WTF is Transformation Matrix?

A transformation matrix transforms a vector of point coordinates in given frame of reference into coordinates in another one (e.g. from child to parent in our beloved display lists or scene graphs) – Captain Obvious.

Many people think of transformation matrices as impenetrable black boxes filled with random numbers that are impossible to make sense of while fiddling with your application. So this 1st post of the year is my attempt to help those people, by providing as short and simple explanation for those numbers as possible.

P.S. Since matrices transform vectors, you need to know a thing or two about vectors 1st.

Grab my balls

So, I have started cross-engine 3D primitives project at github. Just like as3dmod, it plugs seamlessly into your favorite engine to help you out whenever you feel limited by it. Initial commit supports Alternativa3D 8.12, Away3D 4 Alpha, Minko 1 and contains two spheric primitives: UniformSphere and Globe.

UniformSphere is a mesh built from the spiral of vertices uniformly distributed across the sphere. You have two options here: Rakhmanov formula (default, the spiral joins sphere poles) or Bauer formula (the spiral does not contain poles). This primitive was ported from my 3D metaballs code.

Globe is a mesh that minimizes texture distortions for classic plate carré projection. It does so by placing equilateral triangles along the equator and gradually compressing triangles towards each pole (where high triangle density is actually good property in terms of texture distortions). This primitive was ported from my globe component for FP9, which is now also hosted at github.

I plan to add few more primitives over time, both useful and not really, and maybe more engine proxies if there will be any interest.

Inverse homography using drawTriangles()

Flash player version 10+ native drawTriangles() method is perfect for programming homography (2D transformation that preserves lines) because of its support for UVT data. In general case, however, homography requires solving 8×8 linear equations system, so noone can be arsed to code that. Instead they implement one of two special cases - mapping rectangle to or from quadrilateral. So far, only 1st case was implemented using drawTriangles, while 2nd case was done using filters. Well, no longer so – it turned out, inverting T values in Zeh code was sufficient to make it work backwards:

One interesting thing about this code is that noone really knows exactly how it works :) I didn’t have to study all the math behind Zeh code in order to “invert” it, and Zeh… well, let me quote:

I was lucky enough to get what I wanted working after a good amount of crazy trial and error (and beer)

Improving marching squares

Right. Today is the day of improving 2×2 marching squares flavor described in wikipedia. A year ago Sakri Rosenstrom found that it is flawed:

As you can see, two checkerboard-like patterns are ambiguous, but the algorithm ignores it, causing random deadlocks as a side effect. So I tried to find some solution to this by posting the problem at wonderfl; and then I just couldn’t wait :) Well, my solution uses 2 to 4 pixels out of 2×2 window and seems to work on crazy images, so I’m happy. But if you suddenly find some better way, let me know.

Hough transform thing continues…

Previously on «Hough transform»: brave lone coder decides to explore random line detection algorithm; he splits the experiment in parts roughly corresponding to pixel bender kernels that he needs to write; he writes a post with many words that noone reads, and shares the demo that noone finds impressive.

Well, just so that you know, the experiment continues, and the demo is a bit better now: