Relation mathematics: Encyclopedia II - Relation mathematics - Example: coplanarity

Relation mathematics - Example: coplanarity

For lines L in three-dimensional space, there is a ternary relation picking out the triples of lines that are coplanar. This does not reduce to the binary symmetric relation of coplanarity of pairs of lines.

In other words, writing P(L, M, N) when the lines L, M, and N lie in a plane, and Q(L, M) for the binary relation, it is not true that Q(L, M), Q(M, N) and Q(N, L) together imply P(L, M, N); although the converse is certainly true (any pair out of three coplanar lines is coplanar, a fortiori). There are two geometrical reasons for this.

In one case, for example taking the x-axis, y-axis and z-axis, the three lines are concurrrent, i.e. intersect at a single point. In another case, L, M, and N can be three edges of an infinite triangular prism.

What is true is that if each pair of lines intersects, and the points of intersection are distinct, then pairwise coplanarity implies coplanarity of the triple.

Adapted from the Wikipedia article "Example: coplanarity", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki