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Graphing Non-Functions

March 22, 2012

My current gig is winding down, so this will probably be the last update on the function graphing engine for a while.  Although the function graphing engine is architected to graph cartesian or parameteric functions, only cartesian functions were supported in the initial release.  All functions defined in XML must implement an IPlottable interface, which means that evaluation and derivative methods are interpreted with y as a strict function of x.  All functions must be marked as cartesian, since a parametric function will not plot in the initial implementation.

What about non-functions, that is cases where y is not strictly a function of x?  As the saying goes, there is always a back door.  Functions may identify themselves as self-plotting.  In this case, the engine does not sample the function.  Instead, it provides the function with all the relevant information regarding the current graph window and calls the function’s plot() method.  A self-plotting function is welcome to draw anything it pleases, including Bezier curves, conic sections, spirals, or even smiley faces🙂

Problems may arise with derived functions, however, i.e. other functions that derive their visual display from the definition of another function.  A graphical Marker function is a perfect example.  The Marker is an artist-generated, interactive graphic asset that is constrained to the path of another function.  If domain values exist for which there is not a unique range value, then Marker movement is unpredictable.

This example shows how to work around the non-function issue to a degree.  It plots parabolas that open upward, downward, and to the left or right.  More specifically, the parabola directrix is either parallel to the x or y axis.  Parallel to the x-axis is good.  Parallel to the y-axis causes problems.

Suppose we want a Marker constrained to the parabola along with a display of the tangent and normal, as well as the directrix.  The fixed distance from the point on the parabola to the directrix and axis of symmetry is also drawn.

Of the four use cases, the Marker follows y as a function of x in two and x as a function of y in the other two.  We can’t use the function graphing engine to manage the Marker display as it always calls the base function (parabola) eval() method with an input x-coordinate.  The derivative and normal displays won’t work in two cases because dy/dx is not uniquely defined.

In general, the equation of a parabola with vertex (h,k), focus (h,k+p), and directrix y=k-p is given by

(x-h)2 = 4p(y-k)

which allows y to be defined as a function of x, or x as a function of y.  There are really only two use cases requiring consideration, directrix parallel to the y-axis and directrix parallel to the x-axis.  In one case, horizontal mouse movement is used to position the Marker.  Vertical mouse movements control the Marker in the other case.

The custom Marker is composed directly into the custom Parabola function, which is marked as self-plotting.  This function is responsible for creating the Marker and controlling its placement as well as drawing all supporting graphics.

The Parabola function is defined in XML as

<function id=”Parabola” class=”graph.tests.Parabola” params=”h:0,k:0,p:-1,parallelTo:y”>
<lineMetrics thickness=”2″ color=”0xff0000″ />
<data markerParams=”marker:graph.symbols.CustomMarkerSymbol,rolloverColor:0x9ACD32,digits:2″
markerCoord=”1″ >
<directrix thickness=”3″ color=”0x9933CC” alpha=”1″ lineStyle=”line_dashed” dashWidth=”6″ dashSpacing=”4″ />
<lineSet1 thickness=”2″ color=”0x0000ff” />
<lineSet2 thickness=”1″ color=”0x00ff00″ alpha=”0.5″/>
</data>
</function>

and the display is shown below.

The aspect ratio of this graph is not ideal for the example, but it’s adequate for illustrating the example.  Note that the custom Parabola function is welcome to plot not only the parabola, but as many supporting graphics as it likes.  The custom Marker may be dragged along the boundaries of the parabola and rollover displays the current y-coordinate by default.

A simple parameter change,  h:0,k:0,p:1,parallelTo:y, causes the parabola to open to the right.

Parametric function graphing should be supported in the future, providing for a more clean and general-purpose means to plot any general parabola and supporting visuals.  I thought this was an interesting example of how a decent architecture and simple concepts like Composition open up possibilities that do not seem possible based on the defined constraints of an engine.

Next post will be an update on new capability in the Freehand Drawing Library.

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