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To project a right circle, or one, that is perpendicular to the
plane of projection.
5. Hence it is evident, that a line parallel to the plane of the projection is projected into a right line equal to itself; but a line, that is oblique to the plane of projection, is projected into ones, that is less than itself. 6. A plane surface, as ACBD, per
C pendicular to the plane of the projection is projected into the right line, as AB, in which it cuts that plane. A
E Hence it is evident, that the circle ACBD perpendicular to the plane of projection, passing through its centre, is projecied into that diameter AB, in which it cuts the plane of the projection. Also any arc, as Cc, is projected into Oo, equal to ca, the right sine of that arc; and the complemental arc cB is projected into oB, the versed sine of the same arc cB.
7. A circle parallel to the plane of the projection is projected into a circle equal to itself, having its centre the same with the centre of the projection, and its radius equal to the cosine of its distance from the plane. And a circle oblique to the plane of
To project a given oblique circle, Draw the line of measures AB, and take the circle's nearest distance from the primitive,
D and set from B to D up
B ward, if it be above the
F CH G primitive; or downward, if below ; likewise take its greatest distance, and set from A to E, draw ED, and let fall the perpendiculars EF, DG ; bisect FG in H, and erect the perpendicular KHI, making KH= HI= half ED ; then describe an ellipse, whose transverse is IK, and conjugate FG ; and that will represent the given circle.
To find the pole of a given ellipsea
R i primitive draw the conjugate
of the ellipse ; on the extreme points F, G, erect the perpendic
CHPG ulars FE, GD, or set the transverse IK from E to D, bisect ED in R, and let fall RP perpendicu
. lar to AB ; then is P the pole.
the projection is projected into an ellipse, whose greater axis is equal to the diameter of the circle, and its less axis equal to double the cosine of the obliquity of the circle, to a radius cqual 40 half the greater axis.
To measure an arc of a parallel circle ; or to set any number
of degrees on it.
Cor. If the right circle pass through the centre, it is only necessary to raise perpendiculars on it, which will cut the primitive, as required.
PROBLEM PROBLEM VII.
To set any number of degrees on a right circle.
[See Figure under last Problem.]
On ED, the given right circle, describe the semicircle END ; then, by Prob. V. set off NP = the given degrees, and draw PL perpendicular to ED ; then AL contains the degrees required.
To measure an arc of an ellipse ; or to set any number of de
grees on it.
To find the poles of any projected great circle.
1. The poles of the primitive circle.
They are in the centre C.*
* The following are laws of the stereographic projection.
1. In this projection a right circle, or one perpendicular to the plane of projection, and passing through the eye, is projected into a line of half tangents.
2. The projection of all other circles, not passing through the projecting point, whether parallel or oblique, is into circles.
Thus, let ACEDB represent a sphere, cut by a plane RS, passing through the centre I, perpendicular to the di
K ameter EH, drawn from E, the place of the eye ; and let the section of the sphere by
s the plane. RS be the circle CFDL, whose poles are H