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University of Toronto Mississauga
Instructor: Ester Dalvit
Department of Mathematics
TA: Jennifer Vaughan
Classical geometries - MAT402
Problem set 4 - Due October 21, 2014
You are encouraged to work in groups to solve the homework problems. If you do, please
indicate on your solutions who you worked with and if you used any sources. Write up your
final solutions on your own.
Exercise 1. (optional, will not be marked) Find a condition for two lines in the plane, given by cartesian
equations, to be perpendicular.
Find a condition for a plane and a line (given by cartesian equations) to be perpendicular in the space.
Explain how the second configuration is related to finding spherical circles with a given center on the sphere.
Exercise 2. Using analytical 3-dimensional geometry, show that a plane in R3 intersects a sphere in either
no points, one single point, or one circle.
Hint: consider the sphere of center O and radius 1 and a plane perpendicular to one of the coordinate axes.
Explain why this choice does not affect the generality of the discussion.
Exercise 3.
(1) Check that the map ϕ : R2 → R2 defined by ϕ(0, 2) = (2, −3), ϕ(2, −1) = (0, 0) and
ϕ(0, −1) = (2, 0) is an isometry.
(2) Find ϕ, either giving an orthonormal matrix A and a vector b such that ϕ(x) = Ax + b or as a
composition of reflections. Hint: for the latter draw a diagram and mimic the proof of the three
reflections theorem: to simplify calculations do not start with point (0, 2).
(3) What type of isometry is ϕ?
Exercise 4. In R2
(1) The map ϕ : (x, y) 7→ (−x, −y) is the reflection in the origin (it is not a reflection in a line). Show
that ϕ is a rotation of angle π. How can you define a reflection in a generic point? Hint: for the
second part use composition of translations and ϕ.
(2) Show that the set of rotations about a fixed point is a group under composition and that it is
commutative.
(3) (optional, will not be marked) Show that the composition ψ of a reflection in a line l and a
reflection in a line m perpendicular to l yields a rotation of angle π. Hint: show this for l, m the
coordinate axes; then consider lines such that l ∩ m = (0, 0) and say how to write ψ as composition
of ϕ and other rotations about O (using the commutativity shown in (b)); the general case follows
then from (a).
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(4) (optional, will not be marked) Let l, m be two incident lines. Denote sl and sm the reflections
in line l and m respectively. Show that sl and sm commute if and only if l and m are perpendicular.
Exercise 5. The distance in R3 between points P = (x1 , y1 , z1 ) and Q = (x2 , y2 , z2 ) is given by
p
|P Q| = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2
a) Describe the set of points equidistant to two points in R3 , justifying your answer.
b) Show that an isometry of R3 is determined by where it sends 4 points A, B, C, D as long as A, B, C, D
are not coplanar (do not lie on a single plane in R3 ). Hint: mimic the proof of the 3 reflections theorem
for R2 and use part (a).
A reflection in R3 is in a plane. For example, the reflection in the xy-plane is given by
rxy (x, y, z) = (x, y, −z)
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c) Show that any isometry of R is the composition of up to four reflections in planes. Hint: it works
just like the 2 dimensional case.
d) A screw transformation is a rotation about a line followed by a translation along that line (think of
turning a screw!). Describe qualitatively a set of reflections needed to produce a screw transformation
along the z-axis.
Hint: start by characterizing rotations about a line and translations as compositions of reflections
in planes (this is very similar to the 2-dimensional case).
Just for fun 1. An isometry: http: // xkcd. com/ 184/