MATH 20E Parameterizations + Trig

What is the parametrization of a cone?
& Tangent vectors & Normal vector & Norm

Φ(r,θ) = (r cosθ, r sinθ, r) r ∈ [0,R], θ ∈ [0,2π]

Q: Tangent vectors? A: T_r = (cosθ, sinθ, 1) and T_θ = (−r sinθ, r cosθ, 0)

Q: Normal vector? A: T_r × T_θ = (−r cosθ, −r sinθ, r)

Q: Norm? A: r√2

What is the parametrization of a helicoid?

& Tangent vectors & Normal vector & Norm

Φ(r,θ) = (r cosθ, r sinθ, θ) r ∈ [0,R], θ ∈ [0,2π]

Q: Tangent vectors? A: T_r = (cosθ, sinθ, 0) and T_θ = (−r sinθ, r cosθ, 1)

Q: Normal vector? A: T_r × T_θ = (sinθ, −cosθ, r)

Q: Norm? A: √(1 + r²)

What is the parametrization of a paraboloid?

& Tangent vectors & Normal vector & Norm

Φ(r,θ) = (r cosθ, r sinθ, r2 ) r ∈ [0,R], θ ∈ [0,2π]

Q: Tangent vectors? A: T_r = (cosθ, sinθ, 2r) and T_θ = (−r sinθ, r cosθ, 0)

Q: Normal vector? A: T_r × T_θ = (−2r² cosθ, −2r² sinθ, r)

Q: Norm? A: r√(1 + 4r²)

What is the parametrization of a sphere?

& Tangent vectors & Normal vector & Norm

Φ(θ,φ) = (R sinφ cosθ, R sinφ sinθ, R cosφ), θ ∈ [0,2π], φ ∈ [0,π]

Q: Tangent vectors? A: T_θ = (−R sinφ sinθ, R sinφ cosθ, 0) and T_φ = (R cosφ cosθ, R cosφ sinθ, −R sinφ)

Q: Normal vector? A: T_θ × T_φ = (−R² sin²φ cosθ, −R² sin²φ sinθ, −R² sinφ cosφ)

Q: Norm? A: R² sinφ

What is the parametrization of a ellipsoid?

& Tangent vectors & Normal vector & Norm

A: Φ(θ,φ) = (a sinφ cosθ, b sinφ sinθ, c cosφ), θ ∈ [0,2π], φ ∈ [0,π]

Q: Tangent vectors? A: T_θ = (−a sinφ sinθ, b sinφ cosθ, 0) and T_φ = (a cosφ cosθ, b cosφ sinθ, −c sinφ)

Q: Normal vector? A: T_θ × T_φ = (−bc sin²φ cosθ, −ac sin²φ sinθ, −ab sinφ cosφ)

Q: Norm? A: sinφ √(b²c² sin²φ cos²θ + a²c² sin²φ sin²θ + a²b² cos²φ)

What is the parametrization of a hyperboloid?

& Tangent vectors & Normal vector & Norm

Φ(θ,z) = (√(R²+z²) cosθ, √(R²+z²) sinθ, z), θ ∈ [0,2π], z ∈ [−H,H]

Q: Tangent vectors? A: T_θ = (−√(R²+z²) sinθ, √(R²+z²) cosθ, 0) and T_z = (z cosθ/√(R²+z²), z sinθ/√(R²+z²), 1)

Q: Normal vector? A: T_θ × T_z = (√(R²+z²) cosθ, √(R²+z²) sinθ, −z)

Q: Norm? A: √(R² + 2z²)

What is the parametrization of a plane?

& Tangent vectors & Normal vector & Norm

Φ(u,v) = P₀ + u·e₁ + v·e₂, where e₁, e₂ ⊥ (a,b,c)

Q: Tangent vectors? A: T_u = e₁ and T_v = e₂ (unit vectors in the plane)

Q: Normal vector? A: (a, b, c)/√(a²+b²+c²)

Q: Norm? A: 1 (if e₁, e₂ are orthonormal)

What is the parametrization of a cylinder?

& Tangent vectors & Normal vector & Norm

Φ(θ,z) = (R cosθ, R sinθ, z), θ ∈ [0,2π], z ∈ [z₁,z₂]

Q: Tangent vectors? A: T_θ = (−R sinθ, R cosθ, 0) and T_z = (0, 0, 1)

Q: Normal vector? A: T_θ × T_z = (R cosθ, R sinθ, 0) — points outward ✓

Q: Norm? A: R

What is the parametrization of the graph of a function?

& Tangent vectors & Normal vector & Norm

Φ(x,y) = (x, y, g(x,y)), (x,y) ∈ D

Q: Tangent vectors? A: T_x = (1, 0, g_x) and T_y = (0, 1, g_y)

Q: Normal vector? A: T_x × T_y = (−g_x, −g_y, 1) — always points upward ✓

Q: Norm? A: √(1 + g_x² + g_y²)