(a) Write the definition of the cross-ratio (AB, CD) of points A, B, C, D on a line. (See Ogilvy 97-101)
(b) If on the real number line, A = 1, B = 2, C = 3, D = 4, what is the cross-ratio?
(c) If the cross-ratio (AB, CD) = k, what is the cross-ratio (CD, AB)? What is the cross-ratio (BA, CD)?
(d) Suppose CD divides AB harmonically. Write down the definition that tells what this means.
(e) From the previous expression, tell what is the cross-ratio (AB, CD) if CD divides AB harmonically.
Suppose c1, c2, c3 are circles of radius r1, r2, r3 all with the same center O. Let F, G, H denote inversion in c1, c2, c3 respectively. If P is a point, let P' = F(P) and P'' = G(F(P)) and P''' = H(G(F(P))).
(a) Based on the definition of inversion, each of the points P', P'', and P''' are on the ray OP. Using the definition, write formulas for each of |OP'|, |OP''| and |OP'''| in terms of the circle radii and |OP|.
(b) Based on your formula, tell what transformation of the plane is GF (tell the type and the defining details).
(c) Based on your formula, tell what transformation of the plane is HGF (tell the type and the defining details).
In the (x,y) plane, suppose J is inversion in the circle with radius 1 and center (0,0) and K is inversion in the circle with radius 3 and center (5,0). We suppose a point P = (x,y).
(a) Write a formula for J(x, y).
(b) Write a formula for K(x, y).
(c) Write a formula for KJ(x,y).
Since in inversive geometry, circles and lines are two special cases of I-circles or "circles", we can build another version of the Poincaré disk model by inverting the boundary circle of the disk into a line p, along with all the P-points and P-lines. The P-points are the points on one side of line p (the points in a half-plane) and the P-lines in this model are either arcs of Euclidean circles orthogonal to p or rays on lines orthogonal to p. Some things are easier to construct in this model and some things are harder, but all constructions can be figured out from basic circle constructions. (Note: You should be able to figure out the answers to these questions by reasoning, but you can also look up the half-plane model in many places.)
(a) Draw a line m. Given two ("random") P- points A and B in the model, construct the P-line through A and B. Give a second example of a P-line through points C and D for which the P-line is not an arc (i.e. a special case).
(b) Given a P-point E and a P-point F, construct the P-circle with center E through F in the half-plane model.
(c) Given two "random" P- points G and H, construct the P-line that is the P-perpendicular bisector of GH (i.e., the reflection (circle inversion) of G in this P-line is H).