Calculating the Coordinates of an Internal Point
Let us now calculate the coordinates of the internal point P 2 . The slope of the
negative characteristic evaluated at P 1 that joins the points P 1 and P 2 is
dt
dx
C À
¼
5
2R þ þ 3R À
¼
5
3 Â 5:75
ð
Þþ 3 Â À5:75
ð
Þ
¼ À0:869
and the slope of the negative characteristic evaluated at P 2 that joins the points P 1
and P 2 is
dt
dx
C À
¼
5
2R þ þ 3R À
¼
5
3 Â 5:5
ð
Þþ 3 Â À5:75
ð
Þ
¼ À0:8,
hence, the average slope of the negative characteristic is À0.835. The slope of the
positive characteristic at c that joins the points c and P 2 is
dt
dx
C þ
¼
5
3R þ þ 2R À
¼
5
3 Â 5:5
ð
Þþ 2 Â À6
ð
Þ
¼ 1:111
and the slope of the positive characteristic at P 2 that joins the points c and P 2 is
dt
dx
C þ
¼
5
3R þ þ 2R À
¼
5
3 Â 5:5
ð
Þþ 2 Â À5:75
ð
Þ
¼ 1:0,
giving 1.055 as the average slope of the positive characteristic. If (x 2 , t 2 ) is the
coordinates of P 2 then the equation for the slope of the negative characteristic at P 2
can be written as
t 2 À 9:496
x 2 À 10
¼ À0:835
and the equation for the slope of the positive characteristic at P 2 can be written as
t 2 À 9:468
x 2 À 8:521
¼ 1:055:
By solving these simultaneous equations, we find that x 2 ¼ 9.189 and
t 2 ¼ 10.173. This calculation for the location of P 2 is typical of that carried out for
the determination of the coordinates of all internal points within ℜ 4 . Table A.2 gives
the coordinates of all the points in ℜ 4 and these are plotted in Fig. A.7, while an
expanded view of this region is shown in Fig. A.8.
326
Appendix A