II. GEOMETRY 



2.00 Transformation of coordinates in a plane. 



2.001 Change of origin. Let x, y be a system of rectangular or oblique coor- 

 dinates with origin at O. Referred to x, y the coordinates of the new origin O' 

 are a, b. Then referred to a parallel system of coordinates with origin at 0' 

 the coordinates are #', y'. 



x = x f + a 



2.002 Origin unchanged. Directions of axes changed. Oblique coordinates. 

 Let co be the angle between the x y axes measured counter-clockwise from 

 the x- to the y-axis. Let the o/-axis make an angle a. with the #-axis and the 

 /-axis an angle jS with the #-axis. All angles are measured counter-clockwise 

 from the #-axis. Then 



x sin co = x' sin (co a) + y' sin (co /3) 

 y sin co = x' sin a + y f sin j8 

 co' = j8 - a. 



2.003 Rectangular axes. Let both new and old axes be rectangular, the new 

 axes being turned through an angle 6 with respect to the old axes. Then 



co-^.a -0./3-J+0. 



# = #' cos 6 y' sin 6 

 y = x' sin 6 + y' cos 0. 



2.010 Polar coordinates. Let the y-axis make an angle co with the #-axis and 

 let the x-axis be the initial line for a system of polar coordinates r, 6. All angles 

 are measured in a counter-clockwise direction from the x-axis. 



r sin (co 6) 



X = : 



sm co 

 sin 6 

 sin co 



7T 



2.011 If the x, y axes are rectangular, co = , 



x = r cos 6 



y = r sin 6. 

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