This lab will be on a much more open scheme than some other labs, and the results will be turned in as part of Assignment 5.
The spherical distance between two points A and B on a sphere of radius R was introduced in class as an arc length of a great circle, either measured in degrees, or radians, or as an actual length (in cm or inches, etc).
Create a Sketchpad figure with measurements that shows the relationship between distance |AB| between A and B in 3-space and the spherical distance |AB|S.
Problem 5.1 of Assignment 5
Print this figure or paste it into a Word document or create a Sketchpad file to turn in with explanatory text. Use your figure it to derive and prove
(a) the relationship between |AB| and |AB|S measured in degrees or radians
(b) the relationship between |AB| and |AB|S measured in the same distance measure as |AB| (e.g., cm).
We already know the definition of a Euclidean circle on a plane. On a sphere S with center O and radius R, the definition of a spherical circle with radius r and center P is the set of points A such that |PA|S = r.
Problem 5.2 of Assignment 5
(a) Create a Sketchpad drawing that will help you proved that the intersection of any plane p with the sphere S is either empty, or a single point, or a set that is both a plane Euclidean circle and a spherical circle.
(b) If the plane p is distance d from O, show what is the Euclidean radius of the circle in terms of R and d. What is the center of the plane circle?
(c) In the same case, show that the same set is a spherical circle with two centers. Show the location of the centers and the size of the radii of the two circles in terms of the plane and d..
(d) Also, this circle has
a length (the circumference of the plane circle). What is the circumference
of a spherical circle of spherical radius r?
Draw a Sketchpad figure that shows a view of a sphere where the angle between the great circles is seen in the figure.
Problem 5.3 of Assignment 5
(a) Then use this figure to show that the angle between two great circles g and h is the distance between a pole G of g and a pole H of h. (Note: There is some ambiguity here that you should make clear.)
(b) Also use this figure or one like it to show that two great circles g and h are orthogonal if and only if the poles of g lie on h.
A (complete) right circular cone with vertex O can be defined thus: Let p be a plane through a point P distinct from O such that the line OP is perpendicular to p. Let c be a circle in this plane with center P. The set of all lines (not rays) in 3-space that pass through O and through c, forms a surface called a right circular cone. These lines are called generating lines of the cone. The line OP is called the axis of the cone.
The cone is divided into two nappes (or infinite ice cream cones). One nappe consists of all the rays from O that pass through c. The other consists of the opposite rays. (For even more confusion, we often consider finite cones made of segments from O to some point in c. This is the kind of cone that has finite volume.)
A cone is a surface of revolution, as is the sphere. So to study the surface of revolution, it is enough to show a cross-section by a plane containing the axis of revolution.
Problem 5.4 of Assignment 5
(a) Use this figure to explain why the tangent lines to the sphere through P form a right circular cone with vertex P. (Be sure to explain what the circle c is and why it is a circle.) In this case we say that the cone is tangent to the sphere.
(b) Show that all the tangent segments from P to a point of tangency on the sphere all have the same length. If the 3d distance |OP| = d., what is this length?
(c) Construct a Sketchpad figure of such a cross-section by a plane p of a 3d figure made of two disjoint spheres of different radius and one cone tangent to both spheres. In other words, this is the same as the previous figure, except that there are two spheres.
(d) Add to this figure the cross-section of a plane e that is perpendicular to p and that is tangent to both spheres. What is this familiar figure in the plane?