See point at infinity in All languages combined, or Wiktionary
Download JSON data for point at infinity meaning in English (6.0kB)
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"ref": "2005, John Stillwell, The Four Pillars of Geometry, page 92",
"text": "The point where a family of parallels appear to meet is called their “vanishing point” by artists, and their point at infinity by mathematicians. The horizon itself, which consists of all the points at infinity, is called the line at infinity.",
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"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."
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"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.",
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"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.",
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"Any point added to a space to achieve projective completion."
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"(geometry, Euclidean projective geometry) Any point added to a space to achieve projective completion."
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"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": "quotation"
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"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, […] ."
},
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"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.",
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],
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"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."
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},
{
"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": "quotation"
},
{
"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": "quotation"
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],
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"Any point added to a space to achieve projective completion."
],
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"type": "quotation"
},
{
"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.",
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This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-04-24 from the enwiktionary dump dated 2024-04-21 using wiktextract (82c8ff9 and f4967a5). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
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