The Graphics Display Screen

In order to control the graphics display it is useful to understand the following attributes of the calculator's display screen:

If you are using a calculator other than the TI-85 then you should try to determine the corresponding information for your machine. Here are some ideas of how this was done for the TI-85:

  • The width and height of the TI-85 screen in pixels is mentioned in a few places in the TI-85 owner's manual. One can also simply go into the graphics mode and move the cursor one pixel at a time across the screen keeping track of the count. Do this first horizontally and then vertically.

  • The aspect ratio of the TI-85 screen can be estimated by measuring with a ruler. The screen has approximate height 43/32 inches and width 73/32 inches, so the aspect ratio is about 43/73 = .58904109589.

    A more precise approach is to start from the "ZSTD" RANGE window (which is [-10,10]×[-10,10]). Then choose the "ZSQR" window from the ZOOM menu. By definition, this window has no distortion, which means that its compression ratio is 1. (The compression ratio of a RANGE window equals the aspect ratio of the window divided by the aspect ratio of the display screen. So the aspect ratio of this window equals the aspect ratio of the display screen.) The dimensions of this "ZSQR" window are

    [-16.9970674487,16.9970674487]×[-10,10]

    and the height divided by the width equals   .588336783988.

    Yet another approach is the following. Start with any RANGE window (such as "ZSTD") which has the same width as height. From the "DRAW" menu choose "CIRCL" and create a circle centered at the origin with any positive radius. Let A be the x-coordinate of the x-intercept of the circle and let B be the y-coordinate of its y-intercept. Then B/A will be the aspect ratio of the display screen. Note: this approach can't be done with much accuracy as it is difficult to get close approximations to both A and B.

  • Let XP be the width of one pixel (say in inches) and let YP be its height. As before, let X and Y denote the width and height in inches of the display screen. Then X = 127×XP and Y = 63×YP, and so

    YP ÷ XP = (127/63)(Y ÷ X) = (127/63)(.588336783988) = 1.18601224709

    By dividing the measured width X and height Y of the display screen we can estimate the dimensions of each pixel:

    XP = X/127 ~ (73/32)/127 ~ .01796 inches
    and
    YP = Y/62 ~ (43/32)/63 ~ .02133 inches

Here are some ideas of how to use this information to get better control over graphical outputs:

  • Knowing the number of horizontal and vertical pixels in the display screen helps in getting the calculator to accurately represent the graphs of certain functions. For example, the calculator picture of f(x) = 5(x-2)/(x-2) on the RANGE window

    [-10,10]×[-10,10]

    fails to show that 2 is not in the domain of f. On the other hand if we graph this function on any of the RANGE windows

    [-63,63]×[-10,10], [-6.3,6.3]×[-10,10], [-3,7]×[-10,10], or [-6,6]×[-10,10]

    we do get an accurate picture. The reason is that in the first window x=2 was not one of the x-values that the calculator plotted whereas in the latter four windows it was. In general if the RANGE window is [a,b]×[c,d] then the calculator plots the points (x,f(x)) where x has the form

    x = a + i(b-a)/126

    where i is an integer between 1 and 126. Note that the midpoint of a and b is always one of the plotted x-values (take i = 63). And also points which are increments of one sixth, an one seventh, of the distance between a and b are plotted (take i = 21, 42, 63, 84, 105 for the sixths, and i = 18, 36, 54, 72, 90, 108 for the sevenths). This explains why the windows [-3,7]×[-10,10] and [-6,6]×[-10,10] work out nicely for this example. As you can see, the choice of [-6.3,6.3] for [a,b] is often very convenient since (b-a)/126 = .1.

  • Understanding screen compression (see the aspect ratio and screen compression ratio page) is necessary in order to analyze angles, slopes, derivatives, roundness and other geometric information from the calculator's graphical output. The basic principle is that a compression ratio larger than one indicates the picture has been vertically compressed, whereas a compression ratio less than one indicates horizontal compression.

  • On the TI-85 set the RANGE window to

    [-16.9970674487,16.9970674487]×[-10,10]

    which is a window whose compression ratio is 1. Have the calculator sketch the graphs of y = x and y = 1.18601224709 x. These are both straight lines through the origin but the graph of the second line looks nicer. Why? (The answer has to do with the aspect ratio of the individual pixels.)
    In the "ZSTD" window which line through the origin will have the nicest graph? Which line will have the nicest graph in a RANGE window whose compression ratio is a number C? 


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/display.htm