For a surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the two circles whose radii correspond to the principal curvatures. For each point on the surface there will generally be two points, both on the normal to the surface and the set of all these points form the focal surface, which consist of two sheets. At umbilical points the two sheets will come together. At points where the Gaussian curvature is zero one sheet of the focal surface will have a point at infinity corresponding the the zero principal curvature. An open surface with X-, Y-, and Z-contours shown. ... Principal curvature is the inverse of the radius of the osculating circle. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ... Curvature is the amount by which a geometric object deviates from being flat. ...
Special cases
The spheres is the only surfaces where both sheets of the focal surface degenerate to a single point. A sphere is a perfectly symmetrical geometrical object. ...
Both sheets of the focal surface of Cyclides form degenerate circles. For the torus one of these are is the straight line along the axis of symmetry. In mathematics, a cyclide is the geometric inversion of a torus. ... In geometry, a torus (pl. ...
One sheet of the focal surface of a canal surface will forms a degenerate curve. This family includes all surfaces of revolution. A canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve. ... The parabola y=x2 rotated about the z-axis A surface of revolution is a surface created by rotating a curve lying on some plane (the generatrix) around a straight line (the axis of rotation) that lies on the same plane. ...
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