In mathematics, a normal surface is a surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection is a triangle or a quad (see figure). A triangle cuts off a vertex of the tetrahedron while a quad separates pairs of vertices. A normal surface may have many components of intersection, called normal disks, with one tetrahedron, but no two normal disks can be quads that separate different pairs of vertices since that would lead to the surface self-intersecting.
A normal surface intersects a tetrahedron in (possibly many) triangles (see above left) and quads (see above right)
Dually, a normal surface can be considered to be a surface that intersects each handle of a given handle structure on the 3-manifold in a prescribed manner similar to the above.
The concept of normal surface can be generalized to arbitrary polyhedra. There are also related notions of almost normal surface and spun normal surface.
The concept of normal surface is due to Hellmuth Kneser, who utilized it in his proof of the prime decomposition theorem for 3-manifolds. Later Wolfgang Haken extended and refined the notion to create normal surface theory, which is at the basis of many of the algorithms in 3-manifold theory. The notion of almost normal surfaces is due to Hyam Rubinstein. The notion of spun normal surface is due to Bill Thurston.
Regina is software which enumerates normal and almost-normal surfaces in triangulated 3-manifolds, implementing Rubinstein's 3-sphere recognition algorithm, among other things.
In mathematics, a normalsurface is a surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection...
In mathematics, a rational normal scroll is a ruled surface of degree n in projective space of dimension n + 1. Here "rational" means birational to projective...
and Z coordinates, respectively, of the surfacenormal. In 1978 Jim Blinn described how the normals of a surface could be perturbed to make geometrically...
following is a list of the 100 largest lakes of the United States by normalsurface area. The top twenty lakes in size are as listed by the National Atlas...
In mechanics, the normal force F n {\displaystyle F_{n}} is the component of a contact force that is perpendicular to the surface that an object contacts...
the area of a surface described this way. One can recognize the vector in the second-last line above as the normal vector to the surface. Because of the...
Haken manifolds, in normalsurface theory, and in the study of the fundamental groups of 3-manifolds. Let S be a compact surface properly embedded in...
the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted Φ or...
vectors is a normal vector to the tangent plane. Dividing this vector by its length yields a unit normal vector to the parametrized surface at a regular...
use as a normalsurface plate in high-tolerance production applications because of thermal expansion. The nature and use of a master surface, by contrast...
the surface. The normal curvature, kn, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surfacenormal u;...
a unit normal to the surface. Examining the definition shows that the surface gradient is the (conventional) gradient with the component normal to the...
by damming the Kaskaskia River at Shelbyville, Illinois. The lake's normalsurface pool is 11,100 acres (44.9 km2) at an elevation of 183 meters (600.4 ft)...
the true geometric normal of the surface. Commonly, it is computed as the normalized average of the surfacenormals of the faces that contain that vertex...
dimensions. For a finite planar surface of scalar area S and unit normal n̂, the vector area S is defined as the unit normal scaled by the area: S = n ^ S...
Surface force denoted fs is the force that acts across an internal or external surface element in a material body. Normal forces and shear forces between...
wrinkles on the surface of an object. This is achieved by perturbing the surfacenormals of the object and using the perturbed normal during lighting...
unit (AU). Direct normal irradiance (DNI), or beam radiation, is measured at the surface of the Earth at a given location with a surface element perpendicular...
of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have...
is the angle formed between the vector perpendicular (or normal) to the ellipsoidal surface from the point, and the plane of the equator. Two levels of...
Phong shading, Phong interpolation, or normal-vector interpolation shading is an interpolation technique for surface shading invented by computer graphics...
lead to complex patterns of surface faulting, such as in the 1992 Landers earthquake. Surface ruptures associated with normal faults are typically simple...
that contains the normal with the surface will form a curve that is called a normal section, and the curvature of this curve is the normal curvature. For...
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with...
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