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where » =(Á +Ã)/2 and µ =(Á -Ã)/2.
Three linearly independent quadric loci have, in general, eight points in common and
every quadric that passes through seven of these points is a linear combination of the given
quadrics and, hence, contains the eighth point (see [17, p. 340]). These points are therefore
symmetrically related and form a set of eight associated points. The problem of finding
an explicit construction for the eighth associated point from seven arbitrary points was a
rather famous challenge in the 19th century, and it has been solved in many different ways.
Within geometric algebra, the problem is to find an algebraic criterion which determines
whether or not eight given points form a set of associated points. A criterion of this kind
was first determined by Turnbull [18].
According to (5.7), a quadric containing the lines L, M, N is determined by the equation
(((x '" L) (" M) '" N) '" x =0 .
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Let M = a '" b, then, by (5.2),
(x '" L) (" (a '" b) =[xLa]b - [xLb]a,
so the equation for the quadric can be cast in the form
[Lax][Nbx] =[Lbx][Nax] . (5.11)
When the condition that the entire line a '" b lies in the quadric is relaxed, Equation (5.11)
generalizes to
[Lax][Nbx] =»[Lbx][Nax]. (5.12)
Assuming that this quadric passes through a point c, » is given by
[Lac][Nbc]
» = ,
[Lbc][Nac]
and (5.12) becomes
[Lbc][Lax][Nac][Nbx] - [Lac][Lbx][Nbc][Nax] =0. (5.13)
This is the equation for a quadric passing through points a, b, c and lines L, N. Writing
L = a '" b , N = c '" d and setting x = d, we have
[a b bc][a b ad][c d ac][c d bd] - [a b ac][a b bd][c d bc][c d ad] =0 . (5.14)
This is the desired relation which must be satisfied if the eight points a, b, c, d, a , b , c , d
are to form a set of associated points.
5.6. LINE GEOMETRY
Classical line geometry is concerned with families of lines, especially in P3. To show how
the subject can be developed in the language of geometric algebra, we discuss some corre-
sponding families of 2-blades in G4. We limit our attention to the case of real scalars.
The development of line geometry in the 19th century was greatly influenced by the fact
that the action of a single force on a rigid body can be represented geometrically by a
unique oriented line (along which the force acts) and a positive scalar (for the magnitude
of the force). Each such force (action) can be represented by a unique 2-blade in G4. Note
that this differs from the representation of a projective line by a 2-blade only by fixing the
scale of the blade. Thus, there is a one parameter family of forces associated with each line.
The superposition of forces acting on a rigid body determines a unique resultant force, and
this is expressed quantitatively by the addition of the corresponding 2-blades. This idea
is abstracted and generalized in line geometry as a geometrical rule for combining lines.
It requires, however, the introduction of new geometric concepts, for the bivector sum of
2-blades in G4 is not necessarily reducible to a 2-blade. One geometrical name for such a
bivector is screw, since it can be interpreted kinematically as a infinitesimal screw motion
along a line (sometimes called a twist). A line can be regarded as a degenerate screw,
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represented by a bivector with only a single blade. The mechanical analogue of a screw
is sometimes called a wrench. A wrench is a bivector in G4 representing the action of a
system of forces on a rigid body. It represents a single resultant force only if it degenerates
to a 2-blade. In general, a wrench represents a resultant force and couple (see Section 6.1
of [9]).
Generalized line geometry has been treated extensively in terms of Grassmann algebra
by Whitehead [20]. He called it the non-metrical theory of forces. His use of the obsolete
term force is unfortunate, because it distracts from the purely mathematical character of
his treatment. His forces are just general bivectors in G4, and his theory is concerned with
systems of bivectors. It is nonmetrical in its study of bivector properties which do not
depend on signature. Our subject here is the same. We hope that our discussion will renew
interest in Whitehead s important work, which has not received the attention it deserves.
The ideas have applications far beyond classical mechanics, though the applications to
mechanics are still of interest.
Any sum of coplanar lines is a line in the same plane. This relation is well-defined as
bivector addition as long as the arbitrary scale factor is the same for all lines. In general, a
linear combination of lines which are not coplanar is a screw (or bivector) which cannot be
reduced to a line. However, every screw can be expressed as a sum of two lines. In fact, any
screw S can be resolved into the sum of a line passing through a given point p and a line
Iying in a given plane which does not contain p. This can be proved from the identity
(S '" p) · =S(p · ) - (S · ) '" p,
an application of (2.16). Adjusting the magnitude of the vector = I-1 so that p '" =
(p · )I = I, we have
S =(S· ) '" p +(S'"p) · . (5.15)
The first term on the right is obviously a 2-blade because S · is a vector, and the second
term is a 2-blade because the trivector S '" p is necessarily a blade in G4 even if S is not.
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