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Γ
f(x, y) = 0 f : R 2 → R f ∈ C 1 (R 2 )
(x0, y0) (x0, y0) ∈ R 2 Γ
f(x0, y0) = 0
x = ρ cos θ y = ρ sin θ g(ρ, θ) =
f(ρ cos θ, ρ sin θ)
{(ρ cos θ, ρ sin θ) : g(ρ, θ) = 0, ρ ≥ 0, θ ∈ [0, 2π]}.
g(ρ, θ) = 0
ρ ≥ 0 θ g(ρ, θ) = 0 ρ < 0
g(ρ, θ) = 0 ρ = h(θ)
A ⊆ [0, 2π] h(θ) ≥ 0 θ
θ ∗ ∈ A
cos(θ ∗ )y = sin(θ ∗ )x Γ α ∈ A h(α) = 0
cos(α)y = sin(α)x Γ
h ρ
f Γ f ρ Γ
Γ
g(ρ, θ) = ρ 3 − ρ 2 (cos 2 θ + 1)
g(ρ, θ) = 0 ρ = h(θ) h(θ) = cos 2 θ + 1 ρ ≥ 0
(0, 0) ρ = 0 g(0, θ) = 0
ρ = h(θ) g(ρ, θ) = ρ 3 − ρ 2 (cos 2 θ − 1)
g(ρ, θ) = 0 ρ = h(θ) h(θ) = cos 2 θ − 1 ρ ≥ 0
θ = 0, π
A = [0, π/2] A = [0, π/3]
π/3
y = tan(π/3)x π/2 /∈ A Γ
π/2 /∈ A 3π/2 /∈ A Γ