If you have any expertise in Spanish, French, German, or Japanese, I’d appreciate any feedback you can give me about the words I chose in those languages. Would they be the word one would think of when shown the picture? Thanks, in advance.

There are a couple iPhone apps that claim to have voice-to-text: QuickVoice PRO w/Voice2Text, reQall, and Jott. It seems that reQall and Jott require me to sign up for their web services. reQall has both a free and a pay web service. Jott only seems to have a pay service. So, I will probably try them in that order until I find one that I like or get frustrated with the whole process.

It seems that QuickVoice PRO uses SpinVox to do their voice-to-text. I am not impressed with the test message. I said, This is just a test. open parenthesis foo close parenthesis. The email that I got back says:

This is just a test of open(?) friends to see who close friends to see. – spoken through SpinVox

I need to touch up the jaunty logo tiles in the upper left. Some of them are worse for the wear after the perspective transformations and rotations. More to follow….

The Basic Spiral

I am talking about a simple spiral of the form:

That would be an equation for a spiral that starts at the origin and heads outward as increases. For my application though, I want to end at the origin so I need to substitute in for .

The math is also going to work out slightly better if I use in place of in the above equations:

I don’t want the piece to spiral into place when released though. So, really, I am concerned with just the coordinate from the above equations:

To normalize everything, I am going to let . And, since I want my interpolation value to go from zero to one instead of one to zero, I am again going to subtract this all from one:

Parametrizing by Arc Length

If I just plot the spiral as is, then I am stuck with the same problem that I had with damped spring motion: the frequency is constant. The rate at which it would shimmy would not increase. I want it to really settle into place. So, I have to walk through the spiral at a fixed rate. I need to rescale the into something else so that for any given time interval, I cover the same arc length on the spiral.

The first step then is to figure out how much arc length I sweep out with any given . Call this arc length :

Then, to rescale my , I want to use instead so that . So, when , for example, I will need to find the such that the arc length covered in the first seconds is -th of the arc length covered in the whole interval. Solving for in the equation comes out to:

Plugging all of that back into the equation for gives me:

You can see how the period speeds up toward the end. This was a good starting point. However, the first oscillation goes almost as far beyond the target as our initial point was. So, I upped the exponent to :

This was pretty much the effect I wanted. Unfortunately, it involves taking an -th power, an -th root, and a cosine of calculation. I decided that was close enough to one to give it a whirl without needing the -th power.

Here you can see the final result.

In the previous post, I twiddled the equations for damped spring motion until I found something visually pleasing. Last night, I went back to the drawing (a.k.a. white) board.

What if I used an exponential spiral (parameterized by arc length). Then, I could easily adjust the number of times it bounces back and forth. I could use a spiral like and where is some multiple of . Then, I could walk along the spiral getting to the origin when going around the origin times in the process. If I follow the curve at a fixed rate, then I guarantee that my oscillations will pick up speed as I approach the origin.

Rather than have it spiral into the center, I am just using the x-coordinate of the spiral as my new value to interpolate with. I like the effect for the most part. It overshoots a little bit far on the first oscillation. I may tweak it some more before it’s all over. For now though, I am sticking with it.

I will post some graphs soon.

There are a bunch of things that I wanted to do with it for a long time, but it’s been too slow and rigid to make any of those things fun.

Enter Lisp. As soon as it made it through my skull that Lisp is actually compiled (honest-to-goodness your-CPU instructions), I wanted to rewrite the whole thing in Lisp. I have finally gotten started on doing that. And, I just made it to the point where I’m actually tracing rays. Here is a stereo pair of a three-dimensional scene:

It use’s xach‘s ZPNG library for output, my OpenMPI library for sharing work across machines, and a thin layer that I wrote on top of Portable Threads for threading within a machine.

It doesn’t yet do reflections and refractions, directional lights, or most of the shapes that my old raytracer does.
But, it’s already got multithreading, MPI, more meaningful camera parameters, and functional color characteristics.

So, here’s the source code that generated the above images. Note that the one sphere has checkboarded diffuseness and the other has gradated phong-exponent and positional coloring.

(defun make-checkerboard-func (a b)
  #'(lambda (&key position)
      (let ((sum (reduce #'+
			 position
			 :key #'(lambda (vv)
				  (if (< 1.0 (mod vv 2.0)) 1 0)))))
	(if (evenp sum) a b))))
 
(defun my-universe ()
  (rt:universe (:spatial-dimensions 3
		:color-dimensions 3)
    (let ((look-at (rt:v 18.0 0.0 0.0))
	  (eye-offset (rt:v 0.0 0.25 0.0))
	  (aspect (rt:v 16.0 9.0))
	  (field-of-view 120.0)
	  (object-scale (rt:v 3.0 3.0 3.0))
	  (x-axis (rt:v 1.0 0.0 0.0))
	  (y-axis (rt:v 0.0 1.0 0.0))
	  (z-axis (rt:v 0.0 0.0 1.0))
	  (angle 10.0))
      (rt:with-transforms (:translation (rt:v* eye-offset -1.0)
			   :look-at look-at)
	(rt:camera :name :main-camera-l
		   :aspect aspect
		   :field-of-view field-of-view))
      (rt:with-transforms (:translation (rt:v* eye-offset 1.0)
			   :look-at look-at)
	(rt:camera :name :main-camera-r
		   :aspect aspect
		   :field-of-view field-of-view))
      (rt:with-translation (rt:v 0.0 -5.0 -2.0)
	(rt:light :color (rt:v 0.7 0.7 0.7)))
      (rt:with-translation (rt:v -10.0 -5.0 8.0)
	(rt:light :color (rt:v 0.7 0.7 1.0)))
      (rt:with-transforms (:translation look-at
			   :scaling object-scale
			   :look-at (rt:v 0.0 0.0 0.0)
			   :translation (rt:v 0.0 0.0 -1.0)
			   :rotation (x-axis y-axis angle))
	(rt:with-transforms (:translation (rt:v 0.0 -1.5 0.0)
			     :color-scaling (rt:v 0.75 0.75 0.75)
			     :color-rotation (y-axis z-axis angle))
	  (rt:with-color (rt:c (rt:v 0.2 0.8 0.2)
			       :diffuseness (make-checkerboard-func 0.2 0.6)
			       :specularity 0.4)
	    (rt:sphere)))
	(rt:with-transforms (:translation (rt:v 2.0 2.0 1.5)
			     :scaling (rt:v 1.5 1.5 1.5)
			     :color-translation (rt:v 0.0 0.2 0.0))
	  (rt:with-color (rt:c #'(lambda (&key position)
				   (map 'vector #'abs position))
			       :diffuseness 0.6
			       :specularity 0.4
			       :phong-exponent #'(lambda (&key position)
						   (+ 10
						      (abs (* 100
							     (aref position 2)
							     )))))
	      (rt:sphere)))
	(rt:with-rotation (x-axis y-axis angle)
	  (rt:with-translation (rt:v -15.0 0.0 0.0)
	    (rt:with-color (rt:c (rt:v 0.8 0.2 0.2))
	      (rt:halfspace :name :hspace1))))))))
 
#+:openmpi
(mpi:init)
 
(rt:with-workers (1)
   (let ((uu (my-universe))
	 (dpd (rt:v 2.0 2.0)))
     (rt:render-png :dots-per-degree dpd
		    :universe uu
		    :filename #P"output-l.png"
		    :camera-name :main-camera-l)
     (rt:render-png :dots-per-degree dpd
		    :universe uu
		    :filename #P"output-r.png"
		    :camera-name :main-camera-r)))
 
#+:openmpi
(mpi:finalize)

And, here are the same images swapped for those of you who prefer cross-eyed stereo pairs: