This article presents a geometric framework for reconstructing the intersection curve of a parabolic cylinder and a cone of revolution from orthogonal projection data alone. The central contribution is the Elliptic Curve Reconstruction Theorem: the plan-view shadow (orthogonal projection onto the ground plane) of the cone–cylinder intersection is a quartic curve of the form y² = Q(x)²/m² − x², where Q(x) = ax² + bx + c is a quadratic height function and m is the cone slope. Three cross-line measurements at distinct x-positions uniquely determine Q(x) via a Vandermonde system, enabling full curve reconstruction without elevation data. Supporting results include: - Chord–Sagitta Relation: Determination of parabola curvature from chord length and sagitta height - Cross-Line Theorem: Vertical chord lengths in the projection encode the quadratic height function - Projection–Correlation Theorem: Coupling between parabola curvature a, cone radius R, and cone slope m via m = 2aR The geometry is developed from an interactive Desmos 3D model with concrete numerical parameters validated against GeoGebra measurements. All figures use exact coordinate values from the model. A speculative appendix explores transferring the quartic to finite fields, defines the Geometric Decomposition Problem (GDP), and honestly assesses its hardness limitations alongside proposed hardening directions. Interactive Model: https://www.desmos.com/3d/rgp89wkwoa ## Keywords parabolic cylinder, cone of revolution, conic sections, orthogonal projection, curve reconstruction, elliptic curves, quartic curves, Vandermonde system, chord-sagitta relation, descriptive geometry, projection correlation, Desmos 3D ## License Creative Commons Attribution 4.0 International (CC BY 4.0) ## Upload Type Publication → Journal article ## Related Identifiers - https://www.desmos.com/3d/rgp89wkwoa (Supplement: Interactive 3D Model) - https://github.com/dogan1908/ellipticshadow (Supplement: Source Repository)