Difference between revisions of "Point space"
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A mathematical model of the space in which we live is the '''point space'''. Its elements are points. Objects in point space may be single points; finite sets of points like the centres of the atoms of a molecule; infinite discontinuous point sets like the centres of the atoms of an ideal crystal pattern; continuous point sets like straight lines, curves, planes, curved surfaces, etc. | A mathematical model of the space in which we live is the '''point space'''. Its elements are points. Objects in point space may be single points; finite sets of points like the centres of the atoms of a molecule; infinite discontinuous point sets like the centres of the atoms of an ideal crystal pattern; continuous point sets like straight lines, curves, planes, curved surfaces, etc. | ||
| − | Objects in point space are described by means of a coordinate system referred to point chosen as the origin O. An arbitrary point P is then described by its coordinates x, y, z. | + | Objects in point space are described by means of a coordinate system referred to point chosen as the origin O. An arbitrary point P is then described by its coordinates ''x'', ''y'', ''z''. |
| − | The point space used in crystallography is a Euclidean space, i.e. an affine space where the scalar product is defined. | + | The point space used in crystallography is a Euclidean space, ''i.e.'' an affine space where the scalar product is defined. |
Crystal structures are described in point space. The [[vector space]] is a dual of the point space because to each pair of points in point space a vector in vector space can be associated. | Crystal structures are described in point space. The [[vector space]] is a dual of the point space because to each pair of points in point space a vector in vector space can be associated. | ||
Revision as of 17:17, 7 February 2012
A mathematical model of the space in which we live is the point space. Its elements are points. Objects in point space may be single points; finite sets of points like the centres of the atoms of a molecule; infinite discontinuous point sets like the centres of the atoms of an ideal crystal pattern; continuous point sets like straight lines, curves, planes, curved surfaces, etc.
Objects in point space are described by means of a coordinate system referred to point chosen as the origin O. An arbitrary point P is then described by its coordinates x, y, z.
The point space used in crystallography is a Euclidean space, i.e. an affine space where the scalar product is defined.
Crystal structures are described in point space. The vector space is a dual of the point space because to each pair of points in point space a vector in vector space can be associated.
See also
- Chapter 8.1 in the International Tables for Crystallography Volume A
- Matrices, Mappings and Crystallographic Symmetry, teaching pamphlet No. 22 of the International Union of Crystallography