See affine transformation on Wiktionary
Download JSON data for affine transformation meaning in All languages combined (6.7kB)
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"word": "affine transformation matrix"
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"examples": [
{
"text": "An affine transformation does not in general preserve angles between lines or distances between points, but it does preserve ratios of distances between points lying on a straight line.",
"type": "example"
},
{
"text": "Given an affine space X, every affine transformation on X can be represented as the composition of a linear transformation on X and a translation of X.",
"type": "example"
},
{
"text": "Examples of affine transformations include translation, scaling, homothety, similarity, reflection, rotation, shear and compositions of them in any combination and sequence.",
"type": "example"
},
{
"ref": "1965, P. S. Modenov, A. S. Parkhomenko, translated by Michael B. P. Slater, Geometric Transformations, Volume 1: Euclidean and Affine Transformations, Academic Press, page 145",
"text": "Just as for plane transformations, we may show that the set of all affine transformations of space form a group.\nUnder an affine transformation of space, the image of a line is a line, and the image of a plane is a plane.",
"type": "quotation"
},
{
"ref": "1982, George E. Martin, Transformation Geometry, Springer, page 169",
"text": "Theorem 15.2 A transformation such that the images of every three collinear points are themselves collinear is an affine transformation.\nAre the affine transformations the same as those transformations for which the images of any three noncollinear points are themselves noncollinear? We shall see the answer is \"Yes.\"",
"type": "quotation"
},
{
"text": "2004, Solomon Khmelnik, Computer Arithmetic of Geometrical Figures: Algorithms and Hardware Design, Mathematics in Computer Comp., page 8,\nMost striking and well-known examples of affine transformation applications are computer tomography (see for instance [1]) and information compression for telecommunication systems (see [2]).\nThis book describes affine transformations (displacements, turns, scaling, shifts) of n-dimensional figures, where n=1,2,3,4."
}
],
"glosses": [
"A geometric transformation that preserves lines and parallelism, but in general not lengths or angles; (more formally) an automorphism of an affine space: a mapping of an affine space onto itself that preserves both the dimension of any affine subspace and the ratio of the lengths of any pair of parallel line segments."
],
"holonyms": [
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"word": "affine group"
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],
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{
"word": "affine map"
},
{
"word": "affine mapping"
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{
"english": "mappings from one space to a (generally) different one",
"word": "affine homomorphism"
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"word": "linear transformation"
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"id": "en-affine_transformation-en-noun-lL3t4Qz4",
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"raw_glosses": [
"(geometry, linear algebra) A geometric transformation that preserves lines and parallelism, but in general not lengths or angles; (more formally) an automorphism of an affine space: a mapping of an affine space onto itself that preserves both the dimension of any affine subspace and the ratio of the lengths of any pair of parallel line segments."
],
"synonyms": [
{
"sense": "geometric transformation that preserves lines and parallelism",
"word": "affinity"
}
],
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"geometry",
"linear-algebra",
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"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"sense": "geometric transformation that preserves lines and parallelism",
"word": "仿射變換"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "fǎngshè biànhuàn",
"sense": "geometric transformation that preserves lines and parallelism",
"word": "仿射变换"
},
{
"code": "et",
"lang": "Estonian",
"sense": "geometric transformation that preserves lines and parallelism",
"word": "afiinne teisendus"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "geometric transformation that preserves lines and parallelism",
"word": "affiinikuvaus"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "geometric transformation that preserves lines and parallelism",
"word": "affiini kuvaus"
},
{
"code": "fr",
"lang": "French",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"feminine"
],
"word": "transformation affine"
},
{
"code": "de",
"lang": "German",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"feminine"
],
"word": "affine Abbildung"
},
{
"code": "hu",
"lang": "Hungarian",
"sense": "geometric transformation that preserves lines and parallelism",
"word": "affin transzformáció"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"feminine"
],
"word": "vildarvörpun"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"feminine"
],
"word": "vildarmótun"
},
{
"code": "it",
"lang": "Italian",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"feminine"
],
"word": "trasformazione affine"
},
{
"code": "ja",
"lang": "Japanese",
"roman": "afinhenkan",
"sense": "geometric transformation that preserves lines and parallelism",
"word": "アフィン変換"
},
{
"code": "ru",
"lang": "Russian",
"roman": "affínnoje preobrazovánije",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"neuter"
],
"word": "аффи́нное преобразова́ние"
},
{
"code": "es",
"lang": "Spanish",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"feminine"
],
"word": "transformación afín"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
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"word": "affin transformation"
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"wikipedia": [
"affine transformation"
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"word": "affine transformation"
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{
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{
"word": "affine map"
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{
"word": "affine mapping"
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{
"english": "mappings from one space to a (generally) different one",
"word": "affine homomorphism"
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"word": "linear transformation"
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"lang": "English",
"lang_code": "en",
"pos": "noun",
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"examples": [
{
"text": "An affine transformation does not in general preserve angles between lines or distances between points, but it does preserve ratios of distances between points lying on a straight line.",
"type": "example"
},
{
"text": "Given an affine space X, every affine transformation on X can be represented as the composition of a linear transformation on X and a translation of X.",
"type": "example"
},
{
"text": "Examples of affine transformations include translation, scaling, homothety, similarity, reflection, rotation, shear and compositions of them in any combination and sequence.",
"type": "example"
},
{
"ref": "1965, P. S. Modenov, A. S. Parkhomenko, translated by Michael B. P. Slater, Geometric Transformations, Volume 1: Euclidean and Affine Transformations, Academic Press, page 145",
"text": "Just as for plane transformations, we may show that the set of all affine transformations of space form a group.\nUnder an affine transformation of space, the image of a line is a line, and the image of a plane is a plane.",
"type": "quotation"
},
{
"ref": "1982, George E. Martin, Transformation Geometry, Springer, page 169",
"text": "Theorem 15.2 A transformation such that the images of every three collinear points are themselves collinear is an affine transformation.\nAre the affine transformations the same as those transformations for which the images of any three noncollinear points are themselves noncollinear? We shall see the answer is \"Yes.\"",
"type": "quotation"
},
{
"text": "2004, Solomon Khmelnik, Computer Arithmetic of Geometrical Figures: Algorithms and Hardware Design, Mathematics in Computer Comp., page 8,\nMost striking and well-known examples of affine transformation applications are computer tomography (see for instance [1]) and information compression for telecommunication systems (see [2]).\nThis book describes affine transformations (displacements, turns, scaling, shifts) of n-dimensional figures, where n=1,2,3,4."
}
],
"glosses": [
"A geometric transformation that preserves lines and parallelism, but in general not lengths or angles; (more formally) an automorphism of an affine space: a mapping of an affine space onto itself that preserves both the dimension of any affine subspace and the ratio of the lengths of any pair of parallel line segments."
],
"links": [
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"(geometry, linear algebra) A geometric transformation that preserves lines and parallelism, but in general not lengths or angles; (more formally) an automorphism of an affine space: a mapping of an affine space onto itself that preserves both the dimension of any affine subspace and the ratio of the lengths of any pair of parallel line segments."
],
"topics": [
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"mathematics",
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"synonyms": [
{
"sense": "geometric transformation that preserves lines and parallelism",
"word": "affinity"
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],
"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"sense": "geometric transformation that preserves lines and parallelism",
"word": "仿射變換"
},
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "fǎngshè biànhuàn",
"sense": "geometric transformation that preserves lines and parallelism",
"word": "仿射变换"
},
{
"code": "et",
"lang": "Estonian",
"sense": "geometric transformation that preserves lines and parallelism",
"word": "afiinne teisendus"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "geometric transformation that preserves lines and parallelism",
"word": "affiinikuvaus"
},
{
"code": "fi",
"lang": "Finnish",
"sense": "geometric transformation that preserves lines and parallelism",
"word": "affiini kuvaus"
},
{
"code": "fr",
"lang": "French",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"feminine"
],
"word": "transformation affine"
},
{
"code": "de",
"lang": "German",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"feminine"
],
"word": "affine Abbildung"
},
{
"code": "hu",
"lang": "Hungarian",
"sense": "geometric transformation that preserves lines and parallelism",
"word": "affin transzformáció"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"feminine"
],
"word": "vildarvörpun"
},
{
"code": "is",
"lang": "Icelandic",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"feminine"
],
"word": "vildarmótun"
},
{
"code": "it",
"lang": "Italian",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"feminine"
],
"word": "trasformazione affine"
},
{
"code": "ja",
"lang": "Japanese",
"roman": "afinhenkan",
"sense": "geometric transformation that preserves lines and parallelism",
"word": "アフィン変換"
},
{
"code": "ru",
"lang": "Russian",
"roman": "affínnoje preobrazovánije",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"neuter"
],
"word": "аффи́нное преобразова́ние"
},
{
"code": "es",
"lang": "Spanish",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"feminine"
],
"word": "transformación afín"
},
{
"code": "sv",
"lang": "Swedish",
"sense": "geometric transformation that preserves lines and parallelism",
"tags": [
"common-gender"
],
"word": "affin transformation"
}
],
"word": "affine transformation"
}
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-06-04 from the enwiktionary dump dated 2024-05-02 using wiktextract (e9e0a99 and db5a844). 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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