Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate
SUBJECT
| MII0018 | Linear Transformations |
| Teaching Staff in Charge |
Lect. OLAH-GAL Robert, Ph.D., robert.olah-gal cs.ubbcluj.ro |
| Aims |
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The linear transformation is a sub-field of computer science which studies methods for digitally synthesizing and manipulating visual content. |
| Content |
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In geometry, an affine transformation or affine map or an affinity (from the Latin, affinis, $connected with$) between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation:
In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, satisfying certain properties described below. Geometrically, an affine transformation in Euclidean space is one that preserves The collinearity relation between points; i.e., three points which lie on a line continue to be collinear after the transformation Ratios of distances along a line; i.e., for distinct collinear points p1, p2, p3, the ratio | p2 − p1 | / | p3 − p2 | is preserved In general, an affine transformation is composed of linear transformations (rotation, scaling or shear) and a translation (or $shift$). Several linear transformations can be combined into a single one, so that the general formula given above is still applicable. To visualise the general affine transformation of the Euclidean plane, take labelled parallelograms ABCD and A′B′C′D′. Whatever the choices of points, there is an affine transformation T of the plane taking A to A′, and each vertex similarly. Supposing we exclude the degenerate case where ABCD has zero area, there is a unique such affine transformation T. Drawing out a whole grid of parallelograms based on ABCD, the image T(P) of any point P is determined by noting that T(A) = A′, T applied to the line segment AB is A′B′, T applied to the line segment AC is A′C′, and T respects scalar multiples of vectors based at A. [If A, E, F are collinear then the ratio length(AF)/length(AE) is equal to length(A′F′)/length(A′E′).] Geometrically T transforms the grid based on ABCD to that based in A′B′C′D′.Affine transformations don@t respect lengths or angles; they multiply area by a constant factor area of A′ B′ C′ D′ / area of ABCD. 3D projection is any method of mapping three-dimensional points to a two-dimensional plane. As most current methods for displaying graphical data are based on planar two-dimensional media, the use of this type of projection is widespread, especially in computer graphics, engineering and drafting. |
| References |
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1. Wirth, N.: Algoritmusok+Adatstruktúrák = Programok, Műszaki Kiadó, Budapest, 1982.
2. Székely V.- Benkő T-né: Karakterisztikák, diagramok, nomogramok, Műszaki Könyvkiadó, Budapest, 1975. 3. Szabó, J., Számítógépi grafika, Debrecen, 1986 (Egyetemi jegyzet) 4. Szabó J., Feladatok a számítógépi grafikából, Debrecen, 1992 (Egyetemi jegyzet) 5. Moncea, J. - Geometrie descriptiva si desen tehnic, Editura Didactica si Pedagogica, Bucuresti, 1982 6. Tanasescu, A. - Geometrie descriptiva, perspectiva, axonometrie, Editura Didactica si Pedagogica, Bucuresti, 1979 |
| Assessment |
| Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |