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294. Let A be the pole of the primitive BD, and MN a circle to be projected; MN being in the first figure a small circle, and in the second a great circle; then the point M has
for its projection the point m, and n is the projection of N, and the circle mn is the projection of the circle MN. The line AM is the projecting line of the point M, and the plane AMN is the projecting plane of the diameter MN, whose projection is the line mn.
In the stereographic projection, the projection of every circle of the sphere is a circle.
295. PROBLEM I.-To find the locus of the centres of the projections of all the great circles that pass through a given point.
Let F be any given point within the primitive ABCM. Through F draw the diameter BM and AC perpendicular to it; draw AF, and produce it to D; draw the diameter DL; draw AL, and produce it to meet BM in G; bisect FG perpendicularly by II', B and II' is the required locus. Hence, any circle, PFN, passing through F, and having its centre in any point as I in IHI', is the projection of a great circle, and hence it cuts the primitive in two points P, N, diametrically opposite.*
* The problems under this head are proved in the Solid and Spherical Geometry of the Educational Course.
296. PROBLEM II.-Through any two points in the plane of the primitive, to describe the projection of a great circle.
1. When one of the points is in the centre of the primitive.
Draw a diameter passing through the other point, and it will be the required projection. For the great circle passes through the pole of the primitive.
2. When one of the points is in the circumference, and the other is neither in the circumference nor in the centre. Let A and P be the two points, and ACBD the primitive.
Draw the diameter AB, and describe the circle APB through the three points A, P, B; and it is the c required circle.
3. When neither of the points is in the centre or circumference.
Let F, G, be the given points, and ABC the primitive.
Find IH the locus of the centres of all the projections of great circles passing through one of the points, as F (295); join F, G, and bisect FG perpendicularly by KH; and the centre of every circle through F and G is in KH; but the centre of B the required circle is in IH; hence, H is its centre; and a circle, DFG, through the two given points, described from the centre H, is the circle required.
297. PROBLEM III.-About some given point, as a pole, to describe the projection of a great circle.
1. When the given point is the centre of the primitive. The required projection is evidently the primitive itself. 2. When the given point is in the circumference of the primitive.
Draw a diameter through the given point, and another
diameter perpendicular to the former; the latter diameter is the required projection.
For, since the primitive passes through the pole of the required projection, its original circle must pass through the pole of the primitive, and its projection is a diameter.
3. When the given point is neither in the centre nor the circumference of the primitive.
Let P be the given point, and ADBC the primitive.
Through P draw the diameter AB, and another CD perpendicular to it. Draw DP, and produce it to E; make a the arc EF equal to a quadrant; draw DF, cutting AB in G; and the circle CGD, through the points C, G, D, is the required circle.
For, considering APB as the primitive, and D its pole, PG is evidently the projection of a quadrant EF. Now, if ADBC be the primitive, since APB passes through P, the pole of the required circle, it must pass through C, D, the poles of AB. Hence, the required circle must pass through C, G, and D.
COR.-Hence, the method of finding the pole of a projected great circle is evident.
1. When the projection is a diameter of the primitive. The extremities of the diameter perpendicular to it are evidently its poles.
2. When the given projection is inclined to the primitive, as CGD.
Join C, D, and draw the diameter AB perpendicular to CD. Draw DG, and produce it to F; make the arc FE a quadrant; draw DE, cutting AB in P, and P is the pole of the given circle.
298. PROBLEM IV.-To describe the projection of a small circle about some given point as a pole.
1. When the pole is in the centre of the primitive, or the original small circle parallel to the primitive.
Let AB, CD, be two perpendicular diameters of the primitive. Make CE equal to the distance of the small circle
from its pole, as, for example, 34°. Draw DE, cutting AB in F; from P as a centre, with the radius PF, describe the circle FGK, which will be the required projection.
For PF is evidently the projection of CE, and the centre of the required circle is evidently in P.
2. When the given pole is in the circumference of the primitive, or the original circle is perpendicular to the primitive. Let C be the given pole; AB, CD, two perpendicular diameters. Make CE equal to the distance of the circle from its pole. Draw EL a tangent to the primitive at E, and let it meet DC produced in L. A circle described from the centre L, with the radius LE, namely, MNE, is the required. circle.
3. When the pole is neither in the centre nor the circumference of the primitive.
Let P be the given point, and AB, CD, two perpendicular diameters of the primitive. Draw CP, and produce it to E; lay off EF, EG, each equal to the distance of the circle from its pole, for instance, 62°; draw CF, CG, cutting AB in H and I, and on HI, as a diameter, describe the circle HKI, and it is the required projection. For if AB be the primitive, and C its pole; E the pole of a small circle, and F, G, two
points in its circumference, then HI is the diameter of its projection. Hence, if ACBD be the primitive, HI is evidently the diameter of the projected small circle, whose pole is P.
COR.-The method of finding the projected pole of a given projected small circle, is manifest from this problem.
I. When the small circle is concentric with the primitive, the centre of the latter is the projected pole of the former.
2. When the small circle is perpendicular to the primitive, as MNE, its pole is in C, the middle of the arc MCE.
3. When the circle is inclined to the primitive, as HKI, draw a diameter AB through its centre, and CD perpendicular to it; draw CH, CI, cutting the primitive in F, G; bisect FEG in E; draw CE, and P is the required pole.
299. PROBLEM V. To measure any given arc of a projected circle.
1. If the given arc be a part of the primitive, it may be measured as the arc of any other circle (Part I. 138 or 188.)
2. When the given arc is a part of a circle projected into a straight line. Let KL be any given arc of the jected circle AKB; find C its pole, and draw CK, CL, cutting the primitive in F and G, and FG is the measure of KL, and is, in the present instance, 32°.
Let AEB be the primitive, and AIB the other circle, and IAD the angle. Find F and C their centres; draw AC, AF, and the angle CAF measures the given angle. Or find F and P their poles; draw AP, AF, cutting the primitive in G and B, and GB measures the given angle, which is, in the present instance, 40°.
3. When the given circle is inclined to the primitive.
Let HI be the given arc of the projected circle AIB. Find P its pole; draw PH, PI, cutting the primitive in D, E, and DE is the measure of HI, which is therefore, in the present example, 45°.
300. PROBLEM VI. To measure the projection of a spherical angle.
1. When the circles containing the given angle are the primitive and a diameter of it.
The angle is a right angle.
2. When one of the circles is the primitive, and the other is a circle inclined to it.