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51-684: RP , R-P , Rp , R-p , or rp may refer to: [REDACTED] Look up rp in Wiktionary, the free dictionary. Businesses and organizations [ edit ] Rainforest Partnership , an environmental organization based in Austin, Texas RallyPoint , a social network for the US military Reform Party (Singapore) , an opposition party in Singapore led by Kenneth Jeyaretnam Republic Polytechnic ,

102-403: A demigenus (non-orientable genus, Euler genus) of 1. The topological real projective plane can be constructed by taking the (single) edge of a Möbius strip and gluing it to itself in the correct direction, or by gluing the edge to a disk . Alternately, the real projective plane can be constructed by identifying each pair of opposite sides of the square, but in opposite directions, as shown in

153-419: A Calvinist and Anabaptist principle Religious Programs Specialist Retribution principle (RP) Science, technology, and mathematics [ edit ] Biology and medicine [ edit ] Radical prostatectomy Raynaud's phenomenon Retinitis pigmentosa Medical prescription from Latin, also Rp/. Mathematics [ edit ] RP (complexity) , randomized polynomial time,

204-623: A bottom up approach, Rainforest Partnership matches economic development choices to the needs and desires, culture, knowledge, and skills of local communities, and to the opportunities presented by each individual rainforest. The organization functions on a "collective model" in which "much depends on the active consent and ideas of the Latin American partners" describes Michael Barnes of the Austin American-Statesman . Since May 13, 2010, Rainforest Partnership has held

255-756: A certification program of the National Federation of Paralegal Associations Riot Points (used in League of Legends ) Regimental Police or Regimental Provost, soldiers responsible for regimental discipline and unit custody in the British Army Reporting Person or Party, in U.S. law enforcement jargon Registered Psychotherapist within Ontario Role-playing Rating Pending, a rating used by ESRB in promotional games that has lacked

306-529: A class in computational complexity theory Ranked Pairs , a Condorcet voting method Real projective line Real projective plane Real projective space Other uses in science and technology [ edit ] Rapid prototyping , a manufacturing and engineering process Rear projection effect , a film technique Red phosphorus , an allotrope of the element Rendezvous Point in Protocol Independent Multicast,

357-499: A collection of network layer multicast routing protocols Reversed-phase chromatography , a laboratory technique Route Processor, a general-purpose CPU in some Cisco routers RP, a small rock climbing nut , named after Roland Pauligk RP-1 , a rocket propellant RP-3 , a British rocket projectile in World War II Other uses [ edit ] Relief pitcher , a baseball term Registered Paralegal ,

408-471: A column vector ℓ that is orthogonal to x 1 and x 2 . The cross product will find such a vector: the line joining two points has homogeneous coordinates given by the equation x 1 × x 2 . The intersection of two lines may be found in the same way, using duality, as the cross product of the vectors representing the lines, ℓ 1 × ℓ 2 . The projective plane embeds into 4-dimensional Euclidean space. The real projective plane P ( R )

459-406: A constant. In the adjacent image we have divided by 2 so the z value now becomes 0.5. If we walk far enough away what we are looking at becomes a point in the distance. As we walk away we see more and more of the parallel lines. The lines will meet at a line at infinity (a line that goes through zero on the plane at z = 0 ). Lines on the plane when z = 0 are ideal points. The plane at z = 0

510-470: A map P ( R ) → R . Moreover, this map is an embedding. Notice that this embedding admits a projection into R which is the Roman surface . By gluing together projective planes successively we get non-orientable surfaces of higher demigenus . The gluing process consists of cutting out a little disk from each surface and identifying ( gluing ) their boundary circles. Gluing two projective planes creates

561-413: A particular origin point ; in this model, lines through the origin are considered to be the "points" of the projective plane, and planes through the origin are considered to be the "lines" in the projective plane. These projective points and lines can be pictured in two dimensions by intersecting them with any arbitrary plane not passing through the origin; then the parallel plane which does pass through

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612-521: A point outside either plane, for example by photographing a flat painting from an oblique angle, is a projective transformation. The fundamental objects in the projective plane are points and straight lines , and as in Euclidean geometry , every pair of points determines a unique line passing through both, but unlike in the Euclidean case in projective geometry every pair of lines also determines

663-763: A polytechnic in Singapore Rheinische Post , a German newspaper Rhône-Poulenc , a former French chemical company Royal Society of Portrait Painters (London), with membership indicated RP Roma Party ( Romska partija ), a political party in Serbia Welfare Party , or Refah Partisi , in Turkey Chautauqua Airlines (IATA airline designator RP) Registered Plumber, in UK Economics and finance [ edit ] Repurchase agreement ,

714-481: A rating. Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Rp . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Rp&oldid=1196444793 " Category : Disambiguation pages Hidden categories: Articles containing Polish-language text Short description

765-648: A stake in preserving their forests. The mission of a project depends on the nature of the forest and the local community, this includes creating a market in the United States for shade grown crops such as acai berries, cacao, or coffee, medicinal plants, palm trees or for crafts made by local artisans. Rainforest Partnership's first project, in Chipaota, Peru , involved creating a sustainable management plan for harvesting piassaba palms from which to make brooms. In some communities, such as Pampa Hermosa, Peru , it

816-458: A unique point at their intersection (in Euclidean geometry, parallel lines never intersect). In contexts where there is no ambiguity, it is simply called the projective plane ; the qualifier "real" is added to distinguish it from other projective planes such as the complex projective plane and finite projective planes . One common model of the real projective plane is the space of lines in three-dimensional Euclidean space which pass through

867-466: Is a two-dimensional projective space , similar to the familiar Euclidean plane in many respects but without the concepts of distance , circles , angle measure , or parallelism . It is the setting for planar projective geometry , in which the relationships between objects are not considered to change under projective transformations . The name projective comes from perspective drawing : projecting an image from one plane onto another as viewed from

918-399: Is a more degenerate map of the projective plane into 3-space, containing a cross-cap . A polyhedral representation is the tetrahemihexahedron , which has the same general form as Steiner's Roman surface, shown here. Looking in the opposite direction, certain abstract regular polytopes – hemi-cube , hemi-dodecahedron , and hemi-icosahedron – can be constructed as regular figures in

969-556: Is different from Wikidata All article disambiguation pages All disambiguation pages Rainforest Partnership Rainforest Partnership was founded in 2007 by Niyanta Spelman, Hazel Barbour, Jordan Erdos, and Bob Warneke, facilitated by Beth Caplan. In 2008, Rainforest Partnership had established its first major partnership with the community of Chipaota in Peru. Projects aim to create and support sustainable economic alternatives to deforestation and give local communities

1020-531: Is how a real projective plane is formed out of a disk. Therefore, the surface shown in Figure ;1 (cross-cap with disk) is topologically equivalent to the real projective plane RP . The points in the plane can be represented by homogeneous coordinates . A point has homogeneous coordinates [ x  :  y  :  z ], where the coordinates [ x  :  y  :  z ] and [ tx  :  ty  :  tz ] are considered to represent

1071-405: Is identified with the point ( u + π, 1) whose coordinates are ( r cos ⁡ 2 u , r sin ⁡ 2 u , − r cos ⁡ u ) {\displaystyle (r\,\cos 2u,r\,\sin 2u,-r\,\cos u)} . But this means that pairs of opposite points on the rim of the (equivalent) ordinary disk are identified with each other; this

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1122-471: Is more appropriate to develop plans for sustainable logging and for ecotourism . In protecting cloud forests, as the project in Pampa Hermosa aims to do by introducing alternatives to deforestation, local communities are faced with a "win-win" situation according to Ken Young of UT Austin 's Geography department. Animals and wildlife are protected while the needs of local people go unharmed. Through

1173-415: Is shown in Figure 2. Once this exception is made, it will be seen that the sliced cross-capped disk is homeomorphic to a self-intersecting disk, as shown in Figure 3. The self-intersecting disk is homeomorphic to an ordinary disk. The parametric equations of the self-intersecting disk are: where u ranges from 0 to 2 π and v ranges from 0 to 1. Projecting the self-intersecting disk onto

1224-415: Is the quotient of the two-sphere by the antipodal relation ( x , y , z ) ~ (− x , − y , − z ) . Consider the function R → R given by ( x , y , z ) ↦ ( xy , xz , y − z , 2 yz ) . This map restricts to a map whose domain is S and, since each component is a homogeneous polynomial of even degree, it takes the same values in R on each of any two antipodal points on S . This yields

1275-430: Is the line at infinity. The homogeneous point (0, 0, 0) is where all the real points go when you're looking at the plane from an infinite distance, a line on the z = 0 plane is where parallel lines intersect. In the equation x ℓ = 0 there are two column vectors . You can keep either constant and vary the other. If we keep the point x constant and vary the coefficients ℓ we create new lines that go through

1326-418: Is without intersection) in three-dimensional Euclidean space . The proof that the projective plane does not embed in three-dimensional Euclidean space goes like this: Assuming that it does embed, it would bound a compact region in three-dimensional Euclidean space by the generalized Jordan curve theorem . The outward-pointing unit normal vector field would then give an orientation of the boundary manifold, but

1377-650: The Austin American-Statesman 's online counterpart, Austin360.com, listener-supported public radio station KUT , and local news station News 8 YNN Austin . Further articles have appeared in The Austin Chronicle , an alternative weekly newspaper published on Thursdays in Austin. Real projective plane In mathematics , the real projective plane , denoted ⁠ R P 2 {\displaystyle \mathbf {RP} ^{2}} ⁠ or ⁠ P 2 {\displaystyle \mathbb {P} _{2}} ⁠ ,

1428-429: The projective plane; see also projective polyhedra . Various planar (flat) projections or mappings of the projective plane have been described. In 1874 Klein described the mapping: Central projection of the projective hemisphere onto a plane yields the usual infinite projective plane, described below. A closed surface is obtained by gluing a disk to a cross-cap . This surface can be represented parametrically by

1479-652: The United Kingdom Rioplatense Spanish , a dialect spoken in parts of Argentina and Uruguay Places [ edit ] Republic of the Philippines (former two-letter country code) Republic of Poland ( Rzeczpospolita Polska ) Rhineland-Palatinate , one of sixteen German states Région Parisienne or Île-de-France, the area surrounding Paris, France Religion [ edit ] Reformed Presbyterian Church (disambiguation) Regulative principle of worship ,

1530-412: The boundary manifold would be the projective plane , which is not orientable. This is a contradiction, and so our assumption that it does embed must have been false. Consider a sphere , and let the great circles of the sphere be "lines", and let pairs of antipodal points be "points". It is easy to check that this system obeys the axioms required of a projective plane : If we identify each point on

1581-537: The competition every year since 2010 alongside guest judges including: Lisa McWilliams, Michel Scott, and Evan Smith (2010) Elizabeth Avellan and Ed Begley Jr. (2011) Elizabeth Avellan and Philippe Cousteau Jr. (2012) Philippe Cousteau Jr., Jay Duplass, and Dana Wheeler-Nicholson (2013) Sarah Backhouse, Dilly Gent, and Ginger Sledge (2014) Eloise DeJoria, Taylor Ellison, and Kenny Laubbacher (2015) Solly Granastein and Julio Quintana (2016) Michael Cain and Alonso Mayo (2017). On June 22, 2017, Rainforest Partnership launched

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1632-407: The diagram. (Performing any of these operations in three-dimensional space causes the surface to intersect itself.) Projective geometry is not necessarily concerned with curvature and the real projective plane may be twisted up and placed in the Euclidean plane or 3-space in many different ways. Some of the more important examples are described below. The projective plane cannot be embedded (that

1683-403: The equation above can be written in matrix form as: Using vector notation we may instead write x ⋅ ℓ = 0 or ℓ ⋅ x = 0. The equation k ( x ℓ ) = 0 (which k is a non-zero scalar) sweeps out a plane that goes through zero in R and k ( x ) sweeps out a line, again going through zero. The plane and line are linear subspaces in R , which always go through zero. In P

1734-444: The equation of a line is ax + by + cz = 0 and this equation can represent a line on any plane parallel to the x , y plane by multiplying the equation by k . If z = 1 we have a normalized homogeneous coordinate. All points that have z = 1 create a plane. Let's pretend we are looking at that plane (from a position further out along the z axis and looking back towards the origin) and there are two parallel lines drawn on

1785-556: The following equations: where both u and v range from 0 to 2 π . These equations are similar to those of a torus . Figure 1 shows a closed cross-capped disk. A cross-capped disk has a plane of symmetry that passes through its line segment of double points. In Figure 1 the cross-capped disk is seen from above its plane of symmetry z = 0, but it would look the same if seen from below. A cross-capped disk can be sliced open along its plane of symmetry, while making sure not to cut along any of its double points. The result

1836-432: The generator. Because the sphere covers the real projective plane twice, the plane may be represented as a closed hemisphere around whose rim opposite points are identified. The projective plane can be immersed (local neighbourhoods of the source space do not have self-intersections) in 3-space. Boy's surface is an example of an immersion. Polyhedral examples must have at least nine faces. Steiner's Roman surface

1887-592: The inaugural World Rainforest Day . The holiday was started as a means to bring awareness to the importance of tropical rainforests and encourage action to prevent deforestation. Partners for World Rainforest Day include Avoided Deforestation Partners , South by Southwest , Austin EcoNetwork, Earthx Film, Bonobo Conservation Initiative , 2020 or Bust, Earth Day ATX, and Ear to the Earth. Rainforest Partnership has been featured in multiple local media outlets including

1938-474: The origin (a projective "line") is called the line at infinity . (See § Homogeneous coordinates below.) In topology , the name real projective plane is applied to any surface which is topologically equivalent to the real projective plane. Topologically, the real projective plane is compact and non- orientable (one-sided). It cannot be embedded in three-dimensional Euclidean space without intersecting itself. It has Euler characteristic 1, hence

1989-406: The plane can also be represented by homogeneous coordinates. A projective line corresponding to the plane ax + by + cz = 0 in R has the homogeneous coordinates ( a  :  b  :  c ). Thus, these coordinates have the equivalence relation ( a  :  b  :  c ) = ( da  :  db  :  dc ) for all nonzero values of d . Hence a different equation of

2040-417: The plane of symmetry ( z = 0 in the parametrization given earlier) which passes only through the double points, the result is an ordinary disk which repeats itself (doubles up on itself). The plane z = 0 cuts the self-intersecting disk into a pair of disks which are mirror reflections of each other. The disks have centers at the origin . Now consider the rims of the disks (with v = 1). The points on

2091-417: The plane. From where we are standing (given our visual capabilities) we can see only so much of the plane, which we represent as the area outlined in red in the diagram. If we walk away from the plane along the z axis, (still looking backwards towards the origin), we can see more of the plane. In our field of view original points have moved. We can reflect this movement by dividing the homogeneous coordinate by

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2142-400: The point. If we keep the coefficients constant and vary the points that satisfy the equation we create a line. We look upon x as a point, because the axes we are using are x , y , and z . If we instead plotted the coefficients using axis marked a , b , c points would become lines and lines would become points. If you prove something with the data plotted on axis marked x , y , and z

2193-487: The rim of the self-intersecting disk come in pairs which are reflections of each other with respect to the plane z = 0. A cross-capped disk is formed by identifying these pairs of points, making them equivalent to each other. This means that a point with parameters ( u , 1) and coordinates ( r cos ⁡ 2 u , r sin ⁡ 2 u , r cos ⁡ u ) {\displaystyle (r\,\cos 2u,r\,\sin 2u,r\,\cos u)}

2244-413: The sale of securities together with an agreement for the seller to buy back the securities at a later date Reservation price , the highest price a buyer is willing to pay for goods or a service Rupee , common name for the currencies of several countries Rupiah , the official currency of Indonesia Language [ edit ] Received Pronunciation , a standard accent of Standard English in

2295-450: The same argument can be used for the data plotted on axis marked a , b , and c . That is duality. The equation x ℓ = 0 calculates the inner product of two column vectors. The inner product of two vectors is zero if the vectors are orthogonal . In P , the line between the points x 1 and x 2 may be represented as a column vector ℓ that satisfies the equations x 1 ℓ = 0 and x 2 ℓ = 0 , or in other words

2346-402: The same line dax  +  dby  +  dcz  = 0 gives the same homogeneous coordinates. A point [ x  :  y  :  z ] lies on a line ( a  :  b  :  c ) if ax  +  by  +  cz  = 0. Therefore, lines with coordinates ( a  :  b  :  c ) where a ,  b are not both 0 correspond to the lines in

2397-458: The same point, for all nonzero values of t . The points with coordinates [ x  :  y  : 1] are the usual real plane , called the finite part of the projective plane, and points with coordinates [ x  :  y  : 0], called points at infinity or ideal points , constitute a line called the line at infinity . (The homogeneous coordinates [0 : 0 : 0] do not represent any point.) The lines in

2448-469: The short film competition Films for the Forest (F3) in which films between 30 seconds to 3 minutes long are submitted centered around a featured theme. Since 2012, F3 has been featured at SXSW Film Festival Community Screenings. The films held in the competition are sent from around the world, including "countries as far away as Brazil, Italy and India". Richard Linklater has served as the primary judge for

2499-403: The sphere is homeomorphic with the collection of all lines passing through the origin in R . The quotient map from the sphere onto the real projective plane is in fact a two sheeted (i.e. two-to-one) covering map . It follows that the fundamental group of the real projective plane is the cyclic group of order 2; i.e., integers modulo 2. One can take the loop AB from the figure above to be

2550-408: The sphere with its antipodal point, then we get a representation of the real projective plane in which the "points" of the projective plane really are points. This means that the projective plane is the quotient space of the sphere obtained by partitioning the sphere into equivalence classes under the equivalence relation ~, where x ~ y if y = x or y = − x . This quotient space of

2601-407: The usual real plane , because they contain points that are not at infinity. The line with coordinates (0 : 0 : 1) is the line at infinity, since the only points on it are those with  z  = 0. A line in P can be represented by the equation ax + by + cz = 0. If we treat a , b , and c as the column vector ℓ and x , y , z as the column vector x then

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