In Euclidean and projective geometry , five points determine a conic (a degree-2 plane curve), just as two (distinct) points determine a line (a degree-1 plane curve ). There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines.
43-2025: Five Points may refer to: Places [ edit ] Canada [ edit ] The northern half of Oakwood-Vaughan , neighborhood in Toronto, Ontario United States [ edit ] Alabama: Five Points, Alabama , a town Five Points South Historic District , Birmingham, listed on the National Register of Historic Places (NRHP) in Jefferson County Five Points Historic District (Huntsville, Alabama) , NRHP-listed Five Points, California Five Points, Denver , Colorado Florida: Five Points, Florida Five Points (Jacksonville) , Florida Georgia: Five Points (Athens) , Georgia Five Points, Atlanta , Georgia Five Points station , Atlanta, Georgia Little Five Points , community east of downtown Atlanta, Georgia Five Points, Indiana (disambiguation) , multiple locations Five Points, Iowa Five Points, Michigan (disambiguation) , multiple locations Five Points, Minnesota Five Points, Trenton, New Jersey Five Points, Manhattan , New York North Carolina: Five Points, North Carolina Five Points, Asheville Five Points, Franklin County, North Carolina Five Points Historic District (Albemarle) Five Points Historic Neighborhoods (Raleigh) Five Points, Ohio (disambiguation) , multiple locations Five Points, Pennsylvania (disambiguation) , multiple locations Five Points (Columbia, South Carolina) Five Points, Texas Five Points, West Virginia Five Points, Wisconsin (disambiguation) , multiple locations Five Points Historic District (disambiguation) , multiple locations Other uses [ edit ] Five Points ,
86-486: A pencil of circles such as the Apollonian circles . These results seem to run counter the general result since circles are special cases of conics. However, in a pappian projective plane a conic is a circle only if it passes through two specific points on the line at infinity , so a circle is determined by five non-collinear points, three in the affine plane and these two special points. Similar considerations explain
129-675: A 2004 novel by Indian writer Chetan Bhagat, adapted into the 2009 film 3 Idiots Five-point stencil Five Corners (disambiguation) The 5 Point Cafe Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Five Points . 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=Five_Points&oldid=1230000388 " Categories : Disambiguation pages Place name disambiguation pages Quadripoints and higher Hidden categories: Short description
172-488: A conic can be proven by synthetic geometry —i.e., in terms of lines and points in the plane—in addition to the analytic (algebraic) proof given above. Such a proof can be given using a theorem of Jakob Steiner , which states: This can be shown by taking the points X and Y to the standard points [ 1 : 0 : 0 ] {\displaystyle [1:0:0]} and [ 0 : 1 : 0 ] {\displaystyle [0:1:0]} by
215-535: A conic. The second, that the constraints are independent, is significantly subtler: it corresponds to the fact that given five points in general linear position in the plane, their images in P 5 {\displaystyle \mathbf {P} ^{5}} under the Veronese map are in general linear position, which is true because the Veronese map is biregular : i.e., if the image of five points satisfy
258-459: A cubic, if the nine points lie on more than one cubic—i.e., they are the intersection of two cubics—then they are not in general position, and indeed satisfy an addition constraint, as stated in the Cayley–Bacharach theorem . Four points do not determine a conic, but rather a pencil , the 1-dimensional linear system of conics which all pass through the four points (formally, have
301-761: A literary journal published by Georgia State University , named after the area in Atlanta Five Points (TV series) , 2018 Five Points Gang , Manhattan Quincunx , a geometric pattern consisting of five points arranged in a cross Five dots tattoo See also [ edit ] 5 Pointz , former warehouse and mural venue in Queens, New York City Five points determine a conic Five Points Correctional Facility , Romulus, New York Five Points of Calvinism Le Corbusier's Five Points of Architecture Five-point electoral law Five Point Plan , an American band Five Point Someone ,
344-518: A maze for motorists. Oakwood–Vaughan's municipal status/boundaries has changed on several occasions. One of the most significant changes was the result of an amalgamation in 1998. Formerly part of the City of York , York became part of the larger City of Toronto, along with Etobicoke, Scarborough, and North York—all cities in their own right—and the Borough of East York on January 1, 1998. Municipally,
387-453: A projective transformation, in which case the pencils of lines correspond to the horizontal and vertical lines in the plane, and the intersections of corresponding lines to the graph of a function, which (must be shown) is a hyperbola, hence a conic, hence the original curve C is a conic. Now given five points X, Y, A, B, C, the three lines X A , X B , X C {\displaystyle XA,XB,XC} can be taken to
430-413: A relation, then the relation can be pulled back and the original points must also satisfy a relation. The Veronese map has coordinates [ x 2 : x y : y 2 : x z : y z : z 2 ] , {\displaystyle [x^{2}:xy:y^{2}:xz:yz:z^{2}],} and the target P 5 {\displaystyle \mathbf {P} ^{5}}
473-406: A variety of degree d and dimension m , which is a fundamental question in enumerative geometry . A simple case of this is for a hypersurface (a codimension 1 subvariety, the zeros of a single polynomial, the case m = n − 1 {\displaystyle m=n-1} ), of which plane curves are an example. In the case of a hypersurface, the answer is given in terms of
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#1732765894460516-492: Is ( ( n + 1 d ) ) , {\displaystyle \textstyle {\left(\!\!{n+1 \choose d}\!\!\right)},} from which 1 is subtracted because of projectivization: multiplying a polynomial by a constant does not change its zeros. In the above formula, the number of points k is a polynomial in d of degree n, with leading coefficient 1 / n ! {\displaystyle 1/n!} In
559-421: Is dual to the [ A : B : C : D : E : F ] {\displaystyle [A:B:C:D:E:F]} P 5 {\displaystyle \mathbf {P} ^{5}} of conics. The Veronese map corresponds to "evaluation of a conic at a point", and the statement about independence of constraints is exactly a geometric statement about this map. That five points determine
602-490: Is a unique conic passing through them, which will be non- degenerate ; this is true over both the Euclidean plane and any pappian projective plane . Indeed, given any five points there is a conic passing through them, but if three of the points are collinear the conic will be degenerate (reducible, because it contains a line), and may not be unique; see further discussion . This result can be proven numerous different ways;
645-653: Is also home to the Arlington Village Project, along the western border with Humewood-Cedarvale, which works to create more positive community building and connecting events in the area. Starting in 2010, the project has organized outdoor fairs. Built in 1936, the former Mount Zion Apostolic Church of Canada is one of the hundreds of apostolic churches in Toronto. It was relocated to North York. Estimates of ethnic origin by world region in Oakwood Village: The above estimates are based on
688-476: Is bordered by Eglinton Avenue West to the north ( Briar Hill–Belgravia ), Dufferin Street to the west ( Caledonia–Fairbank ), St. Clair Avenue West to the south ( Corso Italia and Bracondale Hill ), and Winona Drive to the east ( Humewood–Cedarvale ). Oakwood–Vaughan was part of the former City of York before the amalgamation of Toronto in 1998. The neighbourhood's northern half is nicknamed Five Points for
731-498: Is different from Wikidata All article disambiguation pages All disambiguation pages Oakwood-Vaughan Oakwood Village , formerly known as Oakwood–Vaughan , is a neighbourhood in Toronto , Ontario , Canada. Located in the former suburb of York , the neighbourhood is a Business Improvement Area (BIA); it has an annual arts festival and a public library built in 1997. The neighbourhood, commonly known as OV ,
774-601: Is located on Oakwood Avenue and Holland Park Avenue. In addition to Oakwood Village branch, the Maria A. Shchuka Public Library, which was re-rebuilt in 2003, is also located on the neighbourhood's periphery at Eglinton Avenue and Northcliffe Boulevard. Both of these libraries are run by the Toronto Public Library . The neighbourhood is west of the Eglinton West station , and has three 24-hour bus routes:
817-454: Is thus determined by a point, which is codimension 2), 2 points determine a line, 5 points determine a conic, 9 points determine a cubic, 14 points determine a quartic, and so forth. While five points determine a conic, sets of six or more points on a conic are not in general position, that is, they are constrained as is demonstrated in Pascal's theorem . Similarly, while nine points determine
860-546: The 2016 Canadian census ; respondents to the census can identify with more than one ethnic origin, so the above figures can sum to more than 100%. Estimates of Household Mean Income: The public school districts serving the community are Toronto District School Board (secular anglophone), Toronto Catholic District School Board (Catholic anglophone), Conseil scolaire Viamonde (secular francophone), and Conseil scolaire de district catholique Centre-Sud (Catholic francophone). Built in 1997, Oakwood Village Public Library
903-523: The Braikenridge–Maclaurin theorem , which is the converse of Pascal's theorem . Pascal's theorem states that given 6 points on a conic (a hexagon), the lines defined by opposite sides intersect in three collinear points. This can be reversed to construct the possible locations for a 6th point, given 5 existing ones. The natural generalization is to ask for what value of k a configuration of k points (in general position) in n -space determines
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#1732765894460946-537: The multiset coefficient , more familiarly the binomial coefficient , or more elegantly the rising factorial , as: This is via the analogous analysis of the Veronese map : k points in general position impose k independent linear conditions on a variety (because the Veronese map is biregular), and the number of monomials of degree d in n + 1 {\displaystyle n+1} variables ( n -dimensional projective space has n + 1 {\displaystyle n+1} homogeneous coordinates)
989-500: The 29 (Dufferin Street), the 32 (Eglinton Avenue West), and the 63 (Oakwood Ave) which runs between Eglinton West station and Liberty Village . There are other buses that operate very frequently, including the 161, which runs along the southern stretch of Oakwood Avenue and along Rogers Road, and the 90, which runs along Vaughan Road to St. Clair West station . The 512 St. Clair streetcar route runs east–west along St. Clair, just below
1032-567: The St. Paul's provincial riding. The official website, however, has an outdated map from 2005, and so lists the area, incorrectly, as part of the Davenport riding. Federally, the area is also part of St. Paul's riding and is represented by MP Dr. Carolyn Bennett . There are several organizations in the Oakwood–Vaughan area that are working to effect positive change and are trying to counteract
1075-428: The above analysis are that the resulting point is a quadratic equation (not a linear equation), and that the constraints are independent. The first is simple: if A , B , and C all vanish, then the equation D x + E y + F = 0 {\displaystyle Dx+Ey+F=0} defines a line, and any 3 points on this (indeed any number of points) lie on a line – thus general linear position ensures
1118-522: The area has a complex history and its boundaries have changed several times. Since the 1998 amalgamation, from Dufferin Street to Winona Drive and north of Rogers Road up to Eglinton Avenue (and beyond), it is a small part of Ward 15: Eglinton—Lawrence. The eastern part of the neighbourhood (east of Oakwood Avenue) also extends into Ward 21: St. Paul's. Residents west of Oakwood Avenue and south of Rogers Road reside in Ward 17: Davenport. A visual representation of
1161-406: The case of plane curves, where n = 2 , {\displaystyle n=2,} the formula becomes: whose values for d = 0 , 1 , 2 , 3 , 4 {\displaystyle d=0,1,2,3,4} are 0 , 2 , 5 , 9 , 14 {\displaystyle 0,2,5,9,14} – there are no curves of degree 0 (a single point is a point and
1204-460: The corner of Oakwood and Vaughan, was completed in October 2010. Standing tall in this oasis is a 4.88 metres (16.0 ft) steel palm tree that represents the strong roots put down by the residents of Oakwood Village. The palm tree, which can be found throughout the world's warmer nations, is a symbol of unity—the coming together of the many diverse groups in the Oakwood–Vaughan community. However,
1247-523: The determinant is visibly a linear combination of the six monomials of degree at most 2. Also, the resulting polynomial clearly vanishes at the five input points (when ( x , y ) = ( x i , y i ) {\displaystyle (x,y)=(x_{i},y_{i})} ), as the matrix has then a repeated row. Synthetically, the conic can be constructed by the Braikenridge–Maclaurin construction , by applying
1290-434: The dimension counting argument is most direct, and generalizes to higher degree, while other proofs are special to conics. Intuitively, passing through five points in general linear position specifies five independent linear constraints on the (projective) linear space of conics, and hence specifies a unique conic, though this brief statement ignores subtleties. More precisely, this is seen as follows: The two subtleties in
1333-510: The five points, the equation for the conic can be found by linear algebra , by writing and solving the five equations in the coefficients, substituting the variables with the values of the coordinates: five equations, six unknowns, but homogeneous so scaling removes one dimension; concretely, setting one of the coefficients to 1 accomplishes this. This can be achieved quite directly as the following determinantal equation: This matrix has variables in its first row and numbers in all other rows, so
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1376-482: The four points as base locus ). Similarly, three points determine a 2-dimensional linear system (net), two points determine a 3-dimensional linear system (web), one point determines a 4-dimensional linear system, and zero points place no constraints on the 5-dimensional linear system of all conics. As is well known, three non-collinear points determine a circle in Euclidean geometry and two distinct points determine
1419-429: The group has coordinated and completed several community projects to strengthen connections between neighbours, politicians, police, and community agencies. A community garden, located at the corner of Belvidere and Oakwood Avenues, was completed with funding from Clean & Beautiful City. Another project, a striking mosaic, is located on the south side of Belvidere Avenue, on the community garden's retaining wall. This
1462-433: The hypothesis of Steiner’s theorem. The resulting conic thus contains all five points, and is the unique such conic, as desired. Given five points, one can construct the conic containing them in various ways. Analytically, given the coordinates ( x i , y i ) i = 1 , 2 , 3 , 4 , 5 {\displaystyle (x_{i},y_{i})_{i=1,2,3,4,5}} of
1505-695: The intersection of Oakwood Avenue, Vaughan Road, and Belvidere Avenue. The western half was called Northcliffe and is more recently known as Northcliffe Village. Oakwood developed as a streetcar suburb of Toronto. Stephen Rogers constructed one of the first houses (on a 2.0-hectare (5-acre) plot) in this neighbourhood at what is now Rogers Road and Oakwood Avenue. The main streets of Oakwood–Vaughan are Oakwood Avenue, Vaughan Road , Rogers Road (named after Stephen Rogers), Dufferin Street, Eglinton Avenue West and St. Clair Avenue West. These main streets (aside from Vaughan Road) generally demarcate borders with other York and old City of Toronto neighbourhoods or borders of
1548-596: The negative, often stereotypical, media coverage of the area. The Oakwood Village BIA, founded in 2009, is actively trying to beautify the area and attract vibrant businesses. Another organization is the Oakwood Village Community Association (formerly 5 Points Community Action) that was founded in 2005. 5 Points Community Action originally took its name from the "5 Point" intersection of Oakwood Ave., Vaughan Rd., and Belvidere Ave. While there are several outstanding issues to be tackled,
1591-710: The smaller than expected number of points needed to define pencils of circles. Instead of passing through points, a different condition on a curve is being tangent to a given line. Being tangent to five given lines also determines a conic, by projective duality , but from the algebraic point of view tangency to a line is a quadratic constraint, so naive dimension counting yields 2 = 32 conics tangent to five given lines, of which 31 must be ascribed to degenerate conics, as described in fudge factors in enumerative geometry ; formalizing this intuition requires significant further development to justify. Another classic problem in enumerative geometry, of similar vintage to conics,
1634-492: The southern border of Oakwood–Vaughan. The bus routes and the 512 St. Clair streetcar route are operated by the TTC . The Rogers Road streetcar line formerly ran along Rogers Road before being replaced by a trolley bus route, in turn later replaced by the route 161 Rogers Road bus. Five points determine a conic Formally, given any five points in the plane in general linear position , meaning no three collinear , there
1677-408: The sub-neighbourhoods and neighbourhoods (Five Points, Northcliffe Village) previously mentioned. A portion of Little Jamaica is located on the northern border of the neighbourhood, situated along the south side of Eglinton Avenue West between Oakwood Avenue and Marlee Avenue. North of Vaughan Road and east of Oakwood Avenue, this part of the neighbourhood has a set of one-way streets that acts as
1720-416: The three lines Y A , Y B , Y C {\displaystyle YA,YB,YC} by a unique projective transform, since projective transforms are simply 3-transitive on lines (they are simply 3-transitive on points, hence by projective duality they are 3-transitive on lines). Under this map X maps to Y, since these are the unique intersection points of these lines, and thus satisfy
1763-566: The tree generated local controversy, as a political hopeful stated that it cost the city $ 350,000 to erect. The actual cost of the tree was $ 4,200. The Oakwood Village Arts Festival, held annually in June, is brings the community together around arts by providing a public platform for local artists and musicians. As a curated festival revolving around the theme of village, this event features theatre, dance, music, creative writing, art exhibits, interdisciplinary works and public discussions. The area
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1806-404: The wards is provided for reference. Provincially, the political boundaries have also changed, as recently as in the 2007 provincial election where it was designated part of the St. Paul's riding. Until 2009, MPP Michael Bryant represented the residents. Bryant stepped down in early 2009, and Eric Hoskins won a by-election, becoming the new MPP. Residents living east of Oakwood Ave. are part of
1849-467: Was completed in partnership with Art Starts. The mosaic illustrates the motif of 'roots' and has “welcome” in many different languages, including Polish, Hebrew, Italian and Portuguese, among others. A more recent project was an art installation that was completed in partnership with Clean & Beautiful City and the City of Toronto. The newly reconstructed Oakwood Village Transit Island, located directly at
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