Name: Student number: METU MATH 471, Midterm 2 ^{1 2 3 4 5 Σ}

Wednesday, December 7, 2011, at 17:40 (90 minutes), totally 30 points

Instructor: S.Finashin

Problem 1. ( pts) (a) Find the distance between points z_{=}i and z_{2} = 1 + 2i in the hyperbolic half-plane model ℍ.

(b) Find the hyperbolic length of the Euclidean (usual) line segment connecting points z_{1} and z_{2}.

Problem 2. ( pts) Consider vectors u = (3, 1, 2), v = (3, 2, 1), and w = (1, 1, 1) in the Lorenz space E^{1,2}.

(a) Which vectors represent point(s) and which ones line(s) in the hyperbolic plane ? Find the point(s) and line(s) and show their position on Klein’s model.

(b) Find the distance between the points.

(c) Find the distance between the points and line(s).

Problem 3. ( pts) (a) Sketch the hyperbolic triangle ABC on the half-plane model ℍ, where A = 0, B = 1 + i, C = −1 + i.

(b) Find the angles of ABC.

(c) Find the area of ABC.

Problem 4. ( pts) (a) Determine the mutual position of the lines on Klein’s model, represented by vectors a = (1, 2, 1) and b = (1, 1, 2) of the Lorenz space (apply the criterion using the Gram matrix).

(b) Sketch these lines. Find the hyperbolic angle between them.

Problem 5. ( pts)