Download F18PA2 Number Theory and Geometry: Tutorial 9

F18PA2 Number Theory and Geometry: Tutorial 9
1.
Consider the isometry f with matrix equation
√ √
2/2
−
x
1
x
√ 2/2
+
.
f:
7→ √
y
1
y
2/2
2/2
(a) Find f (0, 0).
(b) What happens to the the x and y-axes when the map f is applied?
(c) Find the unique fixed point of f .
(d) What is the distance between (1, 1) and f (300, 400) ?
2.
(a) A half-turn is a rotation by angle π with centre some point C = (p, q) ∈ R2 . We
denote it by hC . Find a formula for hC using the following facts:
• hC is a direct isometry,
• the angle θ in the formula for hC is equal to π,
• hC fixes C.
(b) Let P, Q, R be points on the x–axis in R2 . Show that the composition hP ◦ hQ ◦ hR
of half-turns is another half-turn hS and describe S in terms of P, Q, R.
3.
(a) Find a direct isometry f mapping (1, 2) 7→ (3, 5) and (4, 1) 7→ (6, 4).
(b) Find an opposite isometry g that does the same thing.
Sketch the effects of these isometries on the reference triangle.
4.
(a) Find a direct isometry f mapping (2, 3) 7→ (7, 1) and (4, −1) 7→ (3, 3). Show that f
is a rotation, and find its centre.
(b) Find an opposite isometry g that does the same thing. Show that g is a glide, and
find the equation of its axis.
Sketch the effects of these isometries on the reference triangle.
5.
Find an opposite isometry g that maps (1, 1) to (3, 7) and (3, 3) to (5, 5). Show that g is
a glide and find the equation of its axis.
Sketch the effect of this isometry on the reference triangle.
6.
A general opposite isometry g has the form
x
cos θ sin θ
x
a
g
=
+
,
y
sin θ − cos θ
y
b
for some θ, a, b. Write down the matrix form of the equation for the fixed points (x0 , y0 )
of g, and show that for any given θ, this equation either has no solution or infinitely many
solutions, depending on the value of (a, b).
Explain/interpret this result geometrically.
[Hint: apply a bit of standard linear algebra to the two-by-two system of fixed point
equations.]