Mewnmathemateg, maeisometreg yndrawsffurfiad 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. Mewngofod Euclidaidd2 a 3 dimensiwn, os yw dau ffigur (neu ddausiâp) yn perthyn i'w gilydd drwy isometreg, yna dywedir eu bod "yngyfath". Mae'r berthynas hon, sy'n eu cysylltu, naill ai'n symudiad anhyblyg, neu'nadlewyrchiad.[2]
Mae'r gairGroegaiddisos yn golygu "hafal", sy'n cyfeirio at y pellter rhwng yr elfennau.
Gadewch iX aY fod yn ofod metrig, gyda metricsdX adY. Gelwirmapf :X →Y ynisometrig os ceir (ar gyfera,b ∈X)
[3]
Os dynodir gofodau fector normV aW, yna mae'risometreg llinol yn fap llinolf :V →W sy'n cadw neu'n prisyrfio'r norm/au:
ar gyfer pobv o fewnV.[9].
- ↑Coxeter 1969, t. 29
"We shall find it convenient to use the wordtransformation in the special sense of a one-to-one correspondence
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 memberP and a second memberP' 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, anisometry (or "congruent transformation," or "congruence") is a transformation which preserves length..."
- ↑Coxeter 1969, t. 39
3.11Any 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, naGeiriadur 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. https://archive.org/details/sim_science_2000-12-22_290_5500/page/2322.
- ↑Saul, Lawrence K.; Roweis, Sam T. (2003). "Think globally, fit locally:Unsupervised learning ofnonlinear manifolds". Journal of Machine Learning Research (http://jmlr.org/papers/v4/saul03a.html)+4 (June): 119–155. "Quadratic optimisation of
(page 135) such that
" - ↑Zhang, Zhenyue; Zha, Hongyuan (2004). "Principal Manifolds andNonlinear Dimension Reduction viaLocal 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)
