In the geometry of computer graphics, a vertex normal at a vertex of a polyhedron is a directional vector associated with a vertex, intended as a replacement to the true geometric normal of the surface. It is computed as the normalized average of the surface normals of the faces that contain that vertex; the average can be weighted for example by the area of the face or it can be unweighted. Vertex normals can be computed for polygonal approximations to surfaces such as NURBS, or specified explicitly for artistic purposes. Vertex normals are used in Gouraud Phong shading and other lighting models. Using vertex normals, much smoother shading than flat shading can be achieved. Specular highlight Per-pixel lighting
In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane. Angles are formed by the intersection of two planes in Euclidean and other spaces; these are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle is used to designate the measure of an angle or of a rotation; this measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation; the word angle comes from the Latin word angulus, meaning "corner".
Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow". Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, do not lie straight with respect to each other. According to Proclus an angle must be a relationship; the first concept was used by Eudemus. In mathematical expressions, it is common to use Greek letters to serve as variables standing for the size of some angle. Lower case Roman letters are used, as are upper case Roman letters in the context of polygons. See the figures in this article for examples. In geometric figures, angles may be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC is denoted ∠BAC or B A C ^. Sometimes, where there is no risk of confusion, the angle may be referred to by its vertex. An angle denoted, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign.
However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant, no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise angle from B to C, ∠CAB to the anticlockwise angle from C to B. An angle equal to 0° or not turned is called a zero angle. Angles smaller than a right angle are called acute angles. An angle equal to 1/4 turn is called a right angle. Two lines that form a right angle are said to be orthogonal, or perpendicular. Angles larger than a right angle and smaller than a straight angle are called obtuse angles. An angle equal to 1/2 turn is called a straight angle. Angles larger than a straight angle but less than 1 turn are called reflex angles. An angle equal to 1 turn is called complete angle, round angle or a perigon. Angles that are not right angles or a multiple of a right angle are called oblique angles; the names and measured units are shown in a table below: Angles that have the same measure are said to be equal or congruent.
An angle is not dependent upon the lengths of the sides of the angle. Two angles which share terminal sides, but differ in size by an integer multiple of a turn, are called coterminal angles. A reference angle is the acute version of any angle determined by subtracting or adding straight angle, to the results as necessary, until the magnitude of result is an acute angle, a value between 0 and 1/4 turn, 90°, or π/2 radians. For example, an angle of 30 degrees has a reference angle of 30 degrees, an angle of 150 degrees has a reference angle of 30 degrees. An angle of 750 degrees has a reference angle of 30 degrees; when two straight lines intersect at a point, four angles are formed. Pairwise these angles are named according to their location relative to each other. A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles, they are abbreviated as vert. opp. ∠s. The equality of vertically opposite angles is called the vertical angle theorem.
Eudemus of Rhodes attributed the proof to Thales of Miletus. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical Note, w
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is one of the five Platonic solids, it has 6 faces, 12 edges, 8 vertices. The cube is a square parallelepiped, an equilateral cuboid and a right rhombohedron, it is a regular square prism in three orientations, a trigonal trapezohedron in four orientations. The cube is dual to the octahedron, it has octahedral symmetry. The cube is the only convex polyhedron; the cube has four special orthogonal projections, centered, on a vertex, edges and normal to its vertex figure. The first and third correspond to the B2 Coxeter planes; the cube can be represented as a spherical tiling, projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not lengths. Straight lines on the sphere are projected as circular arcs on the plane. For a cube centered at the origin, with edges parallel to the axes and with an edge length of 2, the Cartesian coordinates of the vertices are while the interior consists of all points with −1 < xi < 1 for all i.
In analytic geometry, a cube's surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of edge length a: As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares and second powers. A cube has the largest volume among cuboids with a given surface area. A cube has the largest volume among cuboids with the same total linear size. For a cube whose circumscribing sphere has radius R, for a given point in its 3-dimensional space with distances di from the cube's eight vertices, we have: ∑ i = 1 8 d i 4 8 + 16 R 4 9 = 2. Doubling the cube, or the Delian problem, was the problem posed by ancient Greek mathematicians of using only a compass and straightedge to start with the length of the edge of a given cube and to construct the length of the edge of a cube with twice the volume of the original cube, they were unable to solve this problem, in 1837 Pierre Wantzel proved it to be impossible because the cube root of 2 is not a constructible number.
The cube has three uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123. The cube has three classes of symmetry, which can be represented by vertex-transitive coloring the faces; the highest octahedral symmetry Oh has all the faces the same color. The dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color; the lowest symmetry D2h is a prismatic symmetry, with sides alternating colors, so there are three colors, paired by opposite sides. Each symmetry form has a different Wythoff symbol. A cube has eleven nets: that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the same color, one would need at least three colors; the cube is the cell of the only regular tiling of three-dimensional Euclidean space. It is unique among the Platonic solids in having faces with an number of sides and it is the only member of that group, a zonohedron; the cube can be cut into six identical square pyramids.
If these square pyramids are attached to the faces of a second cube, a rhombic dodecahedron is obtained. The analogue of a cube in four-dimensional Euclidean space has a special name—a tesseract or hypercube. More properly, a hypercube is the analogue of the cube in n-dimensional Euclidean space and a tesseract is the order-4 hypercube. A hypercube is called a measure polytope. There are analogues of the cube in lower dimensions too: a point in dimension 0, a line segment in one dimension and a square in two dimensions; the quotient of the cube by the antipodal map yields the hemicube. If the original cube has edge length 1, its dual polyhedron has edge length 2 / 2; the cube is a special case in various classes of general polyhedra: The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron. These two together form the stella octangula; the int
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within society at large; the press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton, its first book was a new 1912 edition of John Witherspoon's Lectures on Moral Philosophy. Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two existing local publishers, that of the Princeton Alumni Weekly and the Princeton Press; the new press printed both local newspapers, university documents, The Daily Princetonian, added book publishing to its activities. Beginning as a small, for-profit printer, Princeton University Press was reincorporated as a nonprofit in 1910.
Since 1911, the press has been headquartered in a purpose-built gothic-style building designed by Ernest Flagg. The design of press’s building, named the Scribner Building in 1965, was inspired by the Plantin-Moretus Museum, a printing museum in Antwerp, Belgium. Princeton University Press established a European office, in Woodstock, north of Oxford, in 1999, opened an additional office, in Beijing, in early 2017. Six books from Princeton University Press have won Pulitzer Prizes: Russia Leaves the War by George F. Kennan Banks and Politics in America from the Revolution to the Civil War by Bray Hammond Between War and Peace by Herbert Feis Washington: Village and Capital by Constance McLaughlin Green The Greenback Era by Irwin Unger Machiavelli in Hell by Sebastian de Grazia Books from Princeton University Press have been awarded the Bancroft Prize, the Nautilus Book Award, the National Book Award. Multi-volume historical documents projects undertaken by the Press include: The Collected Papers of Albert Einstein The Writings of Henry D. Thoreau The Papers of Woodrow Wilson The Papers of Thomas Jefferson Kierkegaard's WritingsThe Papers of Woodrow Wilson has been called "one of the great editorial achievements in all history."
Princeton University Press's Bollingen Series had its beginnings in the Bollingen Foundation, a 1943 project of Paul Mellon's Old Dominion Foundation. From 1945, the foundation had independent status and providing fellowships and grants in several areas of study, including archaeology and psychology; the Bollingen Series was given to the university in 1969. Annals of Mathematics Studies Princeton Series in Astrophysics Princeton Series in Complexity Princeton Series in Evolutionary Biology Princeton Series in International Economics Princeton Modern Greek Studies The Whites of Their Eyes: The Tea Party's Revolution and the Battle over American History, by Jill Lepore The Meaning of Relativity by Albert Einstein Atomic Energy for Military Purposes by Henry DeWolf Smyth How to Solve It by George Polya The Open Society and Its Enemies by Karl Popper The Hero With a Thousand Faces by Joseph Campbell The Wilhelm/Baynes translation of the I Ching, Bollingen Series XIX. First copyright 1950, 27th printing 1997.
Anatomy of Criticism by Northrop Frye Philosophy and the Mirror of Nature by Richard Rorty QED: The Strange Theory of Light and Matter by Richard Feynman The Great Contraction 1929–1933 by Milton Friedman and Anna Jacobson Schwartz with a new Introduction by Peter L. Bernstein Military Power: Explaining Victory and Defeat in Modern Battle by Stephen Biddle Banks, Eric. "Book of Lists: Princeton University Press at 100". Artforum International. Staff of Princeton University Press. A Century in Books: Princeton University Press, 1905–2005. ISBN 9780691122922. CS1 maint: Uses authors parameter Official website Princeton University Press: Albert Einstein Web Page Princeton University Press: Bollingen Series Princeton University Press: Annals of Mathematics Studies Princeton University Press Centenary Princeton University Press: New in Print
Günter M. Ziegler
Günter Matthias Ziegler is a German mathematician, serving as president of the Free University of Berlin since 2018. Ziegler is known for his research in discrete mathematics and geometry, on the combinatorics of polytopes. Ziegler studied at the Ludwig Maximilian University of Munich from 1981 to 1984, went on to receive his Ph. D. from the Massachusetts Institute of Technology in Cambridge, Massachusetts in 1987, under the supervision of Anders Björner. After postdoctoral positions at the University of Augsburg and the Mittag-Leffler Institute, he received his habilitation in 1992 from the Technical University of Berlin, which he joined as a professor in 1995. Ziegler has since joined the faculty of the Free University of Berlin. Ziegler was awarded the one million Deutschmark Gerhard Hess Prize by the Deutsche Forschungsgemeinschaft in 1994 and the 1.5 million Deutschmark Gottfried Wilhelm Leibniz Prize, Germany's highest research honor, by the DFG in 2001. He was awarded the 2005 Gauss Lectureship by the German Mathematical Society.
In 2006 the Mathematical Association of America awarded Ziegler and Florian Pfender its highest honor for mathematical exposition, the Chauvenet Prize, for their paper on kissing numbers. In 2006 Ziegler became president of the German Mathematical Society for a two-year term. In 2009, the European Research Council awarded Ziegler one of the ERC Advanced Grants in the amount of 1.85 million Euros. In 2012 he became a fellow of the American Mathematical Society. In 2013 Ziegler was granted the Hector Science Award and became a member of the Hector Fellow Academy. Since 2016 Ziegler has been chair of the Berlin Mathematical School. In 2018 he received the Leroy P. Steele Prize for Mathematical Exposition. German Institute for Economic Research, Member of the Board of Trustees Genshagener Kreis, Member of the Board of Trustees Berlin Social Science Center, Member of the Board of Trustees Klaus Tschira Foundation, Member of the Board of Trustees Lesbian and Gay Federation in Germany, Member Proofs from THE BOOK, Berlin, 1998, ISBN 3-540-63698-6 Aigner, Martin.
Björner, Anders. "8 Introduction to greedoids". In White, Neil. Matroid Applications. Encyclopedia of Mathematics and its Applications. 40. Cambridge: Cambridge University Press. Pp. 284–357. Doi:10.1017/CBO9780511662041.009. ISBN 0-521-38165-7. MR 1165537. Drösser, Christoph, "Ein etwas anderer Streber", Die Zeit. Article in German about Ziegler. "Ziegler's homepage at the Free University". Günter M. Ziegler in the German National Library catalogue Günter M. Ziegler at the Mathematics Genealogy Project Günter M. Ziegler publications indexed by Google Scholar
Computer graphics are pictures and films created using computers. The term refers to computer-generated image data created with the help of specialized graphical hardware and software, it is a vast and developed area of computer science. The phrase was coined in 1960, by computer graphics researchers Verne Hudson and William Fetter of Boeing, it is abbreviated as CG, though sometimes erroneously referred to as computer-generated imagery. Some topics in computer graphics include user interface design, sprite graphics, vector graphics, 3D modeling, shaders, GPU design, implicit surface visualization with ray tracing, computer vision, among others; the overall methodology depends on the underlying sciences of geometry and physics. Computer graphics is responsible for displaying art and image data and meaningfully to the consumer, it is used for processing image data received from the physical world. Computer graphics development has had a significant impact on many types of media and has revolutionized animation, advertising, video games, graphic design in general.
The term computer graphics has been used in a broad sense to describe "almost everything on computers, not text or sound". The term computer graphics refers to several different things: the representation and manipulation of image data by a computer the various technologies used to create and manipulate images the sub-field of computer science which studies methods for digitally synthesizing and manipulating visual content, see study of computer graphicsToday, computer graphics is widespread; such imagery is found in and on television, weather reports, in a variety of medical investigations and surgical procedures. A well-constructed graph can present complex statistics in a form, easier to understand and interpret. In the media "such graphs are used to illustrate papers, theses", other presentation material. Many tools have been developed to visualize data. Computer generated imagery can be categorized into several different types: two dimensional, three dimensional, animated graphics; as technology has improved, 3D computer graphics have become more common, but 2D computer graphics are still used.
Computer graphics has emerged as a sub-field of computer science which studies methods for digitally synthesizing and manipulating visual content. Over the past decade, other specialized fields have been developed like information visualization, scientific visualization more concerned with "the visualization of three dimensional phenomena, where the emphasis is on realistic renderings of volumes, illumination sources, so forth with a dynamic component"; the precursor sciences to the development of modern computer graphics were the advances in electrical engineering and television that took place during the first half of the twentieth century. Screens could display art since the Lumiere brothers' use of mattes to create special effects for the earliest films dating from 1895, but such displays were limited and not interactive; the first cathode ray tube, the Braun tube, was invented in 1897 – it in turn would permit the oscilloscope and the military control panel – the more direct precursors of the field, as they provided the first two-dimensional electronic displays that responded to programmatic or user input.
Computer graphics remained unknown as a discipline until the 1950s and the post-World War II period – during which time the discipline emerged from a combination of both pure university and laboratory academic research into more advanced computers and the United States military's further development of technologies like radar, advanced aviation, rocketry developed during the war. New kinds of displays were needed to process the wealth of information resulting from such projects, leading to the development of computer graphics as a discipline. Early projects like the Whirlwind and SAGE Projects introduced the CRT as a viable display and interaction interface and introduced the light pen as an input device. Douglas T. Ross of the Whirlwind SAGE system performed a personal experiment in which a small program he wrote captured the movement of his finger and displayed its vector on a display scope. One of the first interactive video games to feature recognizable, interactive graphics – Tennis for Two – was created for an oscilloscope by William Higinbotham to entertain visitors in 1958 at Brookhaven National Laboratory and simulated a tennis match.
In 1959, Douglas T. Ross innovated again while working at MIT on transforming mathematic statements into computer generated 3D machine tool vectors by taking the opportunity to create a display scope image of a Disney cartoon character. Electronics pioneer Hewlett-Packard went public in 1957 after incorporating the decade prior, established strong ties with Stanford University through its founders, who were alumni; this began the decades-long transformation of the southern San Francisco Bay Area into the world's leading computer technology hub - now known as Silicon Valley. The field of computer graphics developed with the emergence of computer graphics hardware. Further advances in computing led to greater advancements in interactive computer graphics. In 1959, the TX-2 computer was developed at MIT's Lincoln Laboratory; the TX-2 integrated a number of new man-machine interfaces. A light pen could be used to draw sketches on the computer using Ivan Sutherland's revolutionary Sketchpad software.
Using a light pen, Sketchpad allowed one to draw simple shapes on the computer screen, save them and recall them later. The light pen itself had a small photoelectric cell in its tip. T
In elementary geometry, a polygon is a plane figure, described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon; the segments of a polygonal circuit are called its edges or sides, the points where two edges meet are the polygon's vertices or corners. The interior of a solid polygon is sometimes called its body. An n-gon is a polygon with n sides. A simple polygon is one. Mathematicians are concerned only with the bounding polygonal chains of simple polygons and they define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes; the word polygon derives from the Greek adjective πολύς "much", "many" and γωνία "corner" or "angle".
It has been suggested. Polygons are classified by the number of sides. See the table below. Polygons may be characterized by their convexity or type of non-convexity: Convex: any line drawn through the polygon meets its boundary twice; as a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex: a line may be found which meets its boundary more than twice. Equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple: the boundary of the polygon does not cross itself. All convex polygons are simple. Concave. Non-convex and simple. There is at least one interior angle greater than 180°. Star-shaped: the whole interior is visible from at least one point, without crossing any edge; the polygon must be simple, may be convex or concave. All convex polygons are star-shaped. Self-intersecting: the boundary of the polygon crosses itself.
The term complex is sometimes used in contrast to simple, but this usage risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions. Star polygon: a polygon which self-intersects in a regular way. A polygon can not be both star-shaped. Equiangular: all corner angles are equal. Cyclic: all corners lie on a single circle, called the circumcircle. Isogonal or vertex-transitive: all corners lie within the same symmetry orbit; the polygon is cyclic and equiangular. Equilateral: all edges are of the same length; the polygon need not be convex. Tangential: all sides are tangent to an inscribed circle. Isotoxal or edge-transitive: all sides lie within the same symmetry orbit; the polygon is equilateral and tangential. Regular: the polygon is both isogonal and isotoxal. Equivalently, it is both equilateral, or both equilateral and equiangular. A non-convex regular polygon is called a regular star polygon. Rectilinear: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees.
Monotone with respect to a given line L: every line orthogonal to L intersects the polygon not more than twice. Euclidean geometry is assumed throughout. Any polygon has as many corners; each corner has several angles. The two most important ones are: Interior angle – The sum of the interior angles of a simple n-gon is π radians or × 180 degrees; this is because any simple n-gon can be considered to be made up of triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular n-gon is 180 − 360 n degrees; the interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regular p q -gon, each interior angle is π p radians or 180 p degrees. Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex n-gon, the angle "turned" at a corner is external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°.
This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an n-gon in general, the sum of the exterior angles can be any integer multiple d of 360°, e.g. 720° for a pentagram and 0° for an angular "eight" or antiparallelogram, where d is the density or starriness of the polygon. See orbit. In this section, the vertices of the polygon under consideration are taken to be, ( x 1