curves and can be converted into "light curves" only after making proper modifications in accordance with certain well-defined solid geometrical relations. It seems appropriate to give emphasis to this statement by defining the solid geometrical relations referred to, which are equally as simple as plane geometrical or trigonometrical relations. SOLID GEOMETRICAL RELATIONS Of the several space geometrical relations with which an illuminating engineer should be familiar, by far the most important, and happily the simplest, is that existing between the external area or zonal area of a sphere and its diameter or zonal width. This relation is one of direct proportion. That is to say, the external area of a zone of any chosen sphere varies directly with the width of the zone, and the total external area is that of a zone having a width equal to the diameter of the sphere. In almost all cases of application to illumination problems, one is interested in the relative values rather than the actual values of the various zonal areas and the above mentioned proportion is all that he needs to take into consideration. However, one can determine the actual as well as the relative values with extreme simplicity by means of certain plane geometrical or trigonometrical relations applied to the sphere. In Fig. 1, which represents a sphere cut along a vertical plane through the center O, the zone of infinitesimal vertical width ED, along the diameter, has an external area represented by the sloping width at C multiplied by the circumference of the zonal circle passing horizontally through C. Now the circumference of the horizontal circle through C bears to that of the horizontal circle through B (that is, the "great circle" of the sphere), the relation of costo 1. Likewise the sloping width of the zone at C bears to the vertical width ED the inverse ratio, i to cos p. Since these two ratios, one the inverse of the other, are to be multiplied together in determining the zonal area, it is obvious that the external area of the zone having a width ED along the diameter is equal to the product of this width by the circumference of the "great circle." Similarly the total external area of the sphere is found by multiplying the sphere diameter (= total width of all zones) by the circumference of the great circle; or is equal to d Xπd = πd2 = 4πr2 where d is the diameter and r the radius of the sphere. Familiarity with the above fundamental spherical (space) geometrical relations is absolutely essential to a proper understanding of the significance of the curves showing the space distribution of the candle-power of light sources; to the derivation or interpretation of diagrams showing the light from sources whose candle-power curves are known, and to the solution of problems relating to plane surface or extended surface sources. It is noteworthy in this connection that the modern tendency is away from point sources, and point-source candle-power methods. of calculation, towards extended source and lumen-output calculating methods, so that the importance of becoming familiar with space geometrical relations is ever on the increase. UNIT SOLID ANGLE-THE STERADIAN Although the illuminating engineer is seldom called upon to make use of solid angular dimensions expressed in terms of any unit of solid angular measurement, because almost all of the calculations in which he is interested can be based on ratios rather than actual values of solid angles, yet it may at times be found convenient to refer to some solid angular measurement in terms of a unit of measurement. Two distinct units have been employed for this purpose, one represented by the whole sphere and the other by a value ÷ 4 as large. For the former no special name has been standardized, while to the latter the name "steradian" is applied. = From its definition it will be seen that any zone on a sphere having a diametrical width such that W d4, where d is the diameter of the sphere, will subtend a solid angle of one steradian, and that 412.57 +steradians equal one sphere in solid angular measurement. Since the external surface of a sphere of unit radius is equal to 4 units of area, it follows that a steradian is an angle having such a value as to subtend unit area on the surface of a sphere of unit radius, or an area equal numerically to the radius squared on a sphere of any dimension whatsoever expressed in any unit of length or area. It is sometimes stated that the solid angle subtended by a chosen area when viewed from a chosen position can be calculated in steradians by dividing the numerical value of the area by the square of the distance between the point selected and the area. This statement is correct only when applied to an area every infinitesimal element of which occupies the same distance from the point of observation; that is, when the area lies on the circumference of a sphere having its center at the point chosen. RELATION BETWEEN LIGHT AND CANDLE-POWER DISTRIBUTION In order to present most clearly the exact significance of the candle-power curve, explain most readily the diagram for showing the distribution and summation of the light flux (lumens) from the source, and to give proper emphasis to the necessary distinction between candle-power distribution and light distribution use will be made of the curve of candle-power of a source giving light in only one hemisphere. In order definitely to fix ideas it will be assumed that the maximum candle-power of the source is 100 and that the candle-power decreases uniformly according to a cosine function of the angle of vision to zero at 90 degrees from the position of maximum candlepower. The curve showing the distribution of candle-power of such a source (which could be for example, an infinitesimal plane radiating in accordance with the "cosine law" of space distribution of candle-power) is represented in Fig. 2. Assume now that the source is placed at the center of a hollow sphere of unit radius the interior surface of which is illuminated by the source, as indicated in Fig. 2. The illumination on each elementary area of the surrounding sphere will at each point be numeric ally equal to the candle-power of the source when observed from that point-expressed in foot-candles if the radius of the sphere is one foot; in meter-candles if the radius is one meter, etc. Hence to determine the lumens incident upon any chosen section of the surrounding sphere it is necessary merely to multiply the area of that section by the mean candle-power of the source effective over that section. It is convenient not only for present purposes but also for purposes of subsequent comparisons, to express the area of sections of the surrounding sphere in terms of the zones cut off by various angles below (and above) the horizontal. Figs. 2 and 3.-Space distribution of candle-power and light flux from infinitesimal surface source. It should here be observed that, for sake of convenience in derivation and explanation, the angles indicated herein are measured (in both the plus and the minus direction) from the horizontal plane, whereas in actual curves of candle-power distribution the angles of elevation are "counted positively from the nadir as zero to the zenith as 180 degrees." That is to say, whereas in the curves herein shown the vertical angles are measured through zero from minus 90 degrees to plus 90 degrees, it is the more usual plan to make all measurements in the positive direction from zero plotted at the bottom of the curve to 180 degrees at the top. The zonal areas measured from the horizontal plane are as follows: The vertical widths of the separate zones are represented by the vertical line at the extreme right in Fig. 3. Along this line have been erected certain perpendiculars for representing the candlepower values over each part of the zone width. The product of the candle-power at each point by the zone area at that point which bears the constant relation of 27 1 to the vertical width of each zone, gives the lumens over that zone- to a certain scale. Obviously the area of the triangular figure at the right in Fig. 3 represents (to a scale involving the candle-power scale, the distance scale and the constant 2) the total lumens radiated by the source. From this figure, known as the Rousseau diagram, the lumens effective over any chosen zone can be computed at once from the intercepted area on the diagram. This is not an approximate, but an absolutely exact method of calculation. Any errors involved in using the method can be attributed to inaccuracies in measuring or plotting the candle-power or in determining the areas from the diagram; that is, to inexactness in carrying out the method rather than to the method itself. By using the Rousseau diagram merely as an aid in visualizing the problem and resorting to plane or spherical geometrical or trigonometrical calculations for actual determinations, one can often eliminate all inaccuracies other than those inherent in the photometric testing of the lighting source. |