Let I, J, K be the usual unit vectors on the coordinate axes: I = (1, 0, 0), etc.
Let the sphere s be centered at O with radius r. The equation of the sphere is x2 + y2 + z2 = r2.
If P is a point on the sphere, the antipodal point of P is the point -P.
A circle c on a sphere is the intersection of a plane p with the sphere. Let A be a point so that OA is a normal direction vector to the plane.
The planar center of the circle is the foot C of the perpendicular from O to p. This means that C is the intersection of line OA with p.
The spherical centers N and S of the circle c are the intersections of line OA with the sphere. But these points can be computed as unit vectors multiplied by r. Thus let N = (r/|A|)A and S = -N, where A is a normal vector to the plane of the circle.
Equation from 3 points: Given 3 points P, Q, R on the sphere, two simultaneous equations for the circle are the equation of the sphere and the equation of the plane. Since the equation of the sphere is always present, we focus more on the equation of the plane PQR (as derived in examples on equations and normal vectors).
Equation of a great circle: The equation of the plane of a great circle has the form ax + by + cz = 0, since the circle passes through O. Knowing the point 0 on the plane means that we only need two points P and Q (not antipodal) to determine the great circle through P and Q.
Distance between two points P and Qn the sphere: o This is the angle POQ that can be computed using the dot product to find the cosine of the angle.
Angle between two circles: This angle is the dihedral angle between the planes of the circles and thus can be computed as the angle between normal vectors to the planes. Caution: The angle may be the exterior angle and thus the supplement of the disired angle.
1. What is the equation of the sphere of radius 1 and center O = (0,0,0)?
2. If P = (1,2,3), what is the point that is the intersection of the ray OP
and the sphere above?
3. If Q = (-1, 3, 4), what is angle POQ? (Use a calculator to get a numerical
answer.)
4. If A = (1,0,0), B = (0,1,0), and C = (0,0,1), what is the point P at the
center of triangle ABC? What is the opposite (antipodal) point of P?
5. If the equation of a great circle c is x – y + 2z = 0, what are the
poles of this great circle?
6. If the equation of a great circle d is x + y + z = 0, what are points of
intersection of c and d?
7. What is the angle between these two great circles?
Given I, J, K, the triangle IJK is one of the 8 triangles that are faces of an octahedron with vertices on the coordinate axes.
What are are equations of planes of the great circles through adjacent vertices of the octahedron? What are the spherical centers of these great circles?
What is the equation of plane of the the circle though I, J, K? What is the center of this circle on the plane? What are the spherical centers of this circle?
Any cube can be inscribed in a sphere. If the cube is inscribed with edges parallel to the coordinate axes inside the unit sphere (a sphere of radius 1), what are the vertices of the cube? (Hint: Think symmetry. Also think about the spherical centers of the circle through I, J, K.)
What are the equations of the great circles containing pairs of vertices of the cube (the spherical edges will be arcs of these great circles)? How many such circles are there?
What are the poles of the great circles above?
Compute the angles between the great circles using the dot product. Compare with "visual proof" from observing how the cube tesseallates the sphere.
What are the 3D distances between adjacent vertices of this cube?
What are the spherical distances between adjacent vertices of this cube? )
If one chooses a certain 4 vertices of a cube, these 4 points are the vertices of a regular tetrahedron. What are the spatial and spherical distances between pairs of these vertices?
What are the equations of the planes of the great circles through adjacent vertices of the tetrahedron?
What are the equations of the planes of the circles that pass through 3 of
the vertices of the tetrahedron?
What are the space coordinates of these midpoint of segments in space: midpoint M of segment KI, midpoint N of segment KJ, midpoint L of segment IJ? What is the space distance between any two of these midpoints?
What are the space coordinates of the spherical midpoint M' of segment KI,of the spherical midpoint N' of KJ, and of the spherical midpoint L' of IJ on the sphere?
What is the spherical distance between each pair of these spherical midpoints?
What are the vertex angles of spherical triangle M'N'L'?
What are the vertex angles of spherical triangle IL'M'?
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