/* Explanatory note for ctr807.aml
/* ctr807.aml created by Bill Beeman 12/14/94
/*
/* Eugene G. Ellis */
/* Thu Dec 15 14:49:55 MST 1994 */

The following equations are used to determine the color shades in the
ctr807.shd shadeset that is produced by ctr807.aml.


<------ L ------> <------ L ------>
-----------------------------------|
                 |                 |
    ---------    |    ---------    |
    |       |    |    |       |    |
    |<- l ->|    |    |<- l ->|    |
    |       |    |    |       |    |
    |       |<-- x -->|       |    |
    ---------    |    ---------    |
                 |                 |
-----------------------------------|              
                 |                 |
    ---------    |    ---------    |
    |       |    |    |       |    |
    |<- l ->|    |    |<- l ->|    |
    |       |    |    |       |    |
    |       |<-- x -->|       |    |
    ---------    |    ---------    |
                 |                 |
-----------------------------------|              


The side of each large cell is:			L

The side of each small cell is:			l

The distance between two small cells is:	x

	L = l + 2(x/2) = l + x


[Exponents are expressed by the exp() function; e.g., 'Z(exp(2))' means
'Z' to the power of '2', in other words, 'Z squared'.]

The area of each large cell is:			L(exp(2))

The area of each small cell is:			l(exp(2))

	L(exp(2)) = (l + x)exp(2)


The percentage (s = percent saturation) of the area within a large cell that is
filled by a small cell is:

	s = l(exp(2)) / (l + x)exp(2)


(1)	l(exp(2)) = s(l(exp(2)) + 2xl + x(exp(2)))

(2)	0 = s(x(exp(2))) + 2slx + s(l(exp(2))) - l(exp(2))

[Solve for x by determining the roots of the quadratic equation.]

(3)	x = (-2sl +- ((2sl)exp(2) - 4s(s(l(exp(2))) - l(exp(2)))exp(1/2)) / 2s

(4)	x = (-2sl +- (4((s)exp(2))((l)exp(2)) - 4((s)exp(2))((l)exp(2)) +
		4s((l)exp(2)))exp(1/2)) / 2s

(5)	x = (-2sl +- (4s((l)exp(2)))exp(1/2)) / 2s

(6)	x = (-2sl +- (2l((s)exp(1/2)))) / 2s

(7)	x = (-sl +- (l((s)exp(1/2)))) / s

[But only positive values are needed, so ...]

(8)	x = (-sl + (l((s)exp(1/2)))) / s

(9)	x = (l((s)exp(1/2)) / s) - l

This final formula is used to determine the separation between cells with
sides of 'l'.