The Rotation Group

• Autorzy: Karol Pąk
• Języki publikacji: EN

Abstrakty

EN

We introduce length-preserving linear transformations of Euclidean topological spaces. We also introduce rotation which preserves orientation (proper rotation) and reverses orientation (improper rotation). We show that every rotation that preserves orientation can be represented as a composition of base proper rotations. And finally, we show that every rotation that reverses orientation can be represented as a composition of proper rotations and one improper rotation.

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2012
• Tom: 20
• Numer: 1
• Strony: 23-29

Daty

wydano

2012-01-01

online

2012-09-12

Twórcy

autor

Karol Pąk

• Institute of Informatics, University of Białystok, Poland

Bibliografia

• Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
• Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
• Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
• Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
• Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.
• Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
• Czesław Byliński. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
• Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
• Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
• Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
• Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
• Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.
• Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.
• Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.
• Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2(4):475-480, 1991.
• Katarzyna Jankowska. Transpose matrices and groups of permutations. Formalized Mathematics, 2(5):711-717, 1991.
• Artur Korniłowicz and Yasunari Shidama. Intersections of intervals and balls in εn/T. Formalized Mathematics, 12(3):301-306, 2004.
• Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.
• Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275-278, 1992.
• Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
• Anna Lango and Grzegorz Bancerek. Product of families of groups and vector spaces. Formalized Mathematics, 3(2):235-240, 1992.
• Michał Muzalewski. Categories of groups. Formalized Mathematics, 2(4):563-571, 1991.
• Yatsuka Nakamura. Determinant of some matrices of field elements. Formalized Mathematics, 14(1):1-5, 2006, doi:10.2478/v10037-006-0001-4.[Crossref]
• Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
• Karol Pąk. Basic properties of determinants of square matrices over a field. Formalized Mathematics, 15(1):17-25, 2007, doi:10.2478/v10037-007-0003-x.[Crossref]
• Karol Pąk. Basic properties of the rank of matrices over a field. Formalized Mathematics, 15(4):199-211, 2007, doi:10.2478/v10037-007-0024-5.[Crossref]
• Karol Pąk. Block diagonal matrices. Formalized Mathematics, 16(3):259-267, 2008, doi:10.2478/v10037-008-0031-1.[Crossref]
• Karol Pąk. Linear transformations of Euclidean topological spaces. Formalized Mathematics, 19(2):103-108, 2011, doi: 10.2478/v10037-011-0016-3.[Crossref]
• Nobuyuki Tamura and Yatsuka Nakamura. Determinant and inverse of matrices of real elements. Formalized Mathematics, 15(3):127-136, 2007, doi:10.2478/v10037-007-0014-7.[Crossref]
• Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.
• Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.
• Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847-850, 1990.
• Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.
• Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
• Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
• Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.
• Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998.
• Xiaopeng Yue, Xiquan Liang, and Zhongpin Sun. Some properties of some special matrices. Formalized Mathematics, 13(4):541-547, 2005.
• Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205-211, 1992.
• Katarzyna Zawadzka. The product and the determinant of matrices with entries in a field. Formalized Mathematics, 4(1):1-8, 1993.

Identyfikatory

DOI

10.2478/v10037-012-0004-2