Published March 13, 2026
| Version v1
Journal article
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From Shadows to Shapes: Reconstructing Elliptic Intersection Curves from Orthogonal Projection Data
Authors/Creators
Description
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)
Files
From_Shadows_to_Shapes_v3.pdf
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Additional details
Software
- Repository URL
- https://github.com/dogan1908/ellipticshadow
References
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