lew@ihuxr.UUCP (09/30/83)
One really neat application of projective coordinates is the unified description of force and torque by a "bound vector". If a and b are the projective coordinates of two fixed points. The antisymmetric tensor product (wedge product), a^b represents the line through a and b. The equation for points x on the line is: a^b^x = 0. Nice huh? Of course, the equation of a plane defined by a,b, and c is a^b^c^x = 0. a^b has six components: (a1*b2-a2*b1, a1*b3-a3*b1, ..., a3*b4-a4*b3) For points not at infinity, we can pick the canonical coordinates (a1,a2,a3,a4) == (1,ax,ay,az) The components of a^b then become: (bx-ax, by-ay, bz-az, ax*by-ay*bx, ax*bz-az*bx, ay*bz-az*by) This represents a "bound vector", the first three components are the components of the usual displacement, or "free vector". The second three describe the moment of the free vector about the origin. This binds the vector to lie along a certain line, namely the line a^b. This bound vector can describe a force and its moment, a couple. These form a linear space, since the resultant force and torque of a set of couples is their sum. HOWEVER, not every resultant can be produced by the action of a single force. For example, pure torques require at least two forces. If (E12,E13,E14,E23,E24,E34) are the coordinates of an arbitrary couple, the criterion that a single force can produce it is given by the PLUCKER RELATION (that's PLOO ker: the u has an umlaut): E12*E34 - E13*E24 + E14*E23 = 0 The thing I like about this example is that it shows that our usual formulation of forces is not the only one possible. It makes it clear that a force is not a physical thing but rather a mathematical abstraction. You could argue that the bound vector description is really more "natural" since equivalent couples produce equivalent results. The stuff about forces and couples is in Felix Klein's Geometry book that I mentioned before. I learned about Plucker relations from one of my favorite books of all time, "Vector Spaces of Finite Dimension", by G. C. Shephard. This is a beautifully concise little book published by University Mathematical Texts (from England). It touches on topics that the typical undergraduate linear algebra text never gets near, even though it's about one fifth the length. Lew Mammel, Jr. ihuxr!lew