Hyperboloid
In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes. A hyperboloid is a surface that may be obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more of an affine transformation. A hyperboloid is a quadric surface, a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, a hyperboloid is characterized by not being a cone or a cylinder, having a center of symmetry, intersecting many planes into hyperbolas. A hyperboloid has three pairwise perpendicular axes of symmetry, three pairwise perpendicular planes of symmetry. Given a hyperboloid, if one chooses a Cartesian coordinate system whose axes are axes of symmetry of the hyperboloid, origin is the center of symmetry of the hyperboloid the hyperboloid may be defined by one of the two following equations: x 2 a 2 + y 2 b 2 − z 2 c 2 = 1, or x 2 a 2 + y 2 b 2 − z 2 c 2 = − 1.
Both of these surfaces are asymptotic to the cone of equation x 2 a 2 + y 2 b 2 − z 2 c 2 = 0. One has an hyperboloid of revolution if and only if a 2 = b 2. Otherwise, the axes are uniquely defined, but changing inclination v into hyperbolic trigonometric functions: One-surface hyperboloid: v ∈ x = a cosh v cos θ y = b cosh v sin θ z = c sinh v Two-surface hyperboloid: v ∈ [0, ∞) x = a sinh v cos θ y = b sinh v sin θ z = ± c cosh v A hyperboloid of one sheet contains two pencils of lines, it is a doubly ruled surface. If the hyperboloid has the equation x 2 a 2 + y 2 b 2 − z 2 c 2 = 1 the lines g α ±: x → = + t ⋅, t ∈ R, 0 ≤ α ≤ 2 π are contained in the surface. In cas
Cathedral of Saint Mary of the Assumption (San Francisco, California)
The Cathedral of Saint Mary of the Assumption known locally as Saint Mary's Cathedral, is the principal church of the Roman Catholic Archdiocese of San Francisco in San Francisco, California. It is the mother church of the Catholic faithful in the California counties of Marin, San Francisco and San Mateo and is the metropolitan cathedral for the Ecclesiastical province of San Francisco; the Cathedral Clergy includes Reverend Arturo Albano and Pastor, Reverend Sebastine Bula,VC, Parochial Vicar. Reverend Mr. R. Christoph Sandoval, senior Deacon and Reverend Mr. Alex Madero, Deacon; the cathedral is located in the Cathedral Hill neighborhood of San Francisco. The present cathedral replaced one of the same name; the original Cathedral of Saint Mary of the Immaculate Conception was built in 1853–1854 and still stands today. It is now known as Old Saint Mary's Church. In 1883, Archbishop Patrick W. Riordan purchased the northwest corner of Van Ness Avenue and O'Farrell Street in Western Addition. Riordan broke ground in December 1885.
On May 1, 1887 the archbishop placed the cornerstone. Archbishop Riordan dedicated the edifice to Saint Mary of the Assumption on January 11, 1891; the second cathedral served the Archdiocese of San Francisco for seventy-one years. During the episcopal terms of archbishops Riordan, Edward J. Hanna and John J. Mitty. Papal Secretary of State Eugenio Cardinal Pacelli, said Mass at the high altar in October 1936. On April 3, 1962, Joseph T. McGucken was installed as the fifth Archbishop of San Francisco in the cathedral on Van Ness Avenue. Five months the landmark was destroyed by arson on the night of September 7, 1962 The Very Rev. Msgr. John J. Prendergast, V. G. 1891–1913 The Right Rev. Charles Augustus Ramm, 1914–1948 The Most Rev. Hugh Aloysius Donohoe, V. G. 1948–1962 The Rt. Rev. Msgr. Thomas J. Bowe, 1962–1980 Rev. J. O’Shaughnessy, Administrator, 1979–1981, Rector 1981–1986 Rev. Patrick Joseph McGrath, 1986–1989 Rev. Milton T Walsh, 1989–1997 Monsignor John O’Connor, 1997–2002 Rev. Agnel Jose De Heredia, Administrator, 2002–2003 Monsignor John Talesfore, 2005–2015 Most Reverend William J. Justice Administrator, 2015 Reverend Arturo Albano, Current Catholic Directory Archdiocese of San Francisco The present cathedral was commissioned just as Vatican II was convening in Rome.
Monsignor Thomas J. Bowe served as first rector of the new cathedral from 1962 to 1980; the cornerstone was laid on December 13, 1967, the cathedral was completed three years later. On May 5, 1971, the cathedral was blessed and on October 5, 1996, was formally dedicated to the Blessed Virgin Mary under the name of Saint Mary of the Assumption; the first Papal Mass was celebrated by Pope John Paul II in the cathedral in 1987. It ran the private all-female Cathedral High School, in a building adjoined to the present-day cathedral itself. CHS merged with nearby all-male private Sacred Heart High School in 1987. St. Mary's Cathedral still has close ties to the resulting Sacred Heart Cathedral Preparatory, which uses the cathedral as its principal church for masses and other special events, such as graduation; as well, Junipero Serra High School in San Mateo use the cathedral to hold graduation. The cathedral was designed by local architects John Michael Lee, Paul A. Ryan and Angus McSweeney, collaborating with internationally known architects Pier Luigi Nervi and Pietro Belluschi — at the time, the Dean of the School of Architecture at MIT.
Measuring 255 feet square, the cathedral soars to 190 feet high and is crowned with a 55 feet golden cross. Its saddle roof is composed of eight segments of hyperbolic paraboloids, in such a fashion that the bottom horizontal cross section of the roof is a square and the top cross section is a cross; the design process was controversial. A preliminary design reminded one critic of "the effort of a camel and donkey to mate." After adding Bellushi and Nervi to the team, the situation improved, though the architects were accused of plagiarizing the design of the St. Mary's Cathedral in Tokyo, completed several years earlier. Over time, San Francisco's Catholics who had worshipped in traditional churches grew more fond of the modern design; the building was selected in 2007 by the local chapter of the American Institute of Architects for a list of San Francisco's top 25 buildings. In 2017, Architecture Digest named it one of the 10 most beautiful churches in the United States. List of Catholic cathedrals in the United States List of cathedrals in the United States Roman Catholic Marian churches Official Cathedral Site Roman Catholic Archdiocese of San Francisco Official Site Cathedral of Saint Mary of the Assumption via the Archdiocese of San Francisco Sacred Heart Cathedral Preparatory Cathedrals of California
Lee Valley VeloPark
Lee Valley VeloPark is a cycling centre on Queen Elizabeth Olympic Park in Stratford, East London. It is owned and managed by Lee Valley Regional Park Authority, it was opened to the public in March 2014; the facility was one of the permanent venues for Paralympic Games. Lee Valley VeloPark is at the northern end of Queen Elizabeth Olympic Park, it has a velodrome and BMX racing track, which have been used for the Games, as well as a one-mile road course and 5 miles of mountain bike trails. The park replaces the Eastway Cycle Circuit demolished to make way for it; the facilities built for the Olympics were constructed between 2009 and 2011. The first event in the Velopark was the London round of the 2011 UCI BMX Supercross World Cup series. In February 2005 plans were announced for a £22 million VeloPark. Sport England would invest £10.5 million, Lee Valley Regional Park Authority £6 million and the Mayor of London and Transport for London would invest £3 million and £2.5 million respectively. The site was to be 34 hectares on the northern end of the proposed Olympic Park, next to the A12.
The park would include a velodrome seating 1,500, which could be increased to 6,000 if London's bid for the 2012 Olympic and Paralympic Games were successful. The site would have an international competition BMX circuit, a BMX freestyle park, cyclo-cross/cross-country course mountain bike course and an outdoor cycle speedway track; the facilities would be used by internationals as well as those learning to ride. It was estimated that the park would attract 88,000 users a year, replacing the Eastway Cycle Circuit. Eastway Cycle Circuit opened in 1975, it was the first purpose built road cycling venue in Britain; the facility closed in September 2006 to make way for London's VeloPark. The velodrome is the third 250 m covered track in Great Britain. In September 2008 plans for the VeloPark were revealed. However, by March 2007, the VeloPark was revealed to be only a third of its original size, rescaled from 34 to 10 hectares; the decrease in the size of the site led to users of the Eastway cycle circuit to protest to the Mayor of London.
On 12 July 2007, the Olympic Delivery Authority selected the design team: Hopkins Architects, Expedition Engineering, BDSP, Grant Associates, following an architectural design competition managed by RIBA Competitions. The Velopark was scheduled to be completed by the contractor, ISG, in 2011. In 2004, during London's Olympic and Paralympic bid, the estimated cost was £37 million, including £20 million for the velodrome. In 2009, at the time work began on the construction of the velodrome, the estimated cost of that facility alone was £105 million. Work on the velodrome was completed in February 2011, was the first Olympic Park venue to be completed; the roof is designed to reflect the geometry of cycling as well as being lightweight and efficient reflecting a bike. There is a 360-degree concourse level with windows allowing people views of the Olympic Park; the velodrome is energy efficient—rooflights reduce the need for artificial lights, natural ventilation reduces the need for air condition.
Rain water is collected, which reduces the amount of water used from the municipal water system. Designer Ron Webb, who designed the velodrome tracks for the Sydney and Athens Games, was in charge of the design and installation of the track; the 250-metre track was made with 56 km of 350,000 nails. The velodrome was opened by many successful British athletes including Chris Hoy and Victoria Pendleton, it is informally known as "The Pringle" due to its distinctive shape. It was shortlisted for the 2011 RIBA Stirling Prize. and won the 2011 Structural Awards Supreme Award for Structural Engineering. In 2011, it won the Prime Minister's Better Public Building Award at British Construction Industry Awards; the venue was used for the first time in competition during the UCI Track Cycling World Cup in February 2012. The velodrome was used for the 2012 Paralympics; the outdoor BMX racing track was scheduled to have a spectator capacity of 6,000. Work began on its construction in March 2011. After the games the seating was removed and the track reconfigured to accommodate all abilities.
The first competition on the venue was the test event for the Olympic Games, a round of the 2011 UCI Supercross BMX World Cup series. The track for men is 470 metres long and features a berm jump, an S-bend transfer, a box jump and a rhythm section in the final straight; the women's course is 430 metres long featuring three jumps in the opening straight and a tunnel before like the men's including a rhythm section in the final straight. It has been called one of the most challenging BMX tracks to date; the track features an 8-metre high starting ramp and was designed by the UCI with the aim of pushing the boundaries of the sport. 14,000 cubic metres of soil was used to build the track. After the Supercross world cup event, Shanaze Reade called for changes to the track, she stated. Sarah Walker echoed. In preparation for the 2012 Summer Olympics, in 2010 the Dutch National Olympic Committee commissioned a replica of the planned BMX track at their National Sports Centre Papendal, it came into use in March 2011, ahead of the hand over of the London Velopark BMX venue.
The venue was used for the 2012 Olympic and Paralympic track cycling competition was held in the Velodrome with the adjoining BMX track hosting the Olympic BMX competition. Team GB dominated the track cycling competition winning seven out of a possible ten gold medals plus one silver and one bronze; the GB Paralympic track cycling team won a to
Mean curvature
In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The concept was used by Sophie Germain in her work on elasticity theory. Jean Baptiste Marie Meusnier used it in his studies of minimal surfaces, it is important in the analysis of minimal surfaces, which have mean curvature zero, in the analysis of physical interfaces between fluids which, for example, have constant mean curvature in static flows, by the Young-Laplace equation. Let p be a point on the surface S; each plane through p containing the normal line to S cuts S in a curve. Fixing a choice of unit normal gives a signed curvature to that curve; as the plane is rotated by an angle θ that curvature can vary. The maximal curvature κ 1 and minimal curvature κ 2 are known as the principal curvatures of S; the mean curvature at p ∈ S is the average of the signed curvature over all angles θ: H = 1 2 π ∫ 0 2 π κ d θ.
By applying Euler's theorem, this is equal to the average of the principal curvatures: H = 1 2. More for a hypersurface T the mean curvature is given as H = 1 n ∑ i = 1 n κ i. More abstractly, the mean curvature is the trace of the second fundamental form divided by n. Additionally, the mean curvature H may be written in terms of the covariant derivative ∇ as H n → = g i j ∇ i ∇ j X, using the Gauss-Weingarten relations, where X is a smoothly embedded hypersurface, n → a unit normal vector, g i j the metric tensor. A surface is a minimal surface. Furthermore, a surface which evolves under the mean curvature of the surface S, is said to obey a heat-type equation called the mean curvature flow equation; the sphere is the only embedded surface of constant positive mean curvature without boundary or singularities. However, the result is not true when the condition "embedded surface" is weakened to "immersed surface". For a surface defined in 3D space, the mean curvature is related to a unit normal of the surface: 2 H = − ∇ ⋅ n ^ where the normal chosen affects the sign of the curvature.
The sign of the curvature depends on the choice of normal: the curvature is positive if the surface curves "towards" the normal. The formula above holds for surfaces in 3D space defined in any manner, as long as the divergence of the unit normal may be calculated. Mean Curvature may be calculated 2 H = Trace where I and II denote first and second quadratic form matrices, respectively. For the special case of a surface defined as a function of two coordinates, e.g. z = S, using the upward pointing normal the mean curvature expression is 2 H = − ∇ ⋅ = ∇ ⋅ ( ∇ S − ∇ z 1 + | ∇ S |
Parallel (geometry)
In geometry, parallel lines are lines in a plane which do not meet. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel. Parallel planes are planes in the same three-dimensional space. Parallel lines are the subject of Euclid's parallel postulate. Parallelism is a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism; the parallel symbol is ∥. For example, A B ∥ C D indicates that line AB is parallel to line CD. In the Unicode character set, the "parallel" and "not parallel" signs have codepoints U+2225 and U+2226, respectively. In addition, U+22D5 represents the relation "equal and parallel to". Given parallel straight lines l and m in Euclidean space, the following properties are equivalent: Every point on line m is located at the same distance from line l.
Line m is in the same plane as line l but does not intersect l. When lines m and l are both intersected by a third straight line in the same plane, the corresponding angles of intersection with the transversal are congruent. Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in Euclidean space, but the first and third properties involve measurement, so, are "more complicated" than the second. Thus, the second property is the one chosen as the defining property of parallel lines in Euclidean geometry; the other properties are consequences of Euclid's Parallel Postulate. Another property that involves measurement is that lines parallel to each other have the same gradient; the definition of parallel lines as a pair of straight lines in a plane which do not meet appears as Definition 23 in Book I of Euclid's Elements. Alternative definitions were discussed by other Greeks as part of an attempt to prove the parallel postulate. Proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein.
Simplicius mentions Posidonius' definition as well as its modification by the philosopher Aganis. At the end of the nineteenth century, in England, Euclid's Elements was still the standard textbook in secondary schools; the traditional treatment of geometry was being pressured to change by the new developments in projective geometry and non-Euclidean geometry, so several new textbooks for the teaching of geometry were written at this time. A major difference between these reform texts, both between themselves and between them and Euclid, is the treatment of parallel lines; these reform texts were not without their critics and one of them, Charles Dodgson, wrote a play and His Modern Rivals, in which these texts are lambasted. One of the early reform textbooks was James Maurice Wilson's Elementary Geometry of 1868. Wilson based his definition of parallel lines on the primitive notion of direction. According to Wilhelm Killing the idea may be traced back to Leibniz. Wilson, without defining direction since it is a primitive, uses the term in other definitions such as his sixth definition, "Two straight lines that meet one another have different directions, the difference of their directions is the angle between them."
Wilson In definition 15 he introduces parallel lines in this way. Wilson Augustus De Morgan reviewed this text and declared it a failure on the basis of this definition and the way Wilson used it to prove things about parallel lines. Dodgson devotes a large section of his play to denouncing Wilson's treatment of parallels. Wilson edited this concept out of the third and higher editions of his text. Other properties, proposed by other reformers, used as replacements for the definition of parallel lines, did not fare much better; the main difficulty, as pointed out by Dodgson, was that to use them in this way required additional axioms to be added to the system. The equidistant line definition of Posidonius, expounded by Francis Cuthbertson in his 1874 text Euclidean Geometry suffers from the problem that the points that are found at a fixed given distance on one side of a straight line must be shown to form a straight line; this must be assumed to be true. The corresponding angles formed by a transversal property, used by W. D. Cooley in his 1860 text, The Elements of Geometry and explained requires a proof of the fact that if one transversal meets a pair of lines in congruent corresponding angles all transversals must do so.
Again, a new axiom is needed to justify this statement. The three properties above lead to three different methods of construction of parallel lines; because parallel lines in a Euclidean plane are equidistant there is a unique distance between the two parallel lines. Given the equations of two non-vertical, non-horizontal parallel lines, y = m x + b 1 y = m x + b 2
Parabola
In mathematics, a parabola is a plane curve, mirror-symmetrical and is U-shaped. It fits several superficially different other mathematical descriptions, which can all be proved to define the same curves. One description of a parabola involves a line; the focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane, parallel to another plane, tangential to the conical surface; the line perpendicular to the directrix and passing through the focus is called the "axis of symmetry". The point on the parabola that intersects the axis of symmetry is called the "vertex", is the point where the parabola is most curved; the distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola, parallel to the directrix and passes through the focus.
Parabolas can open up, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit on any other parabola—that is, all parabolas are geometrically similar. Parabolas have the property that, if they are made of material that reflects light light which travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel beam, leaving the parabola parallel to the axis of symmetry; the same effects occur with other forms of energy. This reflective property is the basis of many practical uses of parabolas; the parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are used in physics and many other areas; the earliest known work on conic sections was by Menaechmus in the fourth century BC.
He discovered a way to solve the problem of doubling the cube using parabolas. The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes via the method of exhaustion in the third century BC, in his The Quadrature of the Parabola; the name "parabola" is due to Apollonius. It means "application", referring to "application of areas" concept, that has a connection with this curve, as Apollonius had proved; the focus–directrix property of the parabola and other conic sections is due to Pappus. Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity; the idea that a parabolic reflector could produce an image was well known before the invention of the reflecting telescope. Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne, James Gregory; when Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror.
Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers. A parabola can be defined geometrically as a set of points in the Euclidean plane: A parabola is a set of points, such that for any point P of the set the distance | P F | to a fixed point F, the focus, is equal to the distance | P l | to a fixed line l, the directrix: The midpoint V of the perpendicular from the focus F onto the directrix l is called vertex and the line F V the axis of symmetry of the parabola. If one introduces cartesian coordinates, such that F =, f > 0, the directrix has the equation y = − f one obtains for a point P = from | P F | 2 = | P l | 2 the equation x 2 + 2 = 2. Solving for y yields y = 1 4 f x 2; the parabola is U-shaped. The horizontal chord through the focus is called the latus rectum; the latus rectum is parallel to the directrix. The semi-latus
