TI-92 -- Looking at a 2D World with 3D Eyes

Because this is a long module with several parts, this TI-92 window also is long. You may want to use the links below to go directly to one part of this window.

Dot product and magnitude
The intersection of two lines in R^3
Looking at a 3D scene from different points-of-view

You may want to look at the TI-92 help module vectors.

The following screen shows how to compute the dot product of two vectors using dotP and the magnitude or length of a vector from the dot product.

The TI-92 program below can be used to find the point where two lines "intersect" even when they do not exactly intersect because of measurement error.

Missing TI-92 program

As described in the module Looking at a 2D World with 3D Eyes the key step is solving the pair of equations

where

and

P(t) = t L + (1 - t) S

Q(s) = s V + (1 - s) G

Working out the details of the two equations above is a good exercise. You may want to refer to the module differentiating vector-valued functions. The final equations are

t (L - S).(L - S) + s (G - V).(L - S) = (G - S).(L - S)

t (L - S).(G - V) + s (G - V).(G - V) = (G - S).(G - V)

which is a pair of linear equations. The TI-92 program cross uses the TI-92 procedure simult to solve these equations. If you look at the details of the TI-92 program you will notice that the letters x and y are used for the unknowns s and t because the TI-92 does not distinguish between upper case and lower case.

To use the program cross, type cross() and then ENTER in the TI-92 home screen and then follow the instructions. You will be asked for the location of the light source, the shadow, the viewer, and the apparent position on the ground as shown in the screens below. These must be entered as three dimensional vectors in the usual way -- with their coordinates separated by commas and enclosed by square brackets. After you have entered all the necessary information the program cross will find the point that is closest to the two lines and then print the result. The computation takes a few seconds. When the computation is complete press Green-Diamond HOME to return to the home screen.

Now we want to look at a three dimensional scene from various distant points-of-view. The scene is described by a matrix with n rows, one for each rod in the scene. Each row has six columns -- the first three columns are three coordinates of one end of the rod and the last three coordinates are the coordinates of the other end of the rod. The following matrix describes the pyramid used as an example in this section.


-1  0  0  0  1  0
 0  1  0  1  0  0
 1  0  0  0 -1  0
 0 -1  0 -1  0  0
-1  0  0  0  0  1
 1  0  0  0  0  1
 0  1  0  0  0  1
 0 -1  0  0  0  1

The program below will create a matrix, pyramid, that contains this data. If your workstation has a TI-92 Graph Link cable, click on the button below, download the program pyrmdata in the usual way, and then run the program pyrmdata by typing pyrmdata() and then pressing ENTER in your home screen to create this matrix in your TI-92.

If you don't have a TI-Graph Link cable then you can enter the matrix by typing it in.

Missing program listing

The TI-92 program Lookat shown below draws the photograph taken by a camera located at a distant point

(R cos theta, R sin theta, H).

Missing TI-92 program

To use this program type lookat(mat) and then press ENTER in the home screen, where mat is the name of the matrix representing the three dimensional scene. See the screen below.

The program will ask you for theta, R, H, and the range as shown in the creen below.

The range determines the ranges of the x and y axes in the graph window. The y-axis runs from -range to range and the x-axis is chosen so that the scales for the two axes are the same. If range is the largest absolute value of all the coordinates in the matrix then everything will fit on the screen but you may want to make range a little bigger to allow room for margins or smaller if the picture doesn't entirely fill the screen. The screen below shows one photograph of the pyramid.

Use the same program to produce a series of photographs of your Tinkertoy house using your own matrix with the rods and points you found earlier.