Module II·Article III·~1 min read
Quadratic Forms and Classification of Surfaces
Lines and Planes
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Classification of Second Order Surfaces
General Theory
A second order surface in ℝ³: Σᵢⱼ aᵢⱼxᵢxⱼ + Σᵢ bᵢxᵢ + c = 0 (aᵢⱼ = aⱼᵢ).
Classification: by rotating and shifting the coordinates, we bring it to canonical form (using eigenvectors of the quadratic part).
Invariants: I₁ = tr A (sum of eigenvalues), I₂ = sum of 2×2 minors, I₃ = det A — do not depend on the choice of coordinate system.
Surfaces of Revolution
Rotating the curve f(x, z) = 0 about the z-axis: f(√(x²+y²), z) = 0.
A one-sheeted hyperboloid is a surface of revolution of a hyperbola. Any generator is a straight line (ruled surface). Used in architecture (cooling towers, Shukhov tower).
Connection with Complex Numbers
Stereographic projection maps the sphere onto the extended complex plane ℂ ∪ {∞}. Conformal automorphisms of the sphere are fractional linear transformations.
The Four Major Classes
Non-degenerate: ellipsoids, hyperboloids (one/two-sheeted), paraboloids (elliptic/hyperbolic), cones.
Cylinders: elliptic, hyperbolic, parabolic.
Degenerate: planes, lines, a point, the empty set.
The complete classification comprises 17 types, distinguished by the rank of the quadratic part and the extended matrix.
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