INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
May 1951
(3) Burrell, G.A , , and Robertson, I. W., Ibid., 37,2188 (1915). (4)Farrington, P. S., and Sage, B. H., IND.ENQ.CHEM.,41, 1734 (1949). (5) Franois, A. W., and Robbins, G. W., Ibid., 25,4339 (1933). (6) Hanson, G.H., Trans. Am. Inst. Chem. Engrs., 42, 959 (1946). (7) Kilpatrick, J. E., and Pitzer, K. S., J. Research N&. B ~ Standards, 37,163 (1946). (8) Kistiakowsky, J . Am. Chem. Soc., 57,876 (1935). (9)v'L Newittf D' and Ruhemant Roy' (London), 178,507 (1941). 43# IOg8 (10) Maass$ 0 . r and mrright? c* J * Am. Chem* (1921). H*l
H.l
1193
(11) Marchman, H.,Prengle, H. W., and Motard, R. L., IXD.E m . CHEM.,41,2658 (1949). (12) National Bureau of Standards, Letter Circ. LC-736 (Nov. 23, 1943). and Maass, O.,Can. J . Research, 14B,96 (1936). (13) Pall, T.M., ~ (14) . Powell, T. M., and Giaque, W.F., J. Am. Chem. SOC.,61,2366 (1939). (16) Seibert, F. M., and Burrell, G. A.,Ibid., 37,2683 (1915). (16) Trautz, G.,and Winkler, C. A., J . prakt. Chem., 104,37 (1922). (17) Vaughan, W. E.,and Graves, N. R., IND.
[email protected].,32, 1252 (1940). (18) Winkler, C. A., and Maass, 0, Can. J . Research, 9,610 (1933). RECEIVED July 1, 1950.
Construction of Nornographs with Hyperbolic Coordinates GENERAL HYPERBOLIC COORDINATES WALTER HERBERT BURROWS Georgia Institute of Technology, Atlanta, Ga. T h e original investigations leading to the present article were directed toward simplification of the method of constructing nomographs for polynomials through use of a chart based on the hyperbolic scale. Later it was decided to extend this method to other types of nomographs by generalization of the original methods. It soon became apparent that, instead of using the type of chart developed for polynomials, it was desirable to modify the coordinate system, using the determinant form of defining equations. The first method employed a coordinate system modified only on one axis. The present paper involves extension of this modification to both coordinates. The aspects of nomographic construction which are made easier by use of hyperbolic coordinates are (1) selection of the nomographic form best suited to a given formula, (2) adjustment of scale moduli and positions best to accommodate the ranges of the variables to be represented, and (3) projection of irregular forms into rectangular form. *
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bolically into segments whose lengths are x = p,/(pz r.) and y = py/(py T ~ ) the , lengths of the axes being taken as unity, although these lengths may be different for the two axes. Ordinates are located, as previously, by ordinate lines passing through the terminus of the X axis and the points p , on the Y axis, where the value of py is the value of the ordinate (Figure 1). Abscissas are located similarly by abscissa lines passing through the terminus of the Y axis and the points p , on the X axis, the value of p , being the value of the abscissa. T h e values of r z and rv are independent of one another. There is an auxiliary axis, the 2 axis [in a previous paper ( 2 ) , the Z axis], for the location of points with infinite abscissas and/or ordinates. This axis passes through the termini of the X and Y axes, having its origin a t its 2 intercept and its terminus
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REVIOUS papers ( 2 , 3) have described the construction of nomographs by the use of a form of coordinates termed hyperbolic. The form of coordinates now properly called semihyperbolic, dealt with in those articles, consists of one principal axis (the X axis) subdivided hyperbolically and another principal axis (the Y axis) subdivided linearly, as in Cartesian coordinates. A more general form of plane hyperbolic coordinates, consisting of two principal axes both of which are subdivided hyperbolically, will now be described. The properties of this coordinate system, derived b y an analysis similar to that employed for semihyperbolic coordinates, will be freely quoted here without derivation. There are several fields of nomography in which application of this geometrical device simplifies constructions, two of which are those involving first the V-type nomograph, and secondly, the problem of fitting nomographs to a rectangle. HYPERBOLIC PLANE COORDINATES
LOCATION OF POINTS.Both the X and Y axes of the hyperbolic coordinate Bystem are finite in length and divided hyper-
Figure 1. Method of Locating a Point Whose Coordinates Are p. and p ,
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at its y intercept. It is thus finite in length, that length also being taken as unity. This axis is subdivided into segments such that y = p; / ( p ; r; ).- I n order to simplify relations involving the coordinates on the 2 axis and the coordinates of other points in the plane, it is desirable to define the hyperbolic constant of this auk, T ; , in terms of the hyperbolic constants of the principal axes, thus r; = r , lrZ.
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\" I
V
I
Figure 2.
/
are the image distance, object distanze, ~ i n diocal length, r+ spectively, of the lens. I n Figure 3 this nomograph is shown constructed with hyperbolic coordinates employing the values rz = 0.5 and r g = 5.0 to emphasize ranges (in feet) encountered in certain fields of cornmercial photography. The same nomograph is shown in Figure 4 constructed with Cartesian coordinates, using the same defining equation, but having the modulus of the Y axis ten t.imes as great as that of the X axis. Figure 3 has the rather obvious advanhge of greater detail in the lower range, while the flexihility of thc hyperbolic coordinate system makes it possible to ernphnsiac any port,ion of the ranges involved without increasing the length of the scales. A similar effect may be achieved wiOh Cartesian coordinates only by extensive manipulation of the defining equation. 011 the other hand, a nomograph of the form of Figure 4 is quite easily constructed with hyperbolic coordinates if the tiefining equation is written in the form
\
\\
Schematic Nomograph with Straight Index Line
APPLICATIONS TO
Yol. 43, No. 5
I\;OXOGRAPHY. The defining equation of
any nomograph having a straight index line is
the pa's being abscissas and the p,'s ordinates of the u, v , and w scales of the nomograph, as indicated by the letter in parentheses in Equation 1 (Figure 2). Equation 1 is a general expression for points lying in the plane. Special cases appear for points (or, in nomographv, wales) lying on the axes, rows in the determinant becoming 10 p , 11 for points on the X axis, [ p , 0 11 for points on the Y auis, and lp; 1 01 for points on the 2 axis. The value of hyperbolic coordinates in the construction of nomographs lies in the fact that the determinant of Equation 1 is independent of the values of r z and r,, so that the values of these constants may be rhanged, thus changing the form of the nomograph, without any change taking place in the defining equation. The values of r z and r Y , being simple numbers, may be easily changed, whereas changes in the defining equation often become quite tedious because of the complicated nature of the values of the scale coordinate$.
thc t,hree wales being lincar and having their origins a t the trrrnini of the Y and 2 axes. The so scale lies on the Y axis, arid the s.. scale on the 2 axis. For either form of the defining equation tht. s j scale is easily located, as it always passes through the point p = 1 on the t,hird axis, regardless of t,he vttlues of the hyperbolic constants. FITTIXG THE NOMOGRAPH TO h RECTAhGLE
One of the most interesting problems i u nomography is t h a t of modifying the original form of a nomograph so that, it3 two exterior scales terminate a t the corners of a rectangle, that form of the nomograph having the advant,ages of greater accuracy, increased legibility, and more pleasing appearance. The treatment of this problem for cases in which the two exterior scales are already parallel was the theme of a previous paper (2). In the treat,ment of the general case in which the termini of the exterior scales fall on any four points, the hyperbolic coordinate syst,em provides an interesting mode of approach. There are several phases t o the problem: (1) hdjustment of the hyperbolic const,ants of both vertical and horizontal corn-
THE V-TYPE XOMOGRAPH
If the substitutions, p y ( u ) = si, p,(u) = 0, pY(v) = p z ( u ) = = so, are made in Equation 1, the resulting determinant
I, p y ( w ) = 0, and p,(w)
may be taken as the defining equation of a nomograph for the l/sL,in which si, .so, nnd s; familiar lens formula, l/s/ = l/s,
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Figure 3. Nomograph for Lens Formula Constructed with Hyperbolic Coordinates to Emphasize Certain Ranges
May 1951
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and upper termini uz and v2, respectively, the numerical values of pz and p y corresponding to these termini may be taken as Pz(U1) = Sl, p d u z ) = 92, pz(v1) = 8 3 , PZ(VZ) = 84, P , ( U l ) = tl, ~ U ( U Z ) = t z , p,(vi) = ta, p,(vz) = ta. These values are employed to evaluate rz and ry by the following steps:
/
6.
P8P
to
Then
&-
and
THE RETAlNING WALL FORMULA
By way of illustration, this method may be applied to the retaining wall formula, quoted by Allcock and Jones (1):
Figure 4. Nomograph for Lens Formula Constructed with Cartesian Coordinates
ponents of the reference frame may convert any quadrilateral to a parallelogram; ( 2 ) a change in the angle which those components make with one another-e.g., the angle between the X and Y axes of oblique coordinates-may convert the parallelogram into a rectangle; (3) adjustment of the moduli of the axes may provide the desired length-to-width ratio for the rectangle; and (4)rotation of the reference frame may bring the sides of the rectangle into vertical and horizontal positions. The requirements for steps 2, 3, and 4 are obvious, once step 1 has been accomplished, as may be seen in the example following. In a number of cases, step 1 may be achieved simply by trial-and-error adjustment of the values of rz and r y , the resulting nomograph being approximately, if not exactly, a rectangle. For a treatment of step 1 which is a t once accurate and direct, it is necessary to resort to analytical methods. In the classical method, using Cartesian coordinates (S),all four steps are usually merged into one, the solution to the entire problem being accomplished by the introduction of additional variables into the defining equation, the values of those variables then being determined by the solution of a series of simultaneous equations involving the desired dimensions of the nomograph. By this method one arrives a t a new defining equation which gives the desired results, but which is often very tedious to employ. The method which follows avoids any change in the defining equation and eliminates the solution of simultaneous equations. By this method one proceeds from given coefficients directly t o the required values of rz and ry, those values being such as t o give an exact rectangle. An outline of the derivation of this method follows its discussion and application, and may be omitted f o i most practical purposes. If the defining equation for a nomograph relating the variables u, v , and w is
and u and v are the exterior scales with lower termini
UI
and
VI
This may be expressed in determinant form as
h
Y
3
x
Figure 5. Nomograph for Retaining Wall Formula before Adjustment of Moduli and Positions of Scales
For both p and L, the lower limit is 0.50 and the upper limit, 1.00. The corresponding limits of h are 0.777 and 1.00. From a preliminary plot (Figure 5) of the corresponding limits of the scale functions in the above defining equation, it appears that p and L are the exterior scales. The p scale lies between the limits (*/e, 2/3) and (1, l), while the L scale lies between the limits ( - a / 3 , I/$) and (0, 0). If u is then taken t o represent the
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INDUSTRIAL AND ENGINEERING CHEMISTRY DERIVATION OF THE METHOD
(h)
It may be shown that, like the X axis, every line in the hyperbolic plane may be described as having a hyperbolic subdivision such that the distance from the Y axis of any point on that line is given by 2‘ = p z / ( p z T ~ ’ ) where , p , is the hyperbolic abscissa r g ) . In of the point and rr‘ has the value T,’ = ( p ‘ # rp)/(p’; this expression p’, and p’- are the y and intercepts of the line, while r y and r; are the hyperbolic constants of the Y and % . axes, respectively. The length of the line segment bounded by the Y and axes is taken as unit length for any line in the plane, It may also be shown that the necessary and sufficient condition that two lines (a)and ( b ) be parallel is that r,,(a) = r,(b)Le., that their hyperbolic constants be equal. I t is this principle which is applied in determining the values of r y and T (and, in turn, which must be employed to render the opposite sides of a quadrilateral parallel. , The above value of rz is given in terms of its y and Z intercepts. These, however, are known if any two points on the line are known. It is a simple matter to substitute for these intercepts ’ values in terms of the coordinates in the expression for T ~ their of the four given points. There is a pair of such substitutions for each side of the quadrilateral, resulting in a separate expres’ opposite sides of sion for rr’ for each side. The values of T ~ for the quadrilateral are then equated, and the equations thus obtained are solved for r y and r;; r, is given by the expression rz = ru/r;. It is the resulting expressions for rz and r y which are evaluated by the above steps involving the a’s, b’s, and c’s.
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Vol. 43, No. 5
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z
.60
Figure 6. Nomograph for Retaining Wall Formula with Moduli and Positions of Scales Adjusted to Rectangular Form
present function, p , and v to represent the present function, L, the values of these coordinates are inserted in the above formulas to determine the values of the a’s, b’s, and c’s, as follows: al= 0, a2 = - 2 , bl = l / 6 , bz = 25/6, a3 = 0 , a4 = 1/ 3 , b3 = 3/4, bq = - 1 /IZ, cz = -I/a, and c4 = l / 4 . Then r y = 1/2 and rz = - 4 / 7 . The negative sign of r z indicates merely that positive values of p are to be located in the negative region of the scale, and vice T ) 3 -p/( -p - T ) . The nomograph is versa, since p / ( p constructed using these values of T~ and r y . From a preliminary plot it is apparent that an angle of 30” between the X and Y axes will convert the parallelogram to a rectangle and that the length-to-width ratio desired may be achieved by setting the length (or modulus) of the X axis approximately twice as great as that of the Y axis. The nomograph thus constructed and rotated to place the scales in vertical position is shown in Figure 6.
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LITERATURE CITED
(1) Allcock, H. J., and Jones, J. R., “The Nomogram,” p. 107 ff. London, Sir Isaac Pitman & Sons, 1932, (2) Burrows, W. H., IND.ENQ.CHEM.,38, 472 (1946). (3) Ibid., 43, 158 (1951).
R E c n w E D September
25, 1960.
LCARVONE FROM d-LIMONENE CARL BORDENCA AND RUFUS I
