Focal surface

Focal surface

For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the curvature spheres, which are the tangential spheres whose radii are the reciprocals of one of the principal curvatures at the point of tangency. Equivalently it is the surface formed by the centers of the circles which osculate the curvature lines.

As the principal curvatures are the eigenvalues of the second fundamental form, there are two at each point, and these give rise to two points of the focal surface on each normal direction to the surface. Away from umbilical points, these two points of the focal surface are distinct; at umbilical points the two sheets come together. At points where the Gaussian curvature is zero, one sheet of the focal surface will have a point at infinity corresponding to the zero principal curvature.

pecial cases

The sphere is the only surface where both sheets of the focal surface degenerate to a single point.

Both sheets of the focal surface of Dupin cyclides form degenerate circles. For the torus one of these are is the straight line along the axis of symmetry.

One sheet of the focal surface of a channel surface will form a degenerate curve. Such surfaces includes all surfaces of revolution, where the degenerate curve is the axis of revolution.

ee also

*Focus (optics)
*Evolute


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  • focal surface — židinių paviršius statusas T sritis fizika atitikmenys: angl. focal surface vok. Brennfläche, f; Fokalfläche, f rus. фокальная поверхность, f pranc. surface focale, f …   Fizikos terminų žodynas

  • Focal — may refer to:*Focal (lexicographical website), an Irish lexicographical website *FOCAL (programming language) **FOCAL 69 **Focal (HP 41) *Relating to focusee also*Focal point (disambiguation) *Focal adhesion *Focal character *Focal dystonia… …   Wikipedia

  • surface focale — židinių paviršius statusas T sritis fizika atitikmenys: angl. focal surface vok. Brennfläche, f; Fokalfläche, f rus. фокальная поверхность, f pranc. surface focale, f …   Fizikos terminų žodynas

  • focal — focal, ale, aux [ fɔkal, o ] adj. et n. f. • 1812; du lat. focus « foyer » 1 ♦ Phys. Qui concerne le foyer, les foyers d un instrument d optique. ⇒ bifocal. Axe focal, plan focal. N. f. Distance focale. Objectif à focale variable. ⇒ zoom. 2 ♦… …   Encyclopédie Universelle

  • Focal — Fo cal, a. [Cf. F. focal. See {Focus}.] Belonging to,or concerning, a focus; as, a focal point. [1913 Webster] {Focal distance, or length, of a lens or mirror} (Opt.), the distance of the focus from the surface of the lens or mirror, or more… …   The Collaborative International Dictionary of English

  • Focal distance of a telescope — Focal Fo cal, a. [Cf. F. focal. See {Focus}.] Belonging to,or concerning, a focus; as, a focal point. [1913 Webster] {Focal distance, or length, of a lens or mirror} (Opt.), the distance of the focus from the surface of the lens or mirror, or… …   The Collaborative International Dictionary of English

  • Focal distance or length of a lens or mirror — Focal Fo cal, a. [Cf. F. focal. See {Focus}.] Belonging to,or concerning, a focus; as, a focal point. [1913 Webster] {Focal distance, or length, of a lens or mirror} (Opt.), the distance of the focus from the surface of the lens or mirror, or… …   The Collaborative International Dictionary of English

  • focal length — leŋ(k)th n the distance of a focus from the surface of a lens or concave mirror symbol f * * * the distance between a lens and an object from which all rays of light are brought to a focus. Symbol f …   Medical dictionary

  • Focal length — The focal length of an optical system is a measure of how strongly it converges (focuses) or diverges (diffuses) light. A system with a shorter focal length has greater optical power than one with a long focal length. Thin lens approximation For… …   Wikipedia

  • Focal — Distance focale Cet article fait partie de la série Photographie …   Wikipédia en Français

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