The Degrafa spline architecture has been modified to structurally achieve the desired sepration between the computational elements of a spline and the internal Degrafa geometry pipeline.  This is illustrated by the new NaturalCubicSpline.  In MXML, the spline is treated as before,

<NaturalCubicSpline id=”cubicSpline” graphicsTarget=”{[splineLayer]}” knots=”104,299 166,168 217,236 307,225 370,142 440,299 506,309″ >
<stroke>
<SolidStroke weight=”2″ color=”#0000FF”/>
</stroke>
</NaturalCubicSpline>

The actual Degrafa code is

public class NaturalCubicSpline extends BasicSpline
{
 private var _cubicSpline:PlottableCubicSpline;
 public function NaturalCubicSpline( _myPoints:Array=null )
 {
  super(_myPoints);
  _cubicSpline = new PlottableCubicSpline();
  super.spline = _cubicSpline;
 }

 override public function eval(_x:Number):Number { return _cubicSpline.eval(_x); }
 override public function derivative(_x:Number):Number { return _cubicSpline.derivative(_x); }
 }

The computational basis for the spline is in the PlottableCubicSpline class which implements the IPlottableSpline interface.  The integration into the Degrafa geometry pipeline is handled by BasicSpline.  The basic idea is to supply an external utility that conforms to an established interface to handle the spline computations.  A reference to the utility is used to ‘decorate’ BasicSpline in a simplistic sense.  This makes it possible for people to author splines for Degrafa without having to understand much about Flash or the internals of Degrafa.

Currently, BasicSpline is only functional for cartesian splines.  To preparare a set of test cases for parametric splines, the old Singularity Catmull-Rom spline has been ported to Degrafa as two utility functions.  One is a straight Catmull-Rom spline (with auto-closure) that conforms to the IPlottableSpline interface.  The other extends the base utility to include approximate arc-length parameterization for use in path animation or keyframing (where the path is not intended to be drawn).

A simple demo is available to illustrate the utility.  Click in the draw area to define a few points, then hit the space bar.  The spline is plotted old-school (point-to-point) to illustrate the path.   A set of markers is distributed along the path at roughly equal intervals of arc length.  A Degrafa ellipse is drawn and animated along the path.  The ellipse is also oriented to follow the path.  A screen shot is shown below.

Degrafa Arc-Length Parameterized Catmull-Rom Spline

Like I said, it’s a pretty simple demo, but it should give you an overview of how to use the new utility.  An SVN update is required if you want the new source.

View demo.

View source.

A strange comment coming from a Federer fan!  It was, in fact, watching Federer at the US Open over the last couple years that encouraged me to pick up tennis again after being away from the game for more than two decades.  I made a special attempt to watch Federer during the 2008 Aussie Open as well as the French and Wimbledon finals.  Federer was defeated in the semis at the Australian and lost the two other finals; the French very badly.  So, I decided I would not watch Federer play during the US Open.  He won!

I got up early to watch the 2009 Australian Open final and I mean really early.  Once again, a five set heartbreaker.  So, I decided not to watch his matches during the French Open this year.  An incredible win for his career slam!

As a mathematician I know that correlation does not prove causality, but there was a perfect correlation between me watching Federer playing live and the result.  So, I resolved never to watch a Wimbledon match live.  I did not have any involvement in the final, including NO tweets.

The result speaks for itself.  Clearly Federer owes this historical achievment to me   I’ll be looking forward to that autographed picture, Roger.  While I’m waiting, I’ll continue this new tradition of never watching a Federer match live.  May the Fed Express roll on.

And, finally, credit where credit is due.  Congrats. to Roddick for one of the most impressive and courageous perfomances in a final in the history of the game.

If you are using Papervision and want to expand into applications that involve physics engines, then you might be interesting in this 3D Bowling Game tutorial at thetechlabs.  Since it’s not a full-blown game, the concepts and code are easier to deconstruct.  You may either continue to expand on the game concept or take the ideas in a different direction.

Read the tutorial here.

If you are interested in Degrafa and FXG, this presentation on skinning Flex with Degrafa and FXG by James Whittaker may be of interest.  This blog post includes James’ Flash Camp session slides.

View Presentation Here.

Interesting question.  Tennis magazine poses the question of which player’s 14 major championships is a greater achievement; Tiger Woods or Roger Federer?  Readers of this blog know I’m a Federer fan, but I have to confess being a huge Woods fan as well.  In fact, during my golf craze, I was a member at Hank Haney’s golf ranch in McKinney and had the great honor of meeting Tiger.  He’s one of the nicest people you will ever meet.  So, I change my mind on answering this question about every 15 seconds.

In truth, it’s a very difficult question as comparing tennis and golf (at the professional level) is quite hard.  In golf, you truly compete against the course.  It’s rare that two people even play head-to-head in a final round to determine a championship.  The interesting observation about professional golf is the margin of error.  In tennis, you can get away with one or two bad shots and still win a game.  In stroke play in golf, one bad stroke over the course of four days can literally make the difference in a championship.  I think Tiger’s greatest asset is his ability to turn bad shots into par saves that keep the round going without giving up ground.

Tennis, on the other hand, is a direct physical (although not contact) confrontation with the other player.  Golf  affords long careers; tennis in the modern era allows only the smallest window for young players to accomplish the bulk of what they can possibly accomplish in a career.  If Tiger were a tennis player, he would already be on the ’senior’ tour

The rate at which Federer has won majors is mind-boggling in the modern game; 14 majors in six years.  Add to that a record 20 consecutive appearances in a grand slam seminfinal.  If it were not for a bout with mono in 2008, it might have been 20 consecutive finals.

So, what’s my final vote?  Ask me in another 15 seconds

I went through my golf craze in the 90’s and eventually worked my way to a single-digit handicap.  Although I don’t play any more,  I remember a lot of famous courses and holes, including the 15th hole at Bethpage.  At US Open green speeds, this setup is diabolical, but you can’t really get an appreciation for the difficulty of a hole from a few TV shots.  So, I was really interested when I saw this post from Carlos on a 3D recreation of that hole.   Papervision fans should enjoy the 3D and golfers should gain additional appreciation for the difficulty of the hole.

I remember my professor in my first numerical analysis course talking about Lagrange interpolating polynomials and sampling the test function f(x) = 1/(1 + x²).  We already knew about oscillations in higher-order polynomials, so this discussion was about sampling increments.  The natural thought is to sample in equal increments across an interval.  Thus started a discussion about Chebyshev polynomials and Chebyshev nodes.  I actually found a good blog post about this very topic at John Cook’s blog.  John has a great blog on math and computation, btw.

I wanted one more demo showing how to create cubic spline nodes in script and highlight sections of the spline that were not necessarily in-between knots.  The new demo samples the above function at Chebyshev nodes in [-5,5] and fits a natural cubic spline to the nodes.  The area under the curve in [-1,1] is highlighted as shown in the screenshot below (by clicking the ‘Show Highlight’ button.

Natural Cubic Spline Approximation to a Function

Click on the ‘Show Function’ button to show the original function plotted point-to-point to see a visual comparison of the spline approximation.  If you are interested, change the interpolation points and study the change in quality of fit.

Now that my trip down memory lane is complete, next step is to complete the Degrafa spline architecture (which is only about halfway there) and add a parametric spline.  I did correct a typo in the code, so you will need to update SVN.

View demo.

View source.

SuperShapes are a mathematical construction popularized by Paul Bourke, allowing a very wide range of cool (2D and 3D) images to be generated by varying parameters in a single formula.  Originally proposed by Johan Gielis [1], the superformula was a means to describe complex shapes often found in nature.  Wikipedia article is here.

You can read up on the mathematical foundation of 2D SuperShapes here.  For those wanting to experiment with 3D SuperShapes, you can start here.  I also highly recommend visiting Bourke’s geometry page here.

While I’ve been doing mundane stuff with splines, the rest of the Degrafa team has been hard at work doing really cool stuff like adding SuperShapes to Degrafa.  You can read all about it here.

[1] Gielis, Johan “A generic geometric transformation that unifies a wide range of natural and abstract shapes”, American Journal of Botany 90, pp. 333–338

I’m always interested in computer tools for teaching math.  I recently heard about Algebrator by SoftMath and have been checking out the online animated demos.  Seems like this is a good tool for students needing help with basic concepts (kind of like a fixed-cost home tutor) or for teachers to create lessons and use as an interactive teaching aid.  I suppose it might also be useful for technical professionals wishing to quickly solve complex algebra problems, but most people I know already use Mathematica, Maple, or something similar.  Given a cost of about sixty bucks, it seems that individual students and teachers are the primary market for Algebrator.

There does not seem to be an online demo, so I can’t check it out in more detail.  If anyone is using this tool, I’d be interested in your feedback.  Check out Algebrator here.

Watching Roger Federer at the US Open the past few years was a big part of my wanting to get back into tennis after about 25 years away from the game.  As a Federer fan, I was overjoyed at his winning the French Open today.  Not only does this tie Sampras’ slam record at 14, it gives Federer the elusive career slam (winning all four major championships over the course of a career).  His record now legitimately begs the question of GOAT (Greatest Of All Time).

In a phone interview from LA, Sampras stated, “I’m obviously happy for Roger. Now that he has won in Paris, I think it just more solidifies his place in history as the greatest player that played the game, in my opinion.”

As a Federer fan, it’s an easy sell for me, although I have to say that the GOAT title is probably shared equally with Rod Laver.  Comparing players across eras is difficult.  Laver won the grand slam (all four majors consecutively in a single year) not once, but twice.  This mind-boggling achievement is made even more so by the recognition that he played some of his best tennis before the Open era.  And, some might note that Federer might have an asterisk beside his title in that he did not defeat Nadal.  I would point out that Federer can only play the matches in front of him.  Besides, who is more dangerous?  Nadal or the person who knocked Nadal (and three other higher-seeded players) out of the tournament?  And, Federer did defeat Nadal at Madrid (on clay).

Tennis in the current era is a physical drain unlike any time in the past.  Players are in tremendous condition and the tour is an unrelenting grind.  Youth is served at a faster pace then ever in the past.  Just look at Del Potro.  Federer blew him away at the Australian (including a double-bagel), then DP took him to five a few months later at the French.  The window for winning slams seems to be a bit smaller every year.  This makes Nadal’s run at the French even more impressive and equally underscores Federer’s slam record.

Personally, I would like to see TMF retire with 15+ slams.  Given the pace at which young players are rising toward slam contention, this would be a monumental achievement, solidifying what is already a strong G.O.A.T argument.

May the Fed Express roll on!hr