pdf (it, 2.898,97 KB, 25/01/13)

a ∈ R R 2
(x 2 + y 2 ) 3 = 4a 2 x 2 y 2 .
a = 0 a = 0 f(x, y) = (x 2 + y 2 ) 3 −
4a 2 x 2 y 2
f(ρ cos θ, ρ sin θ) = ρ 6 − 4a 2 ρ 4 cos θ sin θ = ρ 4 (ρ 2 − a 2 sin 2 2θ).
(0, 0) ρ 2 = a 2 sin 2 2θ
ρ ≥ 0 θ = π ρ = 0 ρ(θ) = |a|| sin 2θ|
ρ θ = π/4+kπ/2 k = 1, 2, 3, 4
|a| |a|
(|a| √ 2/2, |a| √ 2/2)
| sin(2(θ + π/2))| = | sin(2θ + π)| = | sin(2θ)|
π/2 0 < θ < π/4
x(θ) = ρ(θ) cos θ 0 < θ < π/4
x(θ) = |a| sin 2θ cos θ = 2|a| sin θ cos 2 θ = 2|a|(sin θ − sin 3 θ)
˙x(θ) = 2|a| cos θ(1 − 3 sin 2 θ).
0 < θ < π/4 sin θ = 1/ √ 3 cos θ = 2/3
x θ x ∗ = ρ ∗ cos θ ∗ = ρ(θ ∗ ) cos θ ∗ = |a|4/(3 √ 3)
ρ ∗ sin θ ∗ = |a|(2/3) 3/2 P = (|a|4/(3 √ 3), |a|(2/3) 3/2 )
(±|a|4/(3 √ 3), ±|a|(2/3) 3/2 )
(±|a|(2/3) 3/2 , ±|a|4/(3 √ 3)) 0 < θ < θ ∗
y(θ) x(θ)
Γ (x(θ), y(θ)) 0 < θ < θ ∗ y = ϕ(x)
θ ∗ < θ < π/4
x(θ) y(θ)
P a = 1