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In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are equal, and every tangent vector is a principal direction .
The approximate field of view of an individual human eye (measured from the fixation point, i.e., the point at which one's gaze is directed) varies by facial anatomy, but is typically 30° superior (up, limited by the brow), 45° nasal (limited by the nose), 70° inferior (down), and 100° temporal (towards the temple).
Away from umbilical points, these two points of the focal surface are distinct; at umbilical points the two sheets come together. When the surface has a ridge the focal surface has a cuspidal edge, three such edges pass through an elliptical umbilic and only one through a hyperbolic umbilic. [3]
The set of ridge points form curves on the surface called ridges. The ridges of a given surface fall into two families, typically designated red and blue, depending on which of the two principal curvatures has an extremum. At umbilical points the colour of a ridge will change from red to blue. There are two main cases: one has three ridge lines ...
Apposition eyes work by gathering a number of images, one from each eye, and combining them in the brain, with each eye typically contributing a single point of information. The typical apposition eye has a lens focusing light from one direction on the rhabdom, while light from other directions is absorbed by the dark wall of the ommatidium .
African buffalo herds actually use a form of voting when trying to decide which direction to travel in. One at a time, adult females will stand up and look in a certain direction before sitting ...
The geodesic distance between opposite umbilical points is the same regardless of the initial direction of the geodesic. Whereas the closed geodesics on the ellipses X = 0 and Z = 0 are stable (a geodesic initially close to and nearly parallel to the ellipse remains close to the ellipse), the closed geodesic on the ellipse Y = 0 , which goes ...
A cross section of the surface forms a deltoid. The umbilic torus occurs in the mathematical subject of singularity theory , in particular in the classification of umbilical points which are determined by real cubic forms a x 3 + 3 b x 2 y + 3 c x y 2 + d y 3 {\displaystyle ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}} .