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Movements, leaders, victories and the continuing fight for equality.
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Meet the people whose lives, choices and achievements shaped the journey.
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Katherine Johnson’s mathematical calculations helped guide some of America’s most important early space missions while she confronted the racial and gender barriers faced by Black women in twentieth-century America.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
The Warnock algorithm is a hidden-surface algorithm invented by John Warnock that is typically used in the field of computer graphics.[1] It solves the problem of rendering a complicated image by recursive subdivision of a scene until areas are obtained that are trivial to compute. In other words, if the scene is simple enough to compute efficiently then it is rendered; otherwise it is divided into smaller parts which are likewise tested for simplicity.[2]
This is a divide and conquer algorithm with run-time of [dubious – discuss], where n is the number of polygons and p is the number of pixels in the viewport.
The inputs are a list of polygons and a viewport. The best case is that if the list of polygons is simple, then draw the polygons in the viewport. Simple is defined as one polygon (then the polygon or its part is drawn in appropriate part of a viewport) or a viewport that is one pixel in size (then that pixel gets a color of the polygon closest to the observer). The continuous step is to split the viewport into 4 equally sized quadrants and to recursively call the algorithm for each quadrant, with a polygon list modified such that it only contains polygons that are visible in that quadrant.
Warnock expressed his algorithm in words and pictures, rather than software code, as the core of his PhD thesis, which also described protocols for shading oblique surfaces and other features that are now the core of 3-dimensional computer graphics. The entire thesis was only 26 pages from Introduction to Bibliography.
The algorithm was Warnock's doctoral thesis., 32 pages
Source: Wikipedia. Article content is retrieved live through the MediaWiki API.
The Warnock algorithm is a hidden-surface algorithm invented by John Warnock that is typically used in the field of computer graphics. It solves the problem of rendering a complicated image by recursive subdivision of a scene until areas are obtained that are trivial to compute. In other words, if the scene is simple enough to compute efficiently then it is rendered; otherwise it is divided into smaller parts which are likewise tested for simplicity. This is a divide and conquer algorithm with run-time of O ( n p ) {\displaystyle O(np)} , where n is the number of polygons and p is the number of pixels in the viewport. The inputs are a list of polygons and a viewport. The best case is that if the list of polygons is simple, then draw the polygons in the viewport. Simple is defined as one polygon (then the polygon or its part is drawn in appropriate part of a viewport) or a viewport that is one pixel in size (then that pixel gets a color of the polygon closest to the observer). The continuous step is to split the viewport into 4 equally sized quadrants and to recursively call the algorithm for each quadrant, with a polygon list modified such that it only contains polygons that are visible in that quadrant. Warnock expressed his algorithm in words and pictures, rather than software code, as the core of his PhD thesis, which also described protocols for shading oblique surfaces and other features that are now the core of 3-dimensional computer graphics. The entire thesis was only 26 pages from Introduction to Bibliography.
In mathematical analysis, the Whitney covering lemma, or Whitney decomposition, asserts the existence of a certain type of partition of an open set in a Euclidean space. Originally it was employed in the proof of Hassler Whitney's extension theorem. The lemma was subsequently applied to prove generalizations of the Calderón–Zygmund decomposition. Roughly speaking, the lemma states that it is possible to decompose an open set by cubes each of whose diameters is proportional, within certain bounds, to its distance from the boundary of the open set. More precisely: The main idea of the proof is to use a quadtree (in two dimensions), octree (in three dimensions), or their higher dimensional equivalents. This is a recursive subdivision of space into cubes, which may each be subdivided into 2 d {\displaystyle 2^{d}} smaller cubes, so that the edge lengths are (positive or negative) powers of two and the vertex coordinates are integer multiples of the edge length. To obtain the Whitney covering, one may start with a subdivision of space into unit cubes, and then repeatedly remove any cube exterior to the given set, subdivide any cube that contains both interior and exterior points of the set or that has a too-small neighboring cube, and un-subdivide any cube for which this simplification would leave the result interior to the given set and not having any too-small neighbors, until these operations are exhausted.
In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals. Subdivision rules have been used in architecture, biology, and computer science, as well as in the study of hyperbolic manifolds. Substitution tilings are a well-studied type of subdivision rule.
In the field of 3D computer graphics, a subdivision surface (commonly shortened to SubD surface or Subsurf) is a curved surface represented by the specification of a coarser polygon mesh and produced by a recursive algorithmic method. The curved surface, the underlying inner mesh, can be calculated from the coarse mesh, known as the control cage or outer mesh, as the functional limit of an iterative process of subdividing each polygonal face into smaller faces that better approximate the final underlying curved surface. Less commonly, a simple algorithm is used to add geometry to a mesh by subdividing the faces into smaller ones without changing the overall shape or volume. The opposite operation is called un-subdividing, a type of mesh decimation that works on subdivided geometry.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Shirley Chisholm, elected in 1968.
