DOI Seite / Zitierlink:
https://doi.org/10.11588/diglit.20658#0365
OF PROJECTION. 351
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light line equal to itself; but a line obliquely situated with respect to the plane is projected into
one less than itself.
6th. A plane surface, perpendicular to the plane of projection, is projected into a right line
where it cuts that plane : hence it is evident that a circle perpendicular to the plane of projection,
passing through its centre, is projected into that diameter which cuts the plane. Also any arch
is projected into a space equal to the right sine of that arch, and the complement of the same
arch is projected into its versed sine.
7th. A circle parallel to the plane of projection is projected into a circle equal to itself, having
its centre the same with that of the projection, and its radius equal to the co-sine of its distance
from the plane. And a circle oblique to the plane of 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 co-
sine of the obliquity of the circle, to a radius equal to half the greater axis.
The properties of the stereographic projection are—1st. That in it, a.right circle or the per-
pendicular to the plane of projection, and passing through the eye,, is projected into the line of
half tangents.
2d. Ail other circles, not passing through the projecting point, whether parallel or oblique,
are projected into circles.
3d. The projected diameter of any circle subtends an angle at the eye, equal to the distance
of that circle from its nearest pole, taken on the sphere; and that angle is-bisected by a right,
line joining the eye and that pole.
4th. Any point of a sphere is projected at such a distance from the centre of projection as is
equal to the tangent of half the arc, intercepted, between that point and the pole opposite to the
eye ; the setnidiaineter of the sphere being radius.
5th. The angle made by two projected circles is equal to the angle formed by these circles on
the sphere itself.
6th. The distance between the poles of the primitive circle and an oblique circle, is equal to
the tangent of half the inclination of those circles; and the distance of their centres is equal to
the tangent of the inclination ; the semidiameter of the primitive being radius.
7th. If through any given point in the primitive circle, an oblique circle be described, then the
centres of all other oblique circles passing through that point, will be in a right line drawn
through the centre of the first oblique circle, and perpendicular to a line passing through that
centre, the given point, and the centre of the primitive circle.
8th. Equal arcs of any two great circles of the sphere, will be intercepted between two other
circles drawn on the sphere, through the remotest poles of those great circles.
9th. If lines be drawn from the projected pole of any great circle, cutting the peripheries of
the projected circle and plane of projection ; the intercepted arcs of these peripheries will be
equal.
1.0th. The radius of any lesser circle whose plane is perpendicular to that of the primitive
circle, is equal to the tangent of that lesser circle's distance from its pole; and the secant of that
distance is equal to the distance of the centres of the primitive and lesser circle.
One of the chief applications of this branch of perspective is the construction of universal
maps, or such as exhibit the whole surface of the earth, or of the two hemispheres. Such repre-
sentations of the earth are very ancient, their invention, at least their use, having been attributed
to Anaximander, about 600 years before Christ. Those drawn in modern times, according to
Ptolomy
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light line equal to itself; but a line obliquely situated with respect to the plane is projected into
one less than itself.
6th. A plane surface, perpendicular to the plane of projection, is projected into a right line
where it cuts that plane : hence it is evident that a circle perpendicular to the plane of projection,
passing through its centre, is projected into that diameter which cuts the plane. Also any arch
is projected into a space equal to the right sine of that arch, and the complement of the same
arch is projected into its versed sine.
7th. A circle parallel to the plane of projection is projected into a circle equal to itself, having
its centre the same with that of the projection, and its radius equal to the co-sine of its distance
from the plane. And a circle oblique to the plane of 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 co-
sine of the obliquity of the circle, to a radius equal to half the greater axis.
The properties of the stereographic projection are—1st. That in it, a.right circle or the per-
pendicular to the plane of projection, and passing through the eye,, is projected into the line of
half tangents.
2d. Ail other circles, not passing through the projecting point, whether parallel or oblique,
are projected into circles.
3d. The projected diameter of any circle subtends an angle at the eye, equal to the distance
of that circle from its nearest pole, taken on the sphere; and that angle is-bisected by a right,
line joining the eye and that pole.
4th. Any point of a sphere is projected at such a distance from the centre of projection as is
equal to the tangent of half the arc, intercepted, between that point and the pole opposite to the
eye ; the setnidiaineter of the sphere being radius.
5th. The angle made by two projected circles is equal to the angle formed by these circles on
the sphere itself.
6th. The distance between the poles of the primitive circle and an oblique circle, is equal to
the tangent of half the inclination of those circles; and the distance of their centres is equal to
the tangent of the inclination ; the semidiameter of the primitive being radius.
7th. If through any given point in the primitive circle, an oblique circle be described, then the
centres of all other oblique circles passing through that point, will be in a right line drawn
through the centre of the first oblique circle, and perpendicular to a line passing through that
centre, the given point, and the centre of the primitive circle.
8th. Equal arcs of any two great circles of the sphere, will be intercepted between two other
circles drawn on the sphere, through the remotest poles of those great circles.
9th. If lines be drawn from the projected pole of any great circle, cutting the peripheries of
the projected circle and plane of projection ; the intercepted arcs of these peripheries will be
equal.
1.0th. The radius of any lesser circle whose plane is perpendicular to that of the primitive
circle, is equal to the tangent of that lesser circle's distance from its pole; and the secant of that
distance is equal to the distance of the centres of the primitive and lesser circle.
One of the chief applications of this branch of perspective is the construction of universal
maps, or such as exhibit the whole surface of the earth, or of the two hemispheres. Such repre-
sentations of the earth are very ancient, their invention, at least their use, having been attributed
to Anaximander, about 600 years before Christ. Those drawn in modern times, according to
Ptolomy