regulus, called the associated regulus, of the regulus generated by A, B, C. It follows
that A, B, C are directrices of the regulus generated by L, M, N; hence each regulus is the
associated regulus of the other.
To determine the locus of points lying on the lines of a regulus generated by A, B, C, [ Pobierz całość w formacie PDF ]
zanotowane.pl doc.pisz.pl pdf.pisz.pl aikidobyd.xlx.pl
in G4. By duality, a sum of 3-blades in G4 can always be reduced to a 3-blade. However, a
bivector generated by adding 2-blades in G4 cannot generally be reduced to a 2-blade. This
is the fundamental fact underlying classical line geometry.
A point p or a line A lies on a plane if and only if p("=p or A("=A, respectively.
The line of intersection of two planes = a '" b '" c and = a '" b '" c is given by their
meet (" which can be expanded according to (2.16) and (2.22):
(" = = (a '" b '" c )
=[a ]b '"c +[b ]c '"a +[c ]a '"b . (5.1)
25
Similarly, the intersection point of a plane = a '" b '" c and a line A = p '" q is given by
(" A = A = (p '" q) =[p]q -[q]p (5.2)
or, equivalently,
(" A = A = A (a '"b '"c) =[Aab]c +[Aca]b +[Abc]a. (5.3)
Hence, we have the identity
[abcp]q - [abcq]p =[pqab]c +[pqca]b +[pqbc]a. (5.4)
5.2. PAIRS OF TETRAHEDRONS
The three-dimensional analogue of Desargues theorem says that if the joins of correspond-
ing vertices of two tetrahedra concur, then the lines of intersection of corresponding face
planes are coplanar. To prove this (see [5, p. 108f.]), take a, b, c, d and a , b , c , d as the
vertices of the two tetrahedra. If p is the center of perspectivity, we can write
a = p + a, b = p + b, c = p + c, d = p + d .
Hence
a '" b '" c =(p +a) '" (p + b) '" (p + c)
= p '" (b '" c - a '" c + a '" b) +a '" b '" c.
Therefore, if a point x lies on the line
L = b '" c - a '" c + a '" b
and on the plane a '" b '" c, then x lies on the plane a '" b '" c . In other words, the line L
is the line of intersection of a '" b '" c and a '" b '" c . The remaining lines of intersection of
corresponding face planes are
c '" d - b '" d + b '" c, d '" a - c '" a + c '" d,
a '" b - d '" b + d '" a,
and they all lie in the plane
b '" c '" d - a '" c '" d + a '" b '" d - a '" b '" c.
This concludes the proof.
The following identity was first derived by Forder [5, p. 170f.] who called it the Mbius
identity:
(b '" c '" a ) (" (c '" a '" b ) (" (a '" b '" c ) (" (a '" b '" c )+
26
+(b '"c '"a) ("(c '"a '"b) ("(a '"b '"c) ("(a'"b'"c) =0. (5.5)
The first term on the left side can be calculated by using (5.1) and (5.2) successively to get
(a '" b '" c ) (" (a '" b '" c ) =[abc a ]b '" c +[abc b ]c '" a
and
(c '" a '" b ) (" (a '" b '" c ) (" (a '" b '" c )
= -[abc a ][cab c ]b +[abc b ][cab c ]a - [abc b ][cab a ]c .
Finally, since
(x '" y '" z) (" w =[xyzw] for any x, y, z, w,
we have
(b '" c '" a ) (" (c '" a '" b ) (" (a '" b '" c ) (" (a '" b '" c )
= -[abc a ][cab c ][bca b ] +[a b ca][c a bc][b c ab] .
Exchanging primed and unprimed points, the left side of this equation becomes equal to
the second term on the left side of (5.5), while the right side changes its sign. Thus, (5.5)
is proved.
The Mbius identity may be interpreted as follows: If the planes b '" c '" a , c '" a '" h ,
a '" b '" c meet at a point d lying on the plane a '" b '" c , then the intersection point,
say d , of the planes b '" c '" a, c '" a '" b, a '" b '" c lies in the plane a '" b '" c. Cyclical
permutations of the points in this theorem leads to another theorem, namely: If for two
tetrahedra {a, b, c, d} and {a , b , c , d } the vertices a, b, c, d lie in the face-planes b '"c '"d ,
c '" d '" a , d '" a '" b , a '" b '" c , respectively, and a , b , c lie in b '" c '" d, c '" d '" a,
d '" a '" b, respectively, then d lies in a '" b '" c. This is to say that each tetrahedron has its
vertices on the face-planes of the other. In other words, the two tetrahedra are inscribed
and circumscribed to each other. Such tetrahedra are called Mbius tetrahedra.
5.3. REGULI
The set of all lines joining corresponding points in two projective ranges of points on skew
lines is called a regulus. Let a, a b, b c, c be any pairs of corresponding points such that
c = a + b and c = a + b . Then a + b and a + b correspond to each other for every
" *" {- ", + "}, and their join is
a '" a + (a '" b + b '" a ) +2b '"b .
If we set A = a '" a , B = b '" b , C = c '" c , then
C =(a +b) '"(a +b ) =A+B+(a'"b +b '"a ) ,
and the join of corresponding points is given by the linear combination
(1 - )A +(2 -)B+C .
27
Therefore, the lines of a regulus, called generators, are linear combinations of any three lines
in the system. Every point p on a generator of a regulus is given by a linear combination
of the form
p = a + b + (a + b ) . (5.6)
Any line L that cuts three generators of a regulus cuts all of them. For if L intersects
A, B, C, that is, if L '" A = L '" B = L '" C = 0, then L intersects all linear combinations
of A, B, C. Such a line L is called a directrix of the regulus. If L, M, N are directrices of
a regulus, then any linear combination of them is also a directrix. Thus L, M, N generate
a regulus, called the associated regulus, of the regulus generated by A, B, C. It follows
that A, B, C are directrices of the regulus generated by L, M, N; hence each regulus is the
associated regulus of the other.
To determine the locus of points lying on the lines of a regulus generated by A, B, C, [ Pobierz całość w formacie PDF ]