## Contour Plotting for the TI-85

### Purpose of program:

This program makes contour plots for functions of the form F(x,y). This program is still in development. A friendly interface will probably be added in the future. Unfortunately, currently the program is not much faster than the brute-force pixel scanning method, but I think the results are nicer.

### Using the program:

This version does not have a friendly interface. To use the program,

1. Store a function in y1 using the variables x and y. These must be lower case.
2. Set the range on x and y using the [RANGE] setting on the TI-85.
3. Run the program. Execution takes 2-5 minutes, depending on the function and range. The program will graph around 8 contour levels that are automatically selected. The program can only graph contours that intersect the edge of the screen, so there may be some contours wholly contained on the screen that the program misses.

### Examples to try:

Examples:
(Set these on the [GRAPH][y(x)=] and [GRAPH][RANGE] screens, respectively.) Then run the program and enjoy the show.

• y1=8y^3+12x2-24xyxMin = -2, xMax = 2,  yMin = -2,  yMax = 2
• y1=y^3+3y-3y2-x2y+x2+x^4-1,   xMin = -0.4,  xMax = 0.4,  yMin = 0.46965, yMax = 1.5

### Warnings:

While the program has been tested with some care, it can still crash. Please send SPECIFIC examples that cause crashes, including the exact formula used for the function and the exact range, to mjanebawillamette.edu . Thank you for your assistance in this.

This program cannot make contour plots for functions that are not differentiable functions of x and y. In fact, any function that the TI-85's [der1] command cannot handle will cause a crash. Changing to [NDer] won't help, since the base algorithm depends on differentiability. For example, the program cannot handle y1=abs(x)+abs(y)if the range includes any of the contours' "corners".

I am providing this program in binary format only. I do not anticipate posting an ASCII version. This program is very long and complex (about 2900 keystrokes), and the likelihood of making typos if it were hand-entered is very great. The program is NOT edit-locked, so you are welcome to download it, and read it using TI's Graph-link software. You can download Graph-link from TI and use it to read the program even if you don't have the cable to pass it to your calculator.

### How the program works:

Let's assume we have entered F(x,y) in y1, and the range has been set as desired.

1. Sample values of F are taken around the edge of the screen to determine the range of values. This range is divided by 8, the default number of contour levels. The quotient is called z and is the spacing between contour levels.
2. Starting 4 pixels up from the bottom left of the screen, a contour is started. After that the program scans around the edge of the screen looking for points (x,y) for which the value of F differs from the previous contour level by more than z. When such a point is found, a new contour is begun.
3. To plot a contour, we use the fact that a contour line is perpendicular to the gradient of F. Alternately, you can say that since F(x,y) = c on the contour, then dF = Fxdx+ Fydy = 0, so dy/dx = -Fx/Fy (again, along the contour). The TI-85 can compute this (assuming F is stored in y1) as -der1(y1,x)/der1(y1,y). The program uses a combination of three methods to plot the actual curve:
1. A predictor-corrector method with a variable step-size is used to plot the curve as the solution to a differential equation. This variable step-size makes this method adaptive; if the curve begins to turn tightly, the stepsize is automatically reduced, then lengthened later. This gives the visual impression of the curve "feeling its way" around tight corners in contours as the graph slows down dramatically at such points. More specifically, my method adds adaptive stepsize to what is called a trapezoid method, a modified Euler method, or a second-order Runge-Kutta method by various texts. Using the trapezoid estimate, the program subdivides an interval if the angle between the tangents at the two ends of the current step exceeds 15 degrees. AC is the tangent of the critical angle, and is set at the beginning of the program. This adaptive aspect not only makes sure the endpoints of each plotted segment are close to correct, but also that no corners on a contour are "cut" as a segment is plotted.
2. When the adaptive portion of the program settles on a point, the program evaluates F there, and performs a linear projection along the gradient to move the point so that the value of F is correct for the contour being plotted. This fixed the annoying problem of distinct contours being plotted as crossing each other.
3. Finally, if the slope of the graph exceeds Mc (critical slope, default set at 1.4), the program ceases to plot y as a function of x and instead plots x as a function of y. This solves two problems: First, the base adaptive predictor-corrector method can only plot functions. Contours often fail the vertical-line test. Secondly, to plot those non-function curves, there will be a vertical tangent somewhere. Large slope values, and especially singularities, cause serious errors in predictor-corrector methods, adaptive or not.
4. Contours terminate when the graph wanders off the screen. Points where the contours exit the screen are stored. New contours are not initiated within 4 pixels of an exit point, to keep the program from double-plotting a single contour.
5. This program computes slopes as -der1(y1,x)/der1(y1,y) or -der1(y1,y)/der1(y1,x). Since the denominator could be zero, this is a source of crashes. Further, either der1(y1,x) or der1(y1,y) could themselves be undefined at any given point. To eliminate crashes from common math-course examples, I have perturbed the graphing range variables xMin, xMax, yMin, and yMax by a factor of (1+1E-12) (or just added 1E-12 if the variable is zero). Consequently der1(y1,y) and der1(y1,x)are rarely evaluated at integers. This fix was suggested by a kind contributor to the graph-TI discussion list, Tom Bird of Austin Community College. Thanks, Tom.