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2. If two equal circles, one of which is parallel, and the other inclined to the primitive, be projected, the distances of the pole of the inclined projected circle, from the centres of the projections, will be directly proportional to the radii of these projected circles.
3. If two equal circles, one of which is parallel, and the other inclined to the primitive, be projected, straight lines, drawn through the pole of the inclined projected circle, will intercept corresponding arcs on the projection.
ORTHOGRAPHIC PROJECTION OF THE SPHERE.
In the orthographic projection of the sphere, the projecting point is still supposed to be in the axis of a great circle, assumed as the primitive or plane of projection; but at so great a distance, that a straight line drawn from it to any point of the sphere, may be considered as perpendicular to the plane of the primitive.
The orthographic projection of any point, therefore, is where a perpendicular from that point meets the primitive.
PROPOSITION I. Every great circle, perpendicular to the primitive, is projected into a diameter of the primitive; and every arc of it, reckoned from the pole of the primitive, is projected into its sine.
Let BFD be the primitive, and ABCD a great circle perpendicular to it, passing through its poles A, C; then the diameter BED, which is their line of common sec- :| tion, will be the projection of the B circle ABCD.
For, if from any point as G, in the circle ABC, a perpendicular GH fall upon BD, it will also be perpendicular to the plane of the primitive. Therefore H
is the projection of G. Hence the whole circle is projected into BD, and any arc AG into EH=GI, its sine. Cor. 1.-Every arc of the great circle, reckoned from its
intersection with the primitive, is projected into its
versed sine. CoR. 2.-The orthographic projection of any point on the
surface of the sphere, is, within the primitive, distant from its centre, by the sine of that point's distance from
either pole of the primitive. Cor. 3.-Every small circle, perpendicular to the primi
tive, is projected into its line of common section with the primitive, which is also its own diameter; and every arc of the semicircle above the primitive, reckoned
from the middle point, is projected into its sine. Cor. 4. Every diameter of the primitive is the projec
tion of a great circle, and every other cord the pro
jection of a small circle. Cor. 5.--A straight line, perpendicular to the primitive,
is projected into a point ; a parallel to the primitive, into an equal line; and one inclined to the primitive, into a less line, such that the radius is to the cosine of the
inclination, as the inclined line to its projection. COR. 6.-A spherical angle, at the pole of the primitive,
also any rectilineal angle, whose plane is parallel to the primitive, is projected into an equal angle.
PROPOSITION II. A circle parallel to the primitive is projected into a circle equal to itself, and concentric with the primitive.
Let the small circle FIG be parallel to the plane of the primitive BND. The straight line HE, which joins their centres, is perpendicular to the primitive; therefore E is the projection of H. Let any radius HI and IN perpendicular to the primitive be drawn. Then IN, HE, being parallel, are in the B same plane; therefore IH, NE, the lines of common section of the plane IE, with two parallel planes, are parallel, and the figure IHEN is a paral
lelogram. Hence NE=IH, and consequently FIG is projected into an equal circle KNL, whose centre is E. CoR.--The radius of the projection is the cosine of the
parallel's distance from the primitive, or the sine of its distance from the pole of the primitive.
PROPOSITION III. An inclined circle is projected into an ellipse, whose transverse axis is the diameter of the circle.
Case 1. Let ELF be a great circle, inclined to the primitive EBF, and EF their line of common section. From the centre C, and any other point K, in EF, let the perpendiculars CB, KI, be raised in the plane of the primitive, and CL, KN, Q in the plane of the great circle, meeting the circumference in Le N. Let LG, ND, be perpendi- E A cular to CB, KI; then G, D, are the projections of L, N. And because the triangles LCG, NKD, are equiangular, CL : CG = NKP: DK, or EC: CG2 = EKF.DK therefore the points G, D, are in the curve of an ellipse, of which EF is the transverse, and CG the semi-conjugate axis. Cor. 1.-In a projected great circle, the semi-conjugate
axis is the cosine of the great circle's inclination to the
primitive. COR. 2.-Perpendiculars to the transverse axis intercept
corresponding arcs of the projection and the primitive. Cor. 3.-The eccentricity of the projection is the sine of
the great circle's inclination to the primitive. For LG’=LC2-CG? ECPCG?; LG is therefore equal to the eccentricity, and it is also = Sin LCG.
Case 2. Let AQB be a small circle, inclined to the primitive, and let the great circle LBM, perpendicular to both, intersect them in the lines AB, LM. From the centre 0, and any
other point n, in the diameter
AB, let the perpendiculars TOP, NQ, be drawn in the plane of the small circle, to meet its circumference in T, P, Q. Also, from the points A, N, O, B, let AG, NI, OC, BH, be drawn perpendicular to LM, and from P, Q, T, PE, QD, TF, perpendicular to the primitive; then G, I, C, H, E, D, F, are the projections of these points. Because OP is perpendicular to LBM, and OC, PE, being perpendicular to the primitive, are in the same plane, the plane COPE is perpendicular to LBM. But the primitive is perpendicular to ĈBM. Therefore the line of common section EC is perpendicular to LBM, and to LM. Hence CP is a parallelogram, and EC=OP. In like manner, FC, DI, are proved perpendicular to LM, and equal to OT, NQ. Thus, ECF is a straight line, and equal to the diameter PT or AB. Let QR, DK, be parallel to AB, LM; then RO = NQ= DI = KC, and PR·RT = EK · KF. But AO:CG= NO: CI; therefore AO?:CGʻ = QR2: DK, or EC: CG? = EKKF:DK?. Cor. 1.-The transverse axis is to the conjugate, as ra
dius to the cosine of the circle's inclination to the pri
mitive. Cor. 2.-Half the transverse axis is the cosine of half
the sum of the greatest and least distances of the small
circle from the primitive. CoR. 3.-The extremities of the conjugate axis are in the
line of measures distant from the centre of the primitive, by the cosines of the greatest and least distances
of the small circle from the primitive. CoR. 4.--If, from the extremities of the conjugate axis
of any elliptical projection, perpendiculars be raised in the same direction, if the circle do not intersect the primitive, but, if otherwise, in opposite directions), they will intercept an arc of the primitive, whose cord is equal to the circle’s diameter.
PROPOSITION IV. The projected poles of an inclined circle are, in its line of measures, distant from the centre of the primitive, the sine of the circle's inclination to the primitive.
Let ABCD be a great circle, perpendicular, both to the
primitive and the inclined circle, and intersecting them in the diameters AC, MN. Then ABCD passes through the poles of the inclined circle ; let these
D be P, Q, and let Pp, Q4, be per- M
mer pendicular to AC; p, q, are the projected poles, and it is evident that po = Sin BP, or sin MA, the inclination. - COR. 1.The centre of the
primitive, the centre of pro-
one and the same straight line. COR. 2.--As radius is to the sine of a small circle's ind
clination to the primitive, so is the cosine of its distance from its own pole to the distance of the centre of its projection from the centre of the primitive.
OM: MG = OE:OF.
EXERCISES. 1. To describe the projection of a small circle parallel to the primitive, its distance from the pole of the primitive being given. 1.2. To project an inclined circle, whose distance from its pole and inclination to the primitive, are given. € 3. To find the poles of a given projection.
4. To measure any part of a given projection. 15. In any inclined circle, two diameters, which cut each other at right angles, are projected into conjugate diameters of the ellipse. 2-6. In every elliptical projection, half the transverse axis is to the eccentricity, as radius to the sine of the circle's inclination to the primitive; and half the conjugate axis is to the eccentricity, as radius to the tangent of the same.
7. Through two given points, in the plane of the primitive, to describe the projection of a great circle.
8. Given the distance of a circle from its pole, and the projection of that pole, to describe the projection of the circles