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Every non-degenerate conic section arises from a plane intersecting a double-napped right circular cone. The curve type is determined solely by the cutting angle, and all are unified by eccentricity (e):
- Circle: e = 0 (perfect symmetry, special ellipse)
- Ellipse: 0 < e < 1 (closed, bounded)
- Parabola: e = 1 (marginal escape trajectory)
- Hyperbola: e > 1 (open, dual-branched)
All satisfy the same quadratic equation
Ax² + Bxy + Cy² + Dx + Ey + F = 0
yet produce entirely different topologies.
Green's Theorem states that the line integral around a positively oriented, piecewise smooth, simple closed curve C is equal to the double integral of the curl over the planar region D it encloses:
∮_C (P dx + Q dy) = ∬_D (∂Q/∂x − ∂P/∂y) dx dy
By breaking the boundary into manageable segments (C₁, C₂, C₃, C₄) and integrating over [a, b], we convert the macroscopic flow around the boundary into the total microscopic circulation (rotation) throughout the entire area.