Stereographer
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This collection produces stereographs of mathematical functions, mixed series and regression results. When properly viewed, they provide a true 3D effect. Requirements: Windows 98 or later.
Click on any image for full size screenshot
To view each stereograph, cross your eyes and merge the two images
stereo.gif (78235 bytes) moebius.gif (83747 bytes) hyp_par.gif (84797 bytes) hyperb.gif (90903 bytes) Download Stereographer
Produces stereographs of mathematical functions or of approximations to the 1st and 2nd order partial derivatives of a function of 2 variables.

The above are STEREOSCOPIC GRAPHS which give the realistic sensation of depth like those 3D viewers we all had as children. But these are designed for free viewing and are cross-eyed stereographs, so the image corresponding to the right eye is on the left and vice-versa. See instructions below.

Surfaces or space curves can be stereographed. The representation can be nonparametric:

z = f(x,y)

or parametric:

x = x(u,v)  ;  y = y(u,v)  ;  z = z(u,v)   (where u and v are the parameters)

You can visualize, for instance, the hyperbolic paraboloid

z = (x2/25-y2/16)/6

or the sphere

x = 2*sin(u)*cos(v)  ;  y = 2*sin(u)*sin(v)  ;  z = 2*cos(u)

Pending programs:

STEREOGRAPHER - FOR MIXED SERIES IN 2 VARIABLES stereographs any 2-variable polynomial, trigonometric, or mixed series of the type produced by programs of the APPROXIMATION AND INTERPOLATION collection or by the MIXED SERIES CREATION program of the SUPPORT collection. Also, any partial derivative of the series can be stereographed. In addition, the original points used to find the series can also be stereographed. These are either the original grid points used to approximate a continuous function, or the original tabulated function.

STEREOGRAPHER - FOR REGRESSION ANALYSIS RESULTS stereographs any 2-variable regression surface produced by the programs of the REGRESSION collection (such a file must first be converted by the program which converts files from results of REGRESSION to STEREOGRAPHER, which is in the same diskette). In addition, the original points used to find the regression surface can also be stereographed.

Authentic depth perception requires that each eye receive a slightly different view of an object, that is, from two slightly different angles. The pair of side-by-side graphs in a stereograph look very similar, but they are not identical. Each represents a view of an object from slightly different angles. If you feed one view to one eye and the other view to the other eye, the brain blends this information to give you the sensation of three-dimensionality.

Stereographs produced by this collection can be viewed by most without the aid of a stereoscope. This is because almost everyone can cross their eyes at will since this is precisely what they do when they focus on an object which is only a few inches from their eyes. In these stereographs the view for the right eye is placed on the left, and vice versa, as illustrated in the figure below. The area labeled R holds the view for the RIGHT eye and the area labeled L holds the view for the LEFT eye. We thus loosely call them "cross-eyed" stereographs.

  R   L

You can practice with a stereograph displayed on the screen or with a printout of a stereograph laid flatly on a surface. Aim your sight perpendicularly at the center of the line which separates the two views. Start crossing your eyes until the two images merge.
 
The following is a simple cross-eyed stereograph. If you can merge the two words you will feel a depth sensation.

t    e  r    e  o s    t  e   r   e    o

If you are one of the people that have some difficulty controlling your eyes in the required manner, practice by first viewing the stereographs from a few feet away and move in closer until you are at a normal distance from the screen. In addition, you can develop smaller stereographs using a reduction factor of less than 1, for instance .5 or even .25. With a reduced stereograph, since the two views are closer together, a lesser degree of eye-crossing is needed to view them. You can slowly increase the reduction factor until you are capable of viewing a full size stereograph.

You can use a pencil to exercise the required muscles. Hold it at arm's length, focus your eyes on it, then slowly bring it as close to your nose as possible without loosing the focus. It may help if your head is tilted back somewhat. In fact, if you have a stereograph in the background, it is possible that you would notice the two images merging as the pencil gets closer to your nose. Once you learn to merge the two images, it is easy to keep a lock on them because the brain is accustomed to receiving different information from the two eyes and combining them to give the true three-dimensional sensation we get from real solid objects.

The only skill required is that you be able to cross your eyes. Remember, everyone can do it. In fact, most of the time our eyes are slightly crossed because the objects we look at are not at infinity. If you need some practice, it will be worth it. Stereographs provide a fascinating sensation.

You have full control on the characteristics of the graph since you will place the axes wherever you wish, determine the size of the divisions to be placed on them, set the X and Y ranges (or U and V ranges for the parametric version), set the number of grid lines by setting the number of spacings, select the number of plotting segments per spacing, define the X, Y and Z windows. If you wish to translate and/or rotate the figure, you enter the desired X, Y and Z translations and the rotations about the X, Y and Z axes. If you wish to have the surface represented by dots randomly scattered on it, you select the number of dots you wish generated. You can use grid lines alone, dots alone, or both. You even control the stereoscopic factor, thereby adjusting the magnitude of the three-dimensional effect to suit your perceptions. In addition, reduced size stereographs can be developed if so desired. In the case of the MIXED SERIES and the REGRESSION versions, you can also have plotted the original data thereby visualizing the three-dimensional scatter diagram.

Once you have set up a nice stereograph, you can store all the details in a disk file which you can retrieve at will.


Check out some stereographs produced by the old GWBasic Stereographer!
 
To view each stereograph, cross your eyes and merge the two images
sphere0.gif (15739 bytes) sincos.gif (19729 bytes) torus.gif (16509 bytes) curve.gif (13498 bytes)
Click on any image for full size stereograph
Click and be patient with this one if you have a slow connection. It's an animated gif - 248KB. When fully loaded it'll show a rotating torus.
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