ABSTRACT
Consider the straight line drawn through two points P1 at ( )x y1 1, and P2 at ( )x y2 2, in Figure 6.1. The slope of the line is de—ned as the ratio of the change in the y coordinates to the change in the x coordinates, for points on the line. Designating the slope by β1, we have the following expression:
= =
−
−
=
rise run
y y x x
y x
Note the calculation of the slope is independent of the actual location of the points on the line, but the highest numerical precision will be obtained by selecting the two points as far apart as possible. We can write the above equation in the form y x= β1 , which indicates that the change in y is proportional to the change in x, and the slope β1 is the proportionality factor. This property is called a linear relation. Parallel lines have equal slopes. The signi—cance of parallel lines in modeling acceler-
ated stresses for several cells will be discussed in Chapter 8. Consider the linear expression for any two points ( )x y1 1, and ( )x y2 2, on the line
y y x x2 1 1 2 1− = −β ( ). Fix the point ( )x y1 1, . For any ( )x y, on the line, y y x x− = −1 1 1β ( ) or y x y x= + −β β1 1 1 1( ). Since y x1 1 1− β is —xed, we can set this quantity equal to a constant β0 and write y x= +β β1 0. β0 is called the intercept because at x = 0, we get y = β0, indicating that β0 is the y coordinate at the point where the straight line crosses the y-axis. An alternative expression for a straight line is
Ax By C+ + = 0
where A, B, and C are constants. Any equation in this form (e.g., 3 4 2 0x y− + = ) describes a straight line.
