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The Beauty of Parametric Curves
The application presented in this article consists of two components: a Flash applet that provides a gallery of visually compelling parametric families and a general parametric families plotter. The plotter allows one to graph an arbitrary family of parametric curves and to find families that are appealing to include in your customized gallery.
Creating a custom gallery requires only simple editing of an XML file. We explain the process in detail in the pdf guide included in the zip package. We include in the zip package all Flash source code files ('fla' and 'as'files) as well, although these are not needed to create a custom gallery.
The application presented in this article stemmed from an ActionScript 3 tutorial at our site http://www.flashandmath.com. The applications uses custom AS3 classes by Doug Ensley and Barbara Kaskosz published in the archived MathDL Flash Forum article, Flash Tools for Developers: Graphing Curves in the Plane.
The applications here require the Flash 9 Player plugin, available free for most browsers from Adobe.
Kaskosz, Barbara, "The Beauty of Parametric Curves," Loci (July 2008), DOI: 10.4169/loci002635
Close, but no cigar...
I applaud the author for the presentation material, but personal experience has taught that we are still not where we should be in regard to 3d (and 4d) mathematical graphers. Perhaps the simplest case is the circle. While it is know that the general case in 2d Cartesian coordinates is (x-h)^2 + (y-k)^2 = r^2 for the circle center at (h,k) and radius r, things become more difficult in 3d. We know that we have a center at [x,y,z] and a radius, but now an orientation becomes paramount. Awhile back I asked several mathematics professionals what the equation of a circle in 3d looks like and recieved no satisfactory answer. The correct answer (extensible to higher dimensions) is this: 1. center at [x,y,z,....] in n-space 2. radius = r. 3. two orthogonal vectors, say U,V in n-space which determine a plane circle is the set of all points on the UV plane such that their distance from the center = r. This definition works with 2d and higher dimensional spaces and I would like to suggest that this is the proper formulation to work with when dealing with computer graphics representations. I have found, sadly, that most 3d graphers are limited and require that the user specify the z-dimension in terms of the other 2, usually parametric. One smart 3d graphing program uses implicit equations, I appreciate the thought behind this, but again, it suffers from orientation simplicity. Orientation simplicity means that when we graph a parametic surface, such as a 3d torus, it is aligned along one of the orthogonal axis. In reality, this is hardly ever the case. So why do 3d graphers implicitly enforce(and simplistically assume) this condition? Real 3d objects have a representational form, then we can define rotations and translations. Having worked with both 3d and 4d (and higher dimensional spaces) with geometric objects, sad to say, the computer graphics still suffers. I would like to see some further discussion upon representational forms and proper recognition of the separation of object, dimension, viewer orientation and motion. Each of these plays a vital part in the appearance upon the computer screen. Finally, I do appreciate the symbolic mathematics that has gone into some graphing engines, an attempt is made to be mathematically precise in simulating the entered equation. Thanks for your patience and consideration of my comments.