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can be expressed in terms of meet and join by writing a = s · =-S · =-S (" and
¦=-S'"p so that (S '" p) · =-¦· = ¦· =¦'" . Then (5.15) takes the form
S = a '" p +¦(" . (5.16)
The screw S is a line if and only if the lines on the right side of this decomposition intersect.
A bivector S is a blade (or line) if and only if S '" S = 0. For if S is expressed as a linear
combination
S = »L + M (5.17)
of lines L and M, then S '" S =2»L '" M, which vanishes if and only if L and M intersect.
If S '" S = 0, then the lines L and M are said to be conjugate with respect to S.
Every line L has a unique conjugate M with respect to a given screw S. For if M is
defined in terms of S and L by (5.17), then we require
M '" M =(S-»L) '" (S - »L) =0,
32
which is satisfied only if » has the unique value
1 S '" S
» = . (5.18)
2 L '" S
A line N is said to be a null line of a bivector S if S '" N = 0. (This is not to be confused
with the more common metrical definition of a null line used in relativity.) The set of all
null lines of S is called a linear complex if S '"S = 0 or a special linear complex if S '"S =0.
In the latter case, S is a line and its null lines are those lines which intersect it. In the former
case, if L is any line intersected by N, then the conjugate of L with respect to S is also
intersected by N. This follows immediately from (5.17); thus, M '"N = S'"N -»L'"N =0.
It follows that any line intersecting a pair of lines which are conjugate with respect to S is a
null line of S. On the other hand, it is easily proved that if every null line which intersects
the line L also intersects a line M, then L and M are conjugate.
Two screws S, S are said to be reciprocal if S '" S = 0. In this case, if L is a null
line of S , then its conjugate M with respect to S is also a null line of S . For by (5.17),
S '" M = S '" (S - »L) =0.
The set of all linear combinations of two independent bivectors S = S is called a pencil
of bivectors, or, in geometric parlance, a pencil of linear complexes. A pencil of bivectors
contains two, one, or no lines according as
(S + »S ) '" (S + »S ) =S'"S+»(S'"S +S '"S) +»2S '"S =0 (5.19)
has two, one, or no real roots for ». With respect to the pencil of linear complexes, these
two uniquely determined lines, if they exist, are called directrices and represent the special
linear complexes of the pencil.
A line which is a null line for every linear complex of a pencil is a null line for each of
its special linear complexes; hence it intersects both directrices (if they exist). Conversely,
every line which intersects both directrices is a common null line of all linear complexes of
the pencil. From this we conclude that the directrices of a pencil of linear complexes are
conjugate lines with respect to every complex of the pencil.
The concept of duality enables us to relate properties of linear complexes to properties
of quadrics. If S is a bivector distinct from its dual S = SI-1 then S and S define a pencil
of linear complexes with directrices determined by the equation
(S + »S) '" (S + »S) =S'"S+2»(S· S) +»2S'" S
=S'"S+2»I(S · S) +»2I2S'"S
S · S
=S'"S 1+2»I + »2I2 =0. (5.20)
S '" S
The unit pseudoscalar satisfies I2 = ±1, depending on the signature. Therefore, the solu-
tions of (5.20) are of the form ±»0, +1/»0, and the two directrices D1, D2 are
Dl = S + »0S, D2 = ±»0S + S.
It follows that D1 = D2 and D2 = ±D1; in other words, D1 and D2 are polar lines with
respect to the quadric determined by the nondegenerate inner product. D1 and D2 exist
33
and are distinct if and only if (S · S)2 > I2(S '" S)2; they are coincident if and only if
(S · S)2 = I2(S '" S)2.
We are now able to prove a theorem which plays a significant role in the kinematics
of projective metric spaces. (In fact, for Euclidean and hyperbolic spaces it implies the
existence and uniqueness of a screw axis for any given moment of a motion.) The theorem
says that in any linear complex S with (S · S)2 >I2(S '" S)2 there are exactly two lines
which are conjugate with respect to S as well as polar with respect to a given quadric
(determined by the inner product). In fact, these two lines are the directrices of the pencil
determined by S and S (see above). If there were another such pairs of lines, say E and E,
which are not directrices of the pencil, then we would have S = E +µE, hence S = E ±µE.
Accordingly, E and E would be directrices of the pencil, in contradiction to the assumption.
6. Discussion
Projective geometry can be developed using synthetic or analytic methods. The two meth-
ods seem so different that synthetic and analytic geometries have developed in parallel into
what some regard as independent branches of mathematics. The analytic method turned
out to be the more powerful of the two. While synthetic geometry has stagnated in the
twentieth century, the analytic method has flowered through linear algebra to diverse appli-
cations throughout mathematics. Synthetic geometry has survived, nevertheless, because
it has decided advantages. Unfortunately, those advantages are not readily available to
mathematicians schooled only in the analytic method. We think that the gap between
synthetic and analytic approaches should be regarded as a deficiency in the design of math- [ Pobierz całość w formacie PDF ]
zanotowane.pl doc.pisz.pl pdf.pisz.pl ocenkijessi.opx.pl
can be expressed in terms of meet and join by writing a = s · =-S · =-S (" and
¦=-S'"p so that (S '" p) · =-¦· = ¦· =¦'" . Then (5.15) takes the form
S = a '" p +¦(" . (5.16)
The screw S is a line if and only if the lines on the right side of this decomposition intersect.
A bivector S is a blade (or line) if and only if S '" S = 0. For if S is expressed as a linear
combination
S = »L + M (5.17)
of lines L and M, then S '" S =2»L '" M, which vanishes if and only if L and M intersect.
If S '" S = 0, then the lines L and M are said to be conjugate with respect to S.
Every line L has a unique conjugate M with respect to a given screw S. For if M is
defined in terms of S and L by (5.17), then we require
M '" M =(S-»L) '" (S - »L) =0,
32
which is satisfied only if » has the unique value
1 S '" S
» = . (5.18)
2 L '" S
A line N is said to be a null line of a bivector S if S '" N = 0. (This is not to be confused
with the more common metrical definition of a null line used in relativity.) The set of all
null lines of S is called a linear complex if S '"S = 0 or a special linear complex if S '"S =0.
In the latter case, S is a line and its null lines are those lines which intersect it. In the former
case, if L is any line intersected by N, then the conjugate of L with respect to S is also
intersected by N. This follows immediately from (5.17); thus, M '"N = S'"N -»L'"N =0.
It follows that any line intersecting a pair of lines which are conjugate with respect to S is a
null line of S. On the other hand, it is easily proved that if every null line which intersects
the line L also intersects a line M, then L and M are conjugate.
Two screws S, S are said to be reciprocal if S '" S = 0. In this case, if L is a null
line of S , then its conjugate M with respect to S is also a null line of S . For by (5.17),
S '" M = S '" (S - »L) =0.
The set of all linear combinations of two independent bivectors S = S is called a pencil
of bivectors, or, in geometric parlance, a pencil of linear complexes. A pencil of bivectors
contains two, one, or no lines according as
(S + »S ) '" (S + »S ) =S'"S+»(S'"S +S '"S) +»2S '"S =0 (5.19)
has two, one, or no real roots for ». With respect to the pencil of linear complexes, these
two uniquely determined lines, if they exist, are called directrices and represent the special
linear complexes of the pencil.
A line which is a null line for every linear complex of a pencil is a null line for each of
its special linear complexes; hence it intersects both directrices (if they exist). Conversely,
every line which intersects both directrices is a common null line of all linear complexes of
the pencil. From this we conclude that the directrices of a pencil of linear complexes are
conjugate lines with respect to every complex of the pencil.
The concept of duality enables us to relate properties of linear complexes to properties
of quadrics. If S is a bivector distinct from its dual S = SI-1 then S and S define a pencil
of linear complexes with directrices determined by the equation
(S + »S) '" (S + »S) =S'"S+2»(S· S) +»2S'" S
=S'"S+2»I(S · S) +»2I2S'"S
S · S
=S'"S 1+2»I + »2I2 =0. (5.20)
S '" S
The unit pseudoscalar satisfies I2 = ±1, depending on the signature. Therefore, the solu-
tions of (5.20) are of the form ±»0, +1/»0, and the two directrices D1, D2 are
Dl = S + »0S, D2 = ±»0S + S.
It follows that D1 = D2 and D2 = ±D1; in other words, D1 and D2 are polar lines with
respect to the quadric determined by the nondegenerate inner product. D1 and D2 exist
33
and are distinct if and only if (S · S)2 > I2(S '" S)2; they are coincident if and only if
(S · S)2 = I2(S '" S)2.
We are now able to prove a theorem which plays a significant role in the kinematics
of projective metric spaces. (In fact, for Euclidean and hyperbolic spaces it implies the
existence and uniqueness of a screw axis for any given moment of a motion.) The theorem
says that in any linear complex S with (S · S)2 >I2(S '" S)2 there are exactly two lines
which are conjugate with respect to S as well as polar with respect to a given quadric
(determined by the inner product). In fact, these two lines are the directrices of the pencil
determined by S and S (see above). If there were another such pairs of lines, say E and E,
which are not directrices of the pencil, then we would have S = E +µE, hence S = E ±µE.
Accordingly, E and E would be directrices of the pencil, in contradiction to the assumption.
6. Discussion
Projective geometry can be developed using synthetic or analytic methods. The two meth-
ods seem so different that synthetic and analytic geometries have developed in parallel into
what some regard as independent branches of mathematics. The analytic method turned
out to be the more powerful of the two. While synthetic geometry has stagnated in the
twentieth century, the analytic method has flowered through linear algebra to diverse appli-
cations throughout mathematics. Synthetic geometry has survived, nevertheless, because
it has decided advantages. Unfortunately, those advantages are not readily available to
mathematicians schooled only in the analytic method. We think that the gap between
synthetic and analytic approaches should be regarded as a deficiency in the design of math- [ Pobierz całość w formacie PDF ]