See point at infinity in All languages combined, or Wiktionary
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The horizon itself, which consists of all the points at infinity, is called the line at infinity.", "type": "quote" } ], "glosses": [ "An asymptotic point in 3-dimensional space, viewed from some point, at which parallel lines appear to meet and which in perspective drawing is represented as a vanishing point." ], "id": "en-point_at_infinity-en-noun-01V6UN7E", "links": [ [ "vanishing point", "vanishing point" ] ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Geometry", "orig": "en:Geometry", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" } ], "examples": [ { "text": "1960, Roger A. Johnson, Advanced Euclidean Geometry, Republished 2007, page 45,\nIn the geometry of inversion, therefore, it is usual to sacrifice the line of points at infinity which is so useful in other fields, and adopt the convention of a single point at infinity, the inverse of the center of the circle of inversion." }, { "ref": "1990, Shreeram Shankar Abhyankar, Algebraic Geometry for Scientists and Engineers, page 10:", "text": "As the degree form Y² or X² of a parabola has only a single root, it has only one point at infinity.", "type": "quote" }, { "text": "1997, Susan Addington, Stuart Levy, Lost in the Fun House: An Application of Dynamic Projective Geometry, James King, Doris Schattschneider, Geometry Turned On: Dynamic Software in Learning, Teaching, and Research, page 159,\nTo unify the treatment of concurrent and parallel lines, define a point at infinity for each family of parallel lines, and declare each point at infinity to lie on each of the parallel lines that define it. […] Now parallel lines meet at infinity, and vanishing points are the images of points at infinity." }, { "ref": "2003, Ron Goldman, Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, page 18:", "text": "Projective space is, by definition, the collection of the points in affine space together with the points at infinity.", "type": "quote" } ], "glosses": [ "Any point added to a space to achieve projective completion." ], "id": "en-point_at_infinity-en-noun-5jKU1NMQ", "links": [ [ "geometry", "geometry" ], [ "completion", "completion" ] ], "qualifier": "Euclidean projective geometry", "raw_glosses": [ "(geometry, Euclidean projective geometry) Any point added to a space to achieve projective completion." ], "topics": [ "geometry", "mathematics", "sciences" ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Geometry", "orig": "en:Geometry", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "_dis": "22 16 61", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "22 16 62", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "22 14 64", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" }, { "_dis": "11 13 75", "kind": "other", "langcode": "en", "name": "Infinity", "orig": "en:Infinity", "parents": [], "source": "w+disamb" }, { "_dis": "11 27 62", "kind": "other", "langcode": "en", "name": "Non-Euclidean geometry", "orig": "en:Non-Euclidean geometry", "parents": [], "source": "w+disamb" }, { "_dis": "18 24 58", "kind": "other", "langcode": "en", "name": "Shapes in non-Euclidean geometry", "orig": "en:Shapes in non-Euclidean geometry", "parents": [], "source": "w+disamb" }, { "_dis": "23 29 49", "kind": "topical", "langcode": "en", "name": "Shapes", "orig": "en:Shapes", "parents": [ "Geometry", "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w+disamb" } ], "examples": [ { "ref": "1996, John Stillwell, Sources of Hyperbolic Geometry, page 82:", "text": "Hyperbolic geometry provides each line with two points at infinity. Whether there is a piece of the line beyond the points at infinity, completing the piece lying in the finite to a closed curve, we cannot say, since we cannot move as far as the points at infinity, let alone beyond them.", "type": "quote" }, { "text": "2007, Maurice Margenstern, Cellular Automata in Hyperbolic Spaces, Volume 1: Theory, page 66,\nAs we have seen, points at infinity behave as ordinary points: a single line passes through two distinct points at infinity or through a point in the plane and a point at infinity.\nThis representation of the points at infinity is a difference between the disc and the half-plane models of Poincaré, […] ." }, { "ref": "2012, Norbert A'Campo, Athanase Papadopoulos, “Notes on non-Euclidean Geometry”, in Athanase Papadopoulos, editor, Strasbourg Master Class on Geometry, page 126:", "text": "To describe a parabolic transformation f of the hyperbolic plane, we consider two parallel lines l and l’ and we call ω their common point at infinity.", "type": "quote" } ], "glosses": [ "An ideal point." ], "id": "en-point_at_infinity-en-noun-t03CrsPO", "links": [ [ "geometry", "geometry" ], [ "ideal point", "ideal point" ] ], "raw_glosses": [ "(geometry, hyperbolic geometry) An ideal point." ], "related": [ { "_dis1": "20 25 56", "word": "line at infinity" }, { "_dis1": "20 25 56", "word": "plane at infinity" }, { "_dis1": "20 25 56", "word": "hyperplane at infinity" }, { "_dis1": "20 25 56", "word": "vanishing point" } ], "synonyms": [ { "_dis1": "20 25 56", "tags": [ "excessive" ], "topics": [ "geometry", "mathematics", "sciences" ], "word": "ideal point" }, { "_dis1": "20 25 56", "tags": [ "excessive" ], "topics": [ "geometry", "mathematics", "sciences" ], "word": "omega point" } ], "tags": [ "excessive" ], "topics": [ "geometry", "mathematics", "sciences" ] } ], "wikipedia": [ "point at infinity" ], "word": "point at infinity" }
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The horizon itself, which consists of all the points at infinity, is called the line at infinity.", "type": "quote" } ], "glosses": [ "An asymptotic point in 3-dimensional space, viewed from some point, at which parallel lines appear to meet and which in perspective drawing is represented as a vanishing point." ], "links": [ [ "vanishing point", "vanishing point" ] ] }, { "categories": [ "English terms with quotations", "en:Geometry" ], "examples": [ { "text": "1960, Roger A. Johnson, Advanced Euclidean Geometry, Republished 2007, page 45,\nIn the geometry of inversion, therefore, it is usual to sacrifice the line of points at infinity which is so useful in other fields, and adopt the convention of a single point at infinity, the inverse of the center of the circle of inversion." }, { "ref": "1990, Shreeram Shankar Abhyankar, Algebraic Geometry for Scientists and Engineers, page 10:", "text": "As the degree form Y² or X² of a parabola has only a single root, it has only one point at infinity.", "type": "quote" }, { "text": "1997, Susan Addington, Stuart Levy, Lost in the Fun House: An Application of Dynamic Projective Geometry, James King, Doris Schattschneider, Geometry Turned On: Dynamic Software in Learning, Teaching, and Research, page 159,\nTo unify the treatment of concurrent and parallel lines, define a point at infinity for each family of parallel lines, and declare each point at infinity to lie on each of the parallel lines that define it. […] Now parallel lines meet at infinity, and vanishing points are the images of points at infinity." }, { "ref": "2003, Ron Goldman, Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, page 18:", "text": "Projective space is, by definition, the collection of the points in affine space together with the points at infinity.", "type": "quote" } ], "glosses": [ "Any point added to a space to achieve projective completion." ], "links": [ [ "geometry", "geometry" ], [ "completion", "completion" ] ], "qualifier": "Euclidean projective geometry", "raw_glosses": [ "(geometry, Euclidean projective geometry) Any point added to a space to achieve projective completion." ], "topics": [ "geometry", "mathematics", "sciences" ] }, { "categories": [ "English terms with quotations", "en:Geometry" ], "examples": [ { "ref": "1996, John Stillwell, Sources of Hyperbolic Geometry, page 82:", "text": "Hyperbolic geometry provides each line with two points at infinity. Whether there is a piece of the line beyond the points at infinity, completing the piece lying in the finite to a closed curve, we cannot say, since we cannot move as far as the points at infinity, let alone beyond them.", "type": "quote" }, { "text": "2007, Maurice Margenstern, Cellular Automata in Hyperbolic Spaces, Volume 1: Theory, page 66,\nAs we have seen, points at infinity behave as ordinary points: a single line passes through two distinct points at infinity or through a point in the plane and a point at infinity.\nThis representation of the points at infinity is a difference between the disc and the half-plane models of Poincaré, […] ." }, { "ref": "2012, Norbert A'Campo, Athanase Papadopoulos, “Notes on non-Euclidean Geometry”, in Athanase Papadopoulos, editor, Strasbourg Master Class on Geometry, page 126:", "text": "To describe a parabolic transformation f of the hyperbolic plane, we consider two parallel lines l and l’ and we call ω their common point at infinity.", "type": "quote" } ], "glosses": [ "An ideal point." ], "links": [ [ "geometry", "geometry" ], [ "ideal point", "ideal point" ] ], "raw_glosses": [ "(geometry, hyperbolic geometry) An ideal point." ], "tags": [ "excessive" ], "topics": [ "geometry", "mathematics", "sciences" ] } ], "synonyms": [ { "tags": [ "excessive" ], "topics": [ "geometry", "mathematics", "sciences" ], "word": "ideal point" }, { "tags": [ "excessive" ], "topics": [ "geometry", "mathematics", "sciences" ], "word": "omega point" } ], "wikipedia": [ "point at infinity" ], "word": "point at infinity" }
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