APPENDIX. SPHERICAL PROJECTIONS. DEFINITIONS. 1. The projection of the sphere is the delineation of the circles drawn upon its surface, as seen upon a given plane, called the plane of projection, the position of the eye being also given. There are several kinds of projection, according to the situation of the eye, and the plane to which the circles are referred. 2. In the orthographic projection, the plane of projection passes through the centre of the sphere; and the eye is supposed to be so situated, that all lines drawn to it from any points on the sphere, are perpendicular to the plane of projection. 3. In the stereographic projection, the plane of projection passes through the centre of the sphere, and the eye is situated in the pole of the circle formed by the intersection of this plane and the spherical surface. 4. The primitive circle is that on whose plane the projections are made; or it is the common section of the plane of projection and the surface of the sphere. And the pole of this circle is called the pole of projection. The point in which the eye is supposed to be situated is called the projecting point. 5 The line of measures of any circle, is the common section of the plane of projection and another plane which passes through the axes of the primitive and of that circle. 6. The right line drawn from any point on the sphere to the projecting point, is called a projecting line. 7. A circle of the sphere is called an original circle; and its representative on the plane of projection, a projected circle. SECTION I. The Orthographic Projection of the Sphere. PROP. I. If a right line AB (fig. 40.) be projected upon a plane, it will be projected into a right line; and its length will be to the length of the projection, as radius to the co-sine of its inclination to that plane. For let fall the perpendiculars Aa, Bb upon the plane of projection; then ab will be the line into which it is projected; but, by trigonometry, AB : Ao, or ab: : radius: sine of B, or co-sine of aAB. COROLLARY I. A right line projected upon a plane parallel thereto, is projected into a right line parallel and equal to itself. COROLLARY II. An angle projected upon a plane, which is parallel to the lines forming the angle, is projected into an angle equal to itself. COROLLARY III. Any plane figure projected upon a plane parallel to itself, is projected into a figure similar and equal to itself. COROLLARY IV. Hence also the area of any plane figure, is to the area of its projection :: radius : co-sine of its inclination to the plane of projection. PROP. II. A circle perpendicular to the plane of projection, is projected into a right line equal to its diameter. For projecting lines drawn through all the points of the circle fall in the common section of the planes of the circle and of projection, which is a right line (3. 2. Supp.), and M that line is equal to the diameter of the circle; because the planes intersect in that diameter. 2. E. D. 1 COROLLARY. Hence any plane figure perpendicular to the plane of projection is projected into a right line. For the perpendiculars from every point, will all fall in the common section of the plane of the figure and the plane of projection. PROP. III. A circle parallel to the plane of projection is projected into a circle equal to itself, and concentric with the primitive. The circle being parallel to the primitive, their common axis passes through the projecting point (2. and 4. def.), and every radius of the circle is projected into a line equal to itself (Cor. 1. prop. 1.) 2. E. D. COROLLARY. The radius of the projected circle is the co-sine of the distance of its original circle from the primitive, and the sine of its distance from its own pole. PROP. IV. An inclined circle is projected into an ellipsis whose transverse axis is the diameter of the circle. |