MATH3061 Geometry and Topology – Semester 2, 2022 – Geometry Assignment
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MATH3061 Geometry and Topology – Semester 2, 2022 – Geometry Assignment
Question 1. Let α : ε → ε be defined by α(x, y) = (6 - 2x, -y).
(a) Prove that α is a transformation.
(b) Find all fixed points of α .
(c) Is α an involution, isometry, affine transformation?
Question 2. Let Q = (2, 1) and v = ┌ ┐6(2) .
(a) Find the Cartesian equation of the line l through Q in direction v.
(b) Find the Cartesian equation of the line m such that ρQ, −T/2 = σe σm .
(c) What is the isometry Tv О ρQ, −T/2 О T1 ?
Question 3. Denote by a, b the lines with the Cartesian equations y = 0, x = 0 respectively,
and i = ┌ ┐0(1) , j = ┌ ┐1(0) .
(a) Find the functions f(x, y) and g(x, y) such that γa,i(x, y) = (f(x, y), g(x, y)). (b) If γa,i(P) = Q, show that the midpoint of the segment PQ belongs to a.
(c) What is the isometry γa,iγb,j?
Question 4. Let A, B , C be the vertices of an equilateral triangle in the plane.
(a) Describe all isometries which map the set {A, B, C} to itself. (Your answer should consist
of a list specifying the reflections, rotations, translations and/or glide-reflections with the requested property.)
(b) Is the set ζ of all those isometries a group?
(c) Find all subsets of ζ which are groups.
2022-08-27