Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate
SUBJECT

Code	Subject	
MII0018	Linear Transformations	

Section	Sem.	Hours: C+S+L	Category	Type	
Computer Science - in Hungarian, Miercurea Ciuc	5	2+0+1	speciality	optional	

Teaching Staff in Charge	Lect. OLAH-GAL Robert, Ph.D., robert.olah-gal@cs.ubbcluj.ro	
Aims	The linear transformation is a sub-field of computer science which studies methods for digitally synthesizing and manipulating visual content.	
Content	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 &#8722; p1 | / | p3 &#8722; 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&#8242;B&#8242;C&#8242;D&#8242;. Whatever the choices of points, there is an affine transformation T of the plane taking A to A&#8242;, 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&#8242;, T applied to the line segment AB is A&#8242;B&#8242;, T applied to the line segment AC is A&#8242;C&#8242;, 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&#8242;F&#8242;)/length(A&#8242;E&#8242;).] Geometrically T transforms the grid based on ABCD to that based in A&#8242;B&#8242;C&#8242;D&#8242;.Affine transformations don@t respect lengths or angles; they multiply area by a constant factor
area of A&#8242; B&#8242; C&#8242; D&#8242; / 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.
	
Bibliography	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