Fresnel Wave Surface

Fresnel Wave Surface is an algebraic surface, defined by an implicit equation, immersed.

Open Fresnel Wave Surface in the interactive viewer →

algebraic aperiodic def-implicit immersed implemented tradition-classical tradition-physical

Formula

\left(x^{2} + y^{2} + z^{2}\right) \left(x^{2} + 1.44 y^{2} + 2.25 z^{2}\right) - 3.69 x^{2} - 4.68 y^{2} - 5.49 z^{2} + 3.2399999999999998 = 0

Properties

Family: algebraic
Given by: an implicit equation
Curvature: no condition imposed
Periodicity: not periodic
Ends: 0
Embedding: immersed
Fidelity: exact
Exactness: elementary

Definition

The quartic locus of light-wavefront propagation in a biaxial anisotropic crystal: two sheets, with four conical points that gave conical refraction. Distinct from Fresnel's ELASTICITY surface, which is already implemented.

Sources

Closed form reproduced numerically against the shipped implementation (math_art/surfaces/algebraic.py) over 240 sample points: matches oracle x 1 over 240 points (worst 0).

R. Ferreol, "Encyclopedie des formes mathematiques remarquables", mathcurve.com -- the chapters "parapluie de Cartan", "surface de Cassini", "surface de Titeica". A converted copy of the whole encyclopedia is in research/books/ mathcurve_encyclopedie_formes_mathematiques/.
H. Cartan and H. Whitney, 1957 -- the umbrella; cf. Whitney's own umbrella x^2 = y^2 z, already in the Hauser block above.
G. D. Cassini, the ovals of 1680; the surface is the classical three-dimensional generalisation of them.
G. Titeica, "Sur une nouvelle classe de surfaces", Rend. Circ. Mat. Palermo 25 (1908) 180-187 -- the affine spheres of 1907.
A. Fresnel (1821) -- the wave surface; the chapter "Fresnel's wave surface" prints the quartic and its four singular points.
A. Gray, "Modern Differential Geometry of Curves and Surfaces with Mathematica", 2nd ed., CRC Press (1997) -- named the sine surface.
R. Ferreol, Encyclopedie des formes mathematiques remarquables (mathcurve).
R. Ferreol, "Fresnel's wave surface", Encyclopedie des formes mathematiques remarquables, mathcurve.com; local mirror chapter ch1201_ondes_2 (S:/data/math_art/references/websites/mathcurve/book/surfaces/ch1201_ondes_2.md).
A. Fresnel (1821), per the chapter; the chapter also points to Jules Richard's thesis and [Loria 3d p. 197].