true Implicit-function graphing routine Anonymous Algorithm to graph implicit functions of 2 variables. This is useful for graphing equations f(x,y)=K (where K is a constant) that are difficult or impossible to separate into a 1-variable function y=F(x). This can also be used to make contour plots of a 3-dimensional function z=f(x,y). Examples, 2-D Contour Graphs 1 4 radians decimal true

What kind of problem can this example be used to solve?

Suppose you want to plot an equation that has 2 variables (they might be called x and y) and, if the equation is described as LeftSide = RightSide, neither LeftSide nor RightSide is 0. For example, sin(2*x)*cos(y) = 0.4 Then write this equation with either LeftSide or RightSide subtracted from both sides so that the equation looks like LeftSide-RightSide=0 (or RightSide-LeftSide=0) In the above example, you could subtract 0.4 to get: sin(2*x)*cos(y)-0.4 = 0 Then define a function which equals (in this example) LeftSide-RightSide. When this function = 0, your original equation is true. The equation needs to be written "= 0" because the implicit-function grapher is designed to plot the points where some expression (the function given by you) = 0. If there was originally a function z=Funct(x,y), such as z=sin(2*x)*cos(y) in the above example, then graphing Funct(x,y) = K, which is Funct(x,y)-K = 0, where K is a constant, makes part of a contour plot of Funct.

This example is based on http://twt.mpei.ac.ru/ochkov/Lace/Lace_eng.htm

Definitions:

The function to be plotted:

Specify what region of the input variables you want to plot (generically, we'll call these variables x and y):

Specify the fineness of display resolution of the plot:

How big is the graphing problem?

Based on the above constants, let's find out how many points will be potentially graphed:

301 301 90601

Total number of points potentially graphed

(Scroll down for the implicit-function graphing routine itself)

The implicit-function graphing routine:

It works by computing the value of the function at and near each one of the xsize*ysize points, scanning across the specified region (of the input variables) in grid fashion. The value ("Pvalue") at each point P is multiplied by the value ("Qvalue") at a point Q that is close to point P (distance "xstep" away from P). - If P or Q is a root of the equation, then Pvalue or Qvalue is 0, so Pvalue * Qvalue = 0. - If a root is between P and Q, then, typically, one of Pvalue or Qvalue is positive and the other is negative, so Pvalue * Qvalue < 0. This requires that: (a) xstep is small enough so that no other roots are between P and Q, and (b) there does not happen to be a root between P and Q that is also a local maximum or minimum [example: if f(x)=x^2 and P is at x=-1 and Q is at x=+1, then Pvalue and Qvalue are both positive, even though there is a root x=0 between P and Q]. So if Pvalue * Qvalue <= 0, the coordinates (x,y) of P are added (concatenated) to a list of points to plot. The same procedure that has just been described for points P and Q is also done for points P and R, where R is close in the y-direction (by distance ystep) to point P. If only P and Q were considered, then nearly horizontal parts of the graph might not have as many points, which would make them harder to see. The names P, Q, R are used only in this text, not in the graphing routine itself. P is (x, y) Q is (x-xstep, y) R is (x, y-ystep)

Initialize the list of points to plot; otherwise, when the first point is to be added, the software won't be able to detect the dimensions of the list and will indicate an error.

 301

A dummy "else" clause is used because it was not evident how to specify an "if" with no "else."

Remove the first "point" that was added when the list of points was initialized. This will NOT erase the point (0, 0), if it happened to be found as a root of the equation or found to be near a root - any (0, 0) that was found will still appear in the list.

(Scroll down for the results)

Results:

1188

The number of points that were found to be roots, or to have roots near them.

 Points