« PreviousContinue »
equator, the line of projection from S to b crosses the plane of the equator at 126° (stereographically projected) from the lefthand end of the diameter XY, or 54° from the right-hand end, while the line of projection from S to a would intersect the plane of the equator far out beyond the divided circle, as indicated by the arrow, at a point which could be determined by scale No. 3, figure 3, a being 144° from N~. All possible lines of projection from S to the great circle a b are located on the surface of an imaginary oblique cone with its apex at S. Moreover, it could be proved, as was done in the case of the small circle illustrated by figure 5, A (the figures being lettered the same), that the intersection of the cone with the plane of the equator is in this case also a circle. The stereographic pro
jection of a great circle is illustrated in figure 7, B. The projection of the point b, 54° from the equator on a north and south meridian, as shown by the upper figure, is quickly found on the diameter x y by means of the graduation on the base line of protractor No. I. A circular arc must then be found passing through b, and intersecting the divided circle at antipodal points at right angles to the diameter x y. To facilitate the construction of such circular arcs, scale No. 1 of figure 3 has been constructed, which gives the radii of possible great circles. As the arcs of projected great circles approach iV(the center of the divided circle) they become flatter; hence they are best constructed by means of the curved ruler described later on. As seen from scale No. 1, figure 3, the shortest radius of any stereographically projected great circle is equal to the radius of the divided circle.
To draw the projected arc of a great circle through two points, one of which, p, tigure 8, is on the divided circle and the other, a, within the circle, is a simple matter. The circular arc must intersect the divided circle at p', antipodal to p, and its center must be on a line x y intersecting the divided circle at 90° from p and p'. The center c may be found by a few trials with a pair of dividers, or it may be determined analytically as follows: With dividers opened up to some convenient distance construct two circular arcs u v, and v v, figure 8, having the same radius and draw a line through their intersections. The line thus drawn will be at right angles to the cen
ter point of a line joining a and p', and will intersect the line x y at c, which will be the center of the circular arc p a p'.
To draw the projected arc of a great circle through two points a and b, both of which are within the divided circle, the following principles may be used: That the great circle passing through a and b must also pass through points a' and b', antipodal, respectively, to a and b; also that the great circle must intersect the divided circle at antipodal points p and p'. If, therefore, there are two points, a and b, figure 9, anywhere within the circle, draw a diameter through one of them, a for example, and continue it beyond the circle. Apply the base line of protractor No. I to the diameter, determine the distance, in stereographically projected degrees, of a from the divided circle, and, making use of scale No. 3, figure 3, locate a' just as many projected degrees beyond the divided circle as a is within it. Thus, as measured on a stereograph ically projected north and south meridian, a' is antipodal to a, and the problem of finding the center c and drawing a circular arc through a, b, and a', which is fully illustrated by the figure, is too simple to need more detailed explanation. In some cases it may prove easier to plot a line x y, as illustrated by figure 9, and find upon it the point c by trial, for only one circular arc can be found, which, passing through a and b, intersects the divided circle at antipodal points p and p'. If the two points within the circle are so located that the projected great circle passing through them has a very long radius, the curved ruler described later on can be quickly adjusted so that a circular arc may be drawn passing through them and intersecting the divided circle at antipodal points.
To measure the Angular Distance between any lioo points on a Stereographic Projection.— To measure the Side of a Spherical Triangle.—Let the two points a and b, be anywhere within the divided circle, figure 10. Since the angular distance between any two points on a sphere is measured in degrees along the arc of a great circle, it is first necessary to construct a great circle passing through a and b, and thus locate the antipodal points p and p' on the divided circle. It is now possible to find some projected vertical small circle described about p, which passes through a and serves as a measure of the angular distance p to a; likewise a small circle described about p' and passing through b, which serves to measure the distance from p' to b. Knowing the angular distances p to a and p' to b, the distance a to b is readily determined. In figore 10, p to a = 33° 0' and p' to b = 58° 15'; hence a to b = 88° 45', or the supplement of the sum of the two angles.
Without the aid of the special protractor described in the next paragraph, it is a laborious task to find the two projected vertical circles passing through a and b, figure 10; though by making use of the protractor No. I and scale No. 2, figure 3, they can be plotted without great difficulty.
Stereographic Protractor JVo. II.—To facilitate measurements of the angular distances between points on a stereographic projection, or measurements of the sides of spherical triangles, a special protractor has been devised which is represented in plate I, without reduction, and will be designated as the Stereographic Protractor, or Protractor JVo. II. The essential features of this protractor are as follows: The circle of 14cm diameter, divided into degrees, corresponds with the divided circle shown, much reduced, in figure 3. On one-half of the circle, the projected arcs of vertical small circles, p. 11, have been constructed for every degree. Since on a 14cm-circle the stereographically projected circles are very near together, the even degrees have been represented by full lines and the odd degrees by dashed lines; also the arcs of every fifth and tenth degree are engraved somewhat heavier than the others. These details, however, are not essential, but simply serve to make the use of the protractor somewhat easier. On the other half of the circle only the arcs of every fifth and tenth degree are represented. This half of the protractor is really superfluous, but for approximate measurements it will at times be found more convenient than the other half where the arcs are crowded. For convenience the protractor is numbered in two directions, from 0° to 180°. Further, in order to have the protractor really practical, it should be printed or engraved on some transparent material; transparent celluloid has been found to satisfy every requirement.
To use the protractor in finding the distance from a to b, figure 10, for example, lay it (best with the printed side down) on the drawing, and bring the 0° and 180° points to correspond with the antipodal points p and p' on the divided circle, then note the distances p to a and p to b in degrees and fractions, and the difference between the two readings will equal the angular distance from a to b.
To measure from any given point p on the divided circle to a point a within the circle, it is not necessary to go through the operation of constructing the arc of a great circle through p and a, as represented in figure 8, p. 14; merely place the protractor with its 0° and 180° points on p and p', and then note on the protractor the projected arc which corresponds most nearly to a.
Am. Jocb. Sci.—Fourth Series Vol. XI, No. 1.—January, 1901.
To measure the Angle made by the meeting of Two Great Circles.— To measure the Angle of a Spherical Triangle.— Just as the angle between two meridians at the north pole of a sphere is measured on the equator, so the angle between two great circles crossing at a point A, which may be anywhere within the divided circle, figure 11, is measured on the arc of a great circle at 90° from A. To make the measurement of the angle, draw a diameter of the divided circle through A, and, applying the base line of protractor No. I to the diameter, note the distance in projected degrees from JV to A; then locate a point 11 just as many degrees from the divided circle as A is from N. A and B are thus 90° apart, and, making use of
scale No. 1, figure 3, the arc of a great circle p p', passing through B, can be easily drawn. All points on the great circle thus plotted, including the intersections and y, are 90° from A, and the angle at A is equal to the distance in projected degrees between x and y, as measured with the stereographic protractor on the arc of the great circle p p'.
Provided an angle is located on the divided circle, as at C, all that is necessary to do is to draw a diameter of the divided circle at 90° from C, and measure the angle at G on the projected north and south meridian by means of the graduation on the base line of protractor No. I; for example, from utov = 59°.
Another method of measuring the angles of spherical triangles depends upon a well known and interesting peculiarity of the stereographic projection, to which the writer's attention was called by Prof. G. P. Starkweather of Yale University; namely, that the angle made by the crossing of two circles on