Isometreg

Mewn mathemateg, mae isometreg yn drawsffurfiad lle cedwir yr hyd (neu'r pellter) rhwng y gofod metrig heb ei newid.^[1] Mewn geiriau eraill, mae isometreg yn drawsffurfiad sy'n mapio elfennau o un gofod metrig i un arall, gan barchu hyd y gofod a geir rhwng yr elfennau, yn union. Mewn gofod Euclidaidd 2 a 3 dimensiwn, os yw dau ffigur (neu ddau siâp) yn perthyn i'w gilydd drwy isometreg, yna dywedir eu bod "yn gyfath". Mae'r berthynas hon, sy'n eu cysylltu, naill ai'n symudiad anhyblyg, neu'n adlewyrchiad.^[2]

Mae'r gair Groegaidd isos yn golygu "hafal", sy'n cyfeirio at y pellter rhwng yr elfennau.

Diffiniad

Gadewch i X a Y fod yn ofod metrig, gyda metrics d_X a d_Y. Gelwir map f : X → Y yn isometrig os ceir (ar gyfer a,b ∈ X)

d_{Y}\left(f(a),f(b)\right)=d_{X}(a,b).

^[3]

Enghreifftiau

Mae unrhyw adlewyrchiad, trawsfudiad (translation) a chylchdro yn 'isometreg global'.^[4]
Mae'r map $x\mapsto |x|$ mewn ${\mathbb {R} }$ yn llwybr isometrig ond nid yw'n isometrig.
Mae'r map llinol isometrig o Cⁿ iddo ef ei hun yn cael ei ddynodi gan fatricsau unedol.^[5]^[6]^[7]^[8]

Isometreg llinol

Os dynodir gofodau fector norm V a W, yna mae'r isometreg llinol yn fap llinol f : V → W sy'n cadw neu'n prisyrfio'r norm/au:

\|f(v)\|=\|v\|

ar gyfer pob v o fewn V.^[9].

Cyfeiriadau

↑ Coxeter 1969, t. 29
"We shall find it convenient to use the word transformation in the special sense of a one-to-one correspondence $P\to P'$ among all points in the plane (or in space), that is, a rule for associating pairs of points, with the understanding that each pair has a first member $P$ and a second member $P'$ and that every point occurs as the first member of just one pair and also as the second member of just one pair...
In particular, an isometry (or "congruent transformation," or "congruence") is a transformation which preserves length..."
↑ Coxeter 1969, t. 39
3.11 Any two congruent triangles are related by a unique isometry.
↑ Beckman, F. S.; Quarles, D. A., Jr. (1953). "On isometries of Euclidean spaces". Proceedings of the American Mathematical Society 4: 810–815. doi:10.2307/2032415. MR 0058193. http://www.ams.org/journals/proc/1953-004-05/S0002-9939-1953-0058193-5/S0002-9939-1953-0058193-5.pdf.
↑ geiriaduracademi.org; Dim term Cymraeg am 'global' yng Ngeiriadur Bangor, na Geiriadur yr Academi; adalwyd 29 Rhagfyr 2018.
↑ Roweis, S. T.; Saul, L. K. (2000). "Nonlinear Dimensionality Reduction by Locally Linear Embedding". Science 290 (5500): 2323–2326. doi:10.1126/science.290.5500.2323. PMID 11125150.
↑ Saul, Lawrence K.; Roweis, Sam T. (2003). "Think globally, fit locally: Unsupervised learning of nonlinear manifolds". Journal of Machine Learning Research (http://jmlr.org/papers/v4/saul03a.html)+4 (June): 119–155. "Quadratic optimisation of $\mathbf {M} =(I-W)^{\top }(I-W)$ (page 135) such that $\mathbf {M} \equiv YY^{\top }$ "
↑ Zhang, Zhenyue; Zha, Hongyuan (2004). "Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment". SIAM Journal on Scientific Computing 26 (1): 313–338. doi:10.1137/s1064827502419154.
↑ Zhang, Zhenyue; Wang, Jing (2006). "MLLE: Modified Locally Linear Embedding Using Multiple Weights". Advances in Neural Information Processing Systems 19. https://papers.nips.cc/paper/3132-mlle-modified-locally-linear-embedding-using-multiple-weights. "It can retrieve the ideal embedding if MLLE is applied on data points sampled from an isometric manifold."
↑ Thomsen, Jesper Funch (2017). Lineær algebra [Linear algebra] (yn Danish). Århus: Department of Mathematics, Aarhus University. t. 125.CS1 maint: unrecognized language (link)

[CoxeterIsometryDef-1] Coxeter 1969, t. 29
"We shall find it convenient to use the word transformation in the special sense of a one-to-one correspondence $P\to P'$ among all points in the plane (or in space), that is, a rule for associating pairs of points, with the understanding that each pair has a first member $P$ and a second member $P'$ and that every point occurs as the first member of just one pair and also as the second member of just one pair...
In particular, an isometry (or "congruent transformation," or "congruence") is a transformation which preserves length..."

[2] Coxeter 1969, t. 39
3.11 Any two congruent triangles are related by a unique isometry.

[3] Beckman, F. S.; Quarles, D. A., Jr. (1953). "On isometries of Euclidean spaces". Proceedings of the American Mathematical Society 4: 810–815. doi:10.2307/2032415. MR 0058193. http://www.ams.org/journals/proc/1953-004-05/S0002-9939-1953-0058193-5/S0002-9939-1953-0058193-5.pdf.

[4] riaduracademi.org; Dim term Cymraeg am 'global' yng Ngeiriadur Bangor, na Geiriadur yr Academi; adalwyd 29 Rhagfyr 2018.

[5] Roweis, S. T.; Saul, L. K. (2000). "Nonlinear Dimensionality Reduction by Locally Linear Embedding". Science 290 (5500): 2323–2326. doi:10.1126/science.290.5500.2323. PMID 11125150.

[6] Saul, Lawrence K.; Roweis, Sam T. (2003). "Think globally, fit locally: Unsupervised learning of nonlinear manifolds". Journal of Machine Learning Research (http://jmlr.org/papers/v4/saul03a.html)+4 (June): 119–155. "Quadratic optimisation of $\mathbf {M} =(I-W)^{\top }(I-W)$ (page 135) such that $\mathbf {M} \equiv YY^{\top }$ "

[7] Zhang, Zhenyue; Zha, Hongyuan (2004). "Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment". SIAM Journal on Scientific Computing 26 (1): 313–338. doi:10.1137/s1064827502419154.

[8] Zhang, Zhenyue; Wang, Jing (2006). "MLLE: Modified Locally Linear Embedding Using Multiple Weights". Advances in Neural Information Processing Systems 19. https://papers.nips.cc/paper/3132-mlle-modified-locally-linear-embedding-using-multiple-weights. "It can retrieve the ideal embedding if MLLE is applied on data points sampled from an isometric manifold."

[Thomsen_2017_p125-9] Thomsen, Jesper Funch (2017). Lineær algebra [Linear algebra] (yn Danish). Århus: Department of Mathematics, Aarhus University. t. 125.CS1 maint: unrecognized language (link)

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