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PMATH 321 Non-Euclidean Geometry

创建日期：2024-04-13 17:48:21

所属分类：物理Physical

标签： PMATH 321

Non-Euclidean Geometry

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PMATH 321 Non-Euclidean Geometry

Read Chapter 3 in the Lecture Notes, then solve the following problems.
1: (a) Let L = Span
{
(−1, 1, 3), (2,−1,−2)} ⊆ R3, Let M = Null(1,−2, 2) ⊆ R3 and let N be the plane through
the origin in R3 which contains the line (x, y, z) = (1, 1, 2) + t(1,−1, 0). Let u = M ∩ N , v = N ∩ L and
w = L ∩M . Note that u, v and w are points in P2. Find dP (v, w), dP (w, u) and dP (u, v).
(b) Let U =
{
[x, y, z] ∈ P2 ∣∣ z 6= 0} and let φ : U ⊆ P2 → R2 be the gnomic projection φ([x, y, z]) = (xz , yz ).
Find the perimeter and the area of the inverse image under φ of the region R =
{
(u, v) ∈ R2∣∣0 ≤ u ≤ 1, 1 ≤ v}.
2: (a) Let u = 1√
2
(1,−1, 0) and v = 1√
6
(1, 1, 2). Find p ∈ S2 and θ ∈ [0, pi] such that Ru,piFv = Rp,θ as an
isometry on P2.
(b) Let a = (1, 0, 1), b = (0, 1, 1), c = (1, 1, 0) and d = (0, 1,−1). Find every isometry S on P2 with S([a]) = [c]
and S([b]) = [d]. Express each isometry in the form Rp,θ with p ∈ S2 and θ ∈ [0, pi].
3: (a) Let f(x, y) = y−x3. Find the homogenization F (x, y, z), find the dehomogenizations f1(y, z), f2(x, z) and
f3(x, y), sketch the zero sets Z(f1), Z(f2) and Z(f3) and sketch the zero set Z(F ).
(b) Let C be the cardioid in R2 given in polar coordinates by r = 1 + cos θ, that is
C =
{(
(1+cos θ) cos θ, (1+cos θ) sin θ
) ∈ R2 ∣∣∣ θ ∈ R}.
Show that there exists a polynomial f(x, y) such that C = Z(f).
(c) Find all the points p = [x, y, z] ∈ P2 which lie in Z(yz − x2 + 3xz) ∩ Z(xy + x2 − xz + 2z2).
4: (a) Find a, h ∈ R with h > 0 and θ ∈ (pi4 , pi2 ) such that when p = (a, 0, h) and u = (sin θ, 0, cos θ) and φ = pi4 ,
the double cone V (p, u, φ) intersects the xy-plane along the hyperbola x2 − 2y2 = 1.
(b) Find a, b, h ∈ R with h > 0 and θ ∈ [0, 2pi) such that when p = (a, b,−h) and u = 1√
2
(cos θ, sin θ, 1) and
φ = pi4 , the double cone V (p, u, φ) intersects the xy-plane along the parabola x
2 + y2 − 2xy − 6x+ 2y = 0.
(c) Determine whether there exists u ∈ S2 and φ ∈ (0, pi2 ) such that V (0, u, φ) = Z(x2 − yz).

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