Canal Surface
Canal Surface is a derived surface, derived from another surface in the catalogue, immersed.
Open Canal Surface in the interactive viewer →
aperiodic def-derived derived immersed implemented tradition-classical
Properties
- Family
- derived
- Given by
- another surface in the catalogue
- Curvature
- no condition imposed
- Periodicity
- not periodic
- Ends
- 0
- Embedding
- immersed
- Fidelity
- exact
Definition
The envelope of the one-parameter family of spheres of radius r(s) centred on the spine c(s): each sphere touches the envelope along a characteristic circle of radius r sqrt(1 - r'^2), centred at c - r r' T and tilted asin(r') out of the normal plane. The operator's spine presets are specimens; a selected curve object can also serve as the spine.
Sources
Derived from the Math Art generator registries and cross-referenced to the literature; not transcribed from any gallery's compilation.
- G. Monge, "Application de l'analyse a la geometrie" (1807) -- canal surfaces as envelopes of one-parameter families of spheres, with the characteristic-circle analysis.
- C. Dupin, "Applications de geometrie et de mechanique" (1822) -- the cyclides as the surfaces that are canal surfaces in two ways.
- V. Chandru, D. Dutta and C. M. Hoffmann, "On the geometry of Dupin cyclides", The Visual Computer 5 (1989), 277-290 -- the sphere-family construction and the (a, b, d) quartic normal form used by the self-test.
- R. Ferreol, "Encyclopedie des formes mathematiques remarquables" (mathcurve.com), "Surface tubulaire / surface canal".
- G. Monge; see R. Ferreol, mathcurve, 'surface canal'.