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(0)
60? 1000" tSOtf 200rf Fig: 7.
A logarithmic anamorphosis as illustrated in (c) is so frequently resorted to in practice that paper already ruled with a logarithmic network can be obtained commercially and is largely employed.
6. Graphic Representation of a Three- Variable Formula in Parallel Coordinates. The preceding sections have dealt with Intersec- tion Nomograms in which the answer is read from the intersection of lines in a point. For certain types of formulae, however, a repre- sentation is possible in which the three variables are arranged along three scales, and the answer is read by the alignment of points on these scales. Such an arrangement, an " Alignment Nomogram," is possible only when a diagram for the formula can be constructed, in cartesian coordinates, which consist of three systems of straight lines.
Defining the three systems of straight lines in cartesian coordi- nates by
corresponding to the three variables z lt %, z 3 , we arrive at an equa- tion for the formula which is most conveniently expressed in deter- minant notation
/i gi h.
/2 #2 ! /3 g3 h;
= 0-
-(I).
For further investigation it is necessary to introduce the idea of Parallel Coordinates referred to two parallel axes, so that a point is represented by an equation of the first degree.
These coordinates are denned as follows: If
a straight line MN (fig. 8) cuts two parallel axes AM, Bv (A and B being the origins of the axes) in M and N, the coordinates of the straight line are
= AM, t/ = BN.
Any equation of the first degree
au-\-bv-\-c=o
will represent a point, and to determine this point it is sufficient to know two solutions of the equation, and take the intersection of the straight lines resulting from these two solutions.
Putting v=o, 11=
u = 0, v -r
Along the axes A, Bv (fig. 9) take
AQ=--,
a
b
The intersection of the straight lines AR, BQ in P will then give the point required, the point
au+bv-}-c = o
This correspondence of points to straight lines and vice versa, according as to whether cartesian or parallel coordinates are em- ployed for the geometrical interpretation, is an example of the Principle of Duality. As an alternative to a diagram composed of
straight lines there is a correlative diagram composed of points, and if three straight lines intersect in a point in the first diagram, the three points in the second will lie on a straight line.
Effecting such a dualistic transformation the three systems of straight lines
will now be represented by three systems of points
uf,+vg,+h,=o
Fig: 10.
forming three scales arranged along a straight line or a curve,
according as to whether the straight lines of the correlative system
meet in a point or not, 1 and when three points are taken on these
scales whose graduations correspond to three values of z\, z 2 , z 3
satisfying (i), the three points will lie on a straight line, since the
correlative straight lines meet in a point.
Hence to use such a diagram (shown schematically in fig. 10) we join any two values of two of the variables on their re- spective scales by a straight line, and the point of intersection of this straight line with the third scale gives the corresponding value of the third variable.
It will not be necessary actually to draw the straight line on the diagram; a piece of thread stretched across it will give the align- ment, or a strip of transparent celluloid, having a straight line engraved down the centre, may be employed for the same purpose.
Given a diagram consisting of straight lines only, representing a three-variable formula in cartesian coordinates, the correlative diagram representing the same formula in parallel coordinates can be constructed geometrically without knowing the analytical expression of the formula represented.
Let D (fig. n) be a straight line of the left-hand diagram. Take a point M on this straight line whose cartesian coordi- nates are OH, OK.
The correlative straight line H'K' will be one whose paral- lel coordinates are AH' = OH, BK' = OK.
Taking in this way the cor- relative straight lines to any two points on the straight line D, we get by their intersec- tion the point P, correlative to the straight line D.
Thus we might take BX', AY', the correlatives of X and Y, the points where the straight line D cuts the axes Ox, Oy, making
AX'=OX, BY'=OY.
Proceeding in this way we can replace all the straight lines of the intersection diagram by points.
As an example we have taken the In- tersection Nomogram, fig. 6, and con- structed from it an Alignment Nomogram, fig. 12.
Suppose, for instance, we want to know the value of R for V = 7O m./h., D = 1,100 ft. All that it is necessary to do in fig. 12 is to join 70 on the (V) scale to i.lpoon the (D) scale. This straight line will be found to cut the (R) scale at 15 % (see transverse line in fig. 12), the required value of R.
Comparing the two figures the ad- vantages of the Alignment Nomogram will be evident. The disadvantages re- ferred to in 4 have disappeared, for there is no tracing back along a line to read its graduation, and any interpolation by eye is only neces- sary on simple graduated scales.
Proceeding to the direct construction of Alignment Nomograms, without the preliminary construction of an Intersection Nomo- gram, certain types will now be considered which are particular cases of the general equation (i).
Type A Nomograms with Three Parallel Rectilinear Scales. If the formula to be represented can be put in the form
/l+/ 2 +/3=0 -(2)
the three systems of points (zi), (z 2 ), (z 3 ) can be arranged on three parallel straight lines.
For the systems (zi), (z 2 ) we take the functional scales
2000-
f\f. 12.
(3) -(4)
along the two parallel axes AM, Bv (fig. 13).
Eliminating /i, / 2 between (2), (3) and (4) gives us for (z s )
(5).
It is now convenient to revert to cartesian coordinates, taking as origin O, the midpoint of AB, the axis of x along AB, the axis of y parallel to AM or Bv (see fig. 9). Also let OB be denoted by X.
With these axes (5) will denote the system of points,
._. M1-M2
Mi-rW
y ^MlM2/3
1 Parallel straight lines of course fulfil this condition and lead to a rectilinear scale as they have a common point at infinity.
