Download Sample Problems for Exam 2

Sample Problems for Exam 2
Geometry for Teachers, MTH 623, Fall 2014
Instructor: Abhijit Champanerkar
• Exam 2 will be held in class on Wednesday Dec 3rd.
• Review for Exam 2 will be held on Monday Dec 1st.
• Review the problems in homeworks in addition to the problems below.
• Syllabus for Exam 2: Spherical Geometry.
Spherical Geometry
• S 2 = {(x, y, z) ∈ R3 | x2 + y 2 + z 2 = 1} = {~u = h x, y, z i| ||~u|| = 1} (i.e. unit vectors).
−−→
We identify points with unit vectors i.e. P = (x, y, z) ↔ OP = h x, y, z i and take dot
products and cross products of points.
• A great circle (line) between two non-antipodal points P, Q is given by
LR = S 2 ∩ ( Plane through O, P, Q)
where R is the pole of the line which equals the unit normal to the plane i.e R =
P ×Q
. A great circle can be described by its pole ( unit normal) or an equation
||P × Q||
of the plane.
Problems
1. Find great circles passing through the following pairs of points:
√
√
√
√
(a) P = (1/ 2, 1/ 2, 0) and Q = (0, 1/ 2, 1/ 2)
(b) P = (1, 0, 0) and Q = (0, −1, 0).
2. Explicity find infinitely many great circles passing through the antipodal points P =
(0, 1, 0) and Q = −P = (0, −1, 0).
3. Find the length of any great circle on S 2 .
4. Find the distance between the following pairs of points.
(a) P = (0, 0, 1) and Q = (0, −1, 0)
√
√
√
√
(b) P = (1/ 2, 1/ 2, 0) and Q = (0, 1/ 2, 1/ 2)
√
√
√
√
(c) P = (1/ 3, 1/ 3, 1/ 3) and Q = (1/2, − 3/2, 0)
5. Determine if the following 3 points are collinear on S 2 . (P, Q and R ∈ R3 are coplanar
if P · (Q × R) = 0.)
√
√
(a) P = (1, 0, 0), Q = (0, −1, 0) and R = (1/ 2, 1/ 2, 0)
(b) P = (1, 0, 0), Q = (0, −1, 0) and R = (0, 0, 1)
6. (a) Find a triangle on S 2 all of whose angles are π/2.
(b) Find the area of such a triangle.
(c) Using the cosine law, find the length of all the sides of this triangle. Is is
equilateral on S 2 ?
7. Find the length of sides, angles and area of the triangle with the following vertices.
√
√ √ √
√
√
(a) P = (1/2, 1/3, 23/6), Q = (0, −1/ 3, 2 3) and R = (1/ 2, 1/ 2, 0).
√
√
(b) P = (1, 0, 0), Q = (0, −1, 0) and R = (1/ 2, 1/ 2, 0).
8. State with justification whether the following statements are True or False or Maybe
or Maybe Not.
(a) There exists a spherical triangle with all angles π/3.
(b) There exists a spherical triangle with all angles π/2.
(c) A polyhedron can have 150 vertices, 200 edges and 100 faces.
(d) A polyhedron can have 100 vertices, 150 edges and 52 faces.
(e) Spherical line segements can be arbitrarily long.
(f) There exists a spherical quadrilateral with all angles π/2.
(g) Spherical triangles with equal area are congruent.
9. Prove the following statements:
(a) Any two distinct great circles on S 2 intersect.
(b) LR = L−R .
(c) If P ∈ LR then −P ∈ LR .
(d) Show that there are infinitely many great circles passing through any pair of
antipodal points.
(e) Show that |P Q|S 2 = |QP |S 2 .
(f) State the Spherical Pythagoran Theorem. Use the spherical Cosine Law to prove
it.
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