文档介绍：Chapter 3 24
undamentals
Some fundamental properties and initial concepts related to line-to-line functions will
be developed in this chapter. These will provide the foundation for introducing
concepts concerning direction later.
parisons
Consider the point-to-line function that was previously discussed. By virtue of the
S
A
T
P
Q
igure 5: Distance from a Point to a Line
continuity axiom the distance d(A,Q) will be some differentiable function of d(P,Q)
for Q to the right of P. Does the limit of the derivative of this function as d(P,Q)
tends toward zero have some geometrical significance characteristic of the
intersection at P?
Part of the answer to this question can be obtained by using the geodesic hypothesis
to relate the derivatives of adjacent point-to-line functions. Compare a pair of points,
S
A
B
T
P
Q
igure 6: Collinear Points on One Side of a Line
Chapter 3 25
A and B, on the same side of a line which are collinear with a point, P, on that line.
As in the figure above take B between A and P. Both the distances from these points
to a point, Q , moving about on the line in the vicinity to the right of P will, as
before, be differentiable functions of d(P,Q). In this situation: (1) the triangle
inequality implies that for the small triangle dAQ(,)£+ dAB (,) dBQ (,) and (2) the
following lemma can be used.
Lemma. If f and g are two real valued, n-times ( n ³ 2 ) differentiable functions such
that fg()00= () and otherwise fx()< gx () then, in particular,
(i) fg¢()0= ¢ () 0 and (ii) fg¢¢()00£ ¢¢ ()
and, in general, (iii) either fg()mm()00= () () for all 0 ££mn or there is an even
kn£ such that fg()mm()00= () () for 0 £<mk and fg()kk()00< () () strictly.
Proof. If hx()=- gx () f () x then, according to the hypothesis, this function has an
absolute minimum at zero. Consequently* , h¢()0= 0 and h¢¢()0³ 0 , which are the
particular conclusions, and the general conclusions follow similarly.
Take fdAQdAP=