Aspect Ratio and Screen Compression Ratio 










A window  W = [a,b]×[c,d]  is a rectangular region in the 
(x,y)-plane which is bounded by the two vertical lines  x=a  and  
x=b, and the two horizontal lines y=c and y=d. The 
aspect ratio AR(W) of the window W is defined to be its
height divided by its width:
   
AR(W) = (d-c) ÷  (b-a)  .

Notice that if AR(W) = 1 then the window  W is
square. Otherwise, if AR(W) < 1 then W is elongated in the horizontal
direction, and  if AR(W) > 1 then it's elongated in the vertical
direction.



The aspect ratio can be interpreted geometrically as follows. Imagine  rescaling the 
window  W  by compressing (or stretching) it in the horizontal direction 
so that it has width 1, and then compressing (or stretching) in the vertical 
direction so that it also has height 1. Think of this rescaling as transforming 
the window  W = [a,b]×[c,d] into the window   W' = [0,1]×[0,1]. 
Under the transformation, a circle in the window  W is rescaled
into an ellipse whose axes are horizontal and vertical, and  where the ratio
of the length of the vertical axes divided by the length of the
horizontal is AR(W)-1.
Under the transformation, a line L which passes through the window  
W gets rescaled to a line L' which passes through the window  W'. 
If L has slope m then its not too hard to show that the rescaled line 
L'  will have slope m ÷ AR(W). 



If the calculator's screen were square then AR(W)  would be a direct measure of 
the amount of vertical/horizontal distortion present when a graph whose RANGE window 
is W is drawn on the calculator display. But since the screen is not square, 
we need to apply another 
transformation which rescales the square window  W' = [0,1]×[0,1]
into a window W'' whose aspect ratio is .588336783988  (which, you
will recall from the graphics display screen
page, is the aspect ratio of the TI-85's display screen). This leads us to define the
screen compression ratio CR(W) for the window W to
be


   
CR(W)   =  AR(W) ÷ .588336783988  .
 



A screen compression ratio of CR(W) = 1 indicates there is no distortion in 
the calculator's picture, so that circles look round and the graph of a line with slope 
one meets the horizontal at a 45 degree angle. 
The "ZQR" option under "ZOOM" on the graph menu returns a RANGE window
with compression ratio of 1. If CR(W) > 1 then the picture on 
the calculator display is distorted by vertical compression; 
larger values indicating a higher degree of compression.
If  CR(W) < 1 then the window is distorted by
horizontal compression; values closer to 0 indicating a higher degree of 
horizontal compression. 
A simple TI-85 program which returns the compression ratio for the currently
loaded RANGE window is described at the screen 
compression ratio program page. 




Here's another way to understand the compression ratio. Let W be the current 
RANGE window on the TI-85. Let T denote the angle 
with which the graph of a straight line of slope 1 intersects a horizontal line in 
the calculator display. Then T will equal 45° when there is
no distortion in the window; a vertically compressed window will have T
< 45° and a horizontally compressed window will have T
> 45°.*
In terms of T, the compression ratio of the window can be shown to equal
cot(T). Thus we have

T = arctan( 1 / CR(W) )
    = tan-1( 1 / CR(W) )




















This document was created in September, 1996 
and last revised on August 15, 1998.




Your questions, comments or suggestions are welcomed.
Please direct correspondence to:





Andy Miller

Department of Mathematics

University of Oklahoma

e-mail: amiller@ou.edu





URL: http://www.math.ou.edu/~amiller/ti85/cratio.htm