Home
Math symbols
Jokes
Forum
About us
Links
Contact us
Site map
Search The Site
   
   Program of Lessons
 
 Study Guide
 Topics of problems
 Tests & exams
www.bymath.com Study Guide - Arithmetic Study Guide - Algebra Study Guide - Geometry Study Guide - Trigonometry Study Guide - Functions & Graphs Study Guide - Principles of Analysis Study Guide - Sets Study Guide - Probability Study Guide - Analytic Geometry Select topic of problems Select test & exam Rules Price-list Registration

                                                               

Transformations of coordinates

 

Translation of axes. Turning around origin of coordinates.

Central symmetry. Homothetic transformation. Affine transformation.

 

Consider some transformations tied with a transition from one coordinate system to another. Here ( ) ( '' ) are coordinates of arbitrary point  P in old and new coordinate systems correspondingly.

 

Translation of axes. Let's move the coordinate system  XOY in a plane so, that the axes OX and  OY  are parallel to themselves, and the origin of coordinates    moves to the point   ' ( a, b ). We'll receive the new coordinate system  X'O'Y' ( Fig.1 ):

                                      

 

Coordinates of the point  P  in the new and old coordinate systems are tied by the equations:

 

                                                                                                                

 

Turning around origin of coordinates. Let's turn the coordinate system  XY  in a plane by an angle   ( Fig.2 ).

 

                                                  

 

Now coordinates of the point  P  in the new and old coordinate systems are tied by the equations:

 

                                                                                

 

In the particular case:  =   we'll receive a central symmetry relatively the origin of coordinates O :

 

                                                                                 

 

A homothetic transformation with a center  O ( a , b )  and a coefficient  k 0 :

 

                                                                                 

An affine transformation:

                                                                                  

 

An affine transformation transfers straight lines to straight lines, intersecting lines to intersecting lines, parallel straight lines to parallel straight lines. All above mentioned transformations of coordinates are affine transformations.

 

Back


| Home | About us | Links | Contact us |

Copyright 2002-2007 Dr. Yury Berengard.  All rights reserved.