Chapter V. The Sphere 48. The equation of the sphere Center at , radius r is Any equation of the form May be written in the form

Notice that if r=0, the sphere is called point sphere. If r<0, the sphere is an imaginary sphere. Vector form:

50. Tangent plane Let be a point on the sphere The plane passing through perpendicular to the line joining to the center of the sphere is the tangent plane to the sphere at . The normal vector n coincide with

If the sphere is Then the tangent plane at is

51. The angle between two sphere The angle between two sphere at a point on their curve of intersection is defined as equal to the angle between the tangent planes to the sphere at O’ O

If the two spheres are as follows: Then the tangent planes are

Therefore, (Notice that the angle between two planes is equal to the angle between two directed normals to the planes, P.16, art.15) Notice that lies on both spheres, namely, satisfies both equation of the spheres)

Hint: the above formula is independent of the coordinates of , we have Theorem. Two spheres intersect at the same angle at all points of their intersection curve. The orthogonal condition is

52. Spheres satisfying the given conditions. Homogeneous in five coefficients, a,f,g,h,k. Satisfies four conditions, e.g., pass through four points, intersect four given spheres at given angles,……. Exercises: P. 56, No. 2,4, 6.

53. Linear systems of spheres. Let S, S’ be two spheres: The equation Also represent a sphere for any numbers of

Every sphere of the system contains the curve of intersection of S=0 and S’=0. If the sphere is composite (reduce to a plane). It consists of the plane at infinity and the radical plane:

Stereographic projection. Let O be a fixed point on the surface of a sphere of radius r, and be the plane tangent to the sphere at the opposite end of the diameter passing through O. The intersection with of the joining O to any point on the surface is called the stereographic projection of . Z O X Y P

Take the plane be the plane z=0, O=(0,0,2r) Take the plane be the plane z=0, O=(0,0,2r). The equation of sphere is then The equation of line joining O to on the sphere are To determine the equation of P=(x,y,0)

Conversely, the equations can be solved for Conversely, the equations can be solved for by using the fact that lies on the sphere

Theorem I. The stereographic projection of a circle is a circle Proof. Let the equation of the plane of the given circle on the sphere be Ax+By+Cz+D=0, then we have Substituting the coordinates of into P=(x,y,0),we obtain Which represents a circle in XY-plane.

Definition. The angle between two intersection curves is defined as the angle between their tangents at the point of intersection. Theorem III. The angle between two intersecting curves on the sphere is equal to the angle between their stereographic projections. * The importance of the theorem is that we can measure the angle of intersecting curve on sphere in turns of their stereographic projections

Proof. It suffices if we prove the theorem for great circles: For, let be any two curves whatever on the sphere having a point P’ in common. The great circles whose planes pass through the tangents to at P’ are tangent to , respectively, at P’. The stereographic projections of the great circles are tangent to , respectively, at so that the angle between them is the angle between

The condition that a circle is a great circle is that its plane Ax+By+Cz+D=0 Pass through the center (0,0,r) of the sphere, so that Cr+D=0. Then the equation of stereographic projection Reduces to

The angle between two great circles is equal to the angle between their planes, since the tangents to the circles at their common points are perpendicular to the line of intersection of their planes. The angle between two planes Is defined by art. 15 or the angle between their normals

We can verify that the angle between the stereographic projections of the circle are the same as above. Exercises: P59. No. 1