Figure 4- 1 Projection sphere and projection plane
In stereographic projection, the geometric features of the object are placed in the center of the sphere. The light emitted from the center of the sphere projects all points, straight lines and planes from the center of the sphere to the sphere, and the spherical projection of points, straight lines and planes is obtained. Because the directions of points, lines and surfaces on the sphere and the angular distance between them are not easy to observe and express. Then the spherical projection of points, lines and surfaces (points and lines) is projected to the equatorial plane again with the south pole or north pole of the projected sphere as the emission point. This kind of projection is called stereographic projection, and the resulting projection of points, lines and surfaces on the equatorial plane is called stereographic projection.
Fig. 4-2a shows the stereoscopic schematic diagram of the stereoscopic projection principle. The outer circle on the diagram represents the projected sphere, and the O point is the center of the sphere. The plane NESW is an equatorial plane, and its intersection with the spherical surface is a circle NESW, which is called the great equatorial circle. The plane NASB is an inclined plane passing through the center O of the projected sphere, with the strike of SN, the inclination of E and the inclination of α. The intersection SN between this plane and the equatorial plane is its strike line. Since this plane passes through the center of the projection sphere, its intersection with the projection sphere, that is, its sphere, is projected as a circle NASB with a diameter equal to that of the projection sphere. The semi-circular arc NAS is its spherical projection in the upper hemisphere, and the semi-circular arc SBN is its spherical projection in the lower hemisphere. From the south pole of the projection sphere to the spherical projection of the plane, these rays have a series of intersections with the equatorial plane (as shown in Figure 4-2a, the intersection of the rays emitted from the south pole to point A and the equatorial plane is point C, and the intersection of the rays emitted to point B and the equatorial plane is point D), and the connecting lines of these intersections constitute the polar stereographic projection NCSD of the plane. It can be proved that it is a circle, CD is its diameter, and the point that bisects CD is its drawing center. As can be seen from Figure 4-2a, the arc NCS is an epipolar stereographic projection of the hemispherical projection (semi-arc NAS) on the plane, which is located in the equatorial circle. The arc SDN is the stereographic projection of the hemispherical projection (semi-arc SBN) under the plane, which is located outside the equatorial circle. Fig. 4-2b is a perspective view of this plane.
Figure 4-2b shows the global stereographic projection of NASB plane. In practical application, most of them only do hemispherical projection. According to the different habits and customs of expressing the purpose, or the projection of the upper hemisphere, rays are emitted by the spherical pole (South Pole) of the lower hemisphere; Or make a projection of the lower hemisphere, and the rays are emitted from the spherical pole (North Pole) of the upper hemisphere. The advantage of hemispherical projection is that the projected points and lines are on the hemispherical surface opposite to the emission point, and their stereographic projections are all in the equatorial circle, which is convenient for drawing, comparison and interpretation. The projection of the upper hemisphere is commonly used, and this representation is adopted in this book. As shown in Figure 4-2b, the solid arc NCS in the equatorial circle represents the stereographic projection of plane NASB in the upper hemisphere.
Fig. 4-2 Projection principle through the center plane of the sphere
On the stereographic projection, the outer circle is an equatorial circle, representing the equatorial plane (i.e. the horizontal plane), and its top, bottom, left and right respectively represent the four directions of north, south, east and west, which are divided by 360 azimuth. The arc NCS is the polar stereographic projection of the above plane. The connecting line between point N and point S represents the strike line of the plane, and its orientation is divided and read by the orientation of point N (or point S) on the equatorial circle. The direction pointed by the concave part of the arc NCS represents the inclined direction of the plane, in which the connecting line between the C point and the O center is the oblique line of the plane. Extend the intersection of CO and the great equatorial circle at point e, and the position of point e on the great equatorial circle is the inclined position of the plane. Connect point S and point C and extend the intersection with the equatorial circle at point F. Extend the intersection of OC and equatorial circle at point W, and the azimuth included between two points F and W is the inclination angle α of the plane.
Figure 4-2 shows that the global polar stereographic projection passing through the inclined plane of the projected spherical center is a great circle with a diameter larger than that of the equatorial great circle, and its hemispherical polar stereographic projection is an arc in which the great circle passes through both ends of the equatorial great circle diameter and is located in the equatorial great circle. Therefore, if the spatial direction of the plane is known, we can find the center of the drawing with compasses and rulers according to the projection principle, and then draw its stereographic projection.