Orthologia description, notes and code

    The Orthologia twist animation is derived from the "Thue-Morse in rabbit land" geometry and similarity transform, with the bugit (Orthologia spiralis) image. The animation is looping, using a pair of acceleration and deceleration phases with the sign of scaling changing at the temporal midpoint.

Thue-Morse in rabbit land

Orthologia twist animation Ortho_twist_640_640.swf  10 MB, 640 x 640 px. Ortho_twist_640_480.wmv  7 MB, 640 x 480 px.

Othologia trispiralis

Geometry
Conceptual
Parameters
Programs and code
Animation assembly
Possible improvements

Geometry

    The basis image geometry is an 8x2 checkerboard division of the plane by logarithmic spirals with a pitch of +/- pi/4 radians. The checkerboard is colored (orthogonal to the contours) by the third order (2^3 = 8 element) Thue-Morse sequence [0 1 1 0 1 0 0 1]. The relative phase between the two sets of opposite chirality spirals rotates through pi/2 radians; in this example the phase of one set is fixed.

Conceptual

5/10/07 This animation embodies several ideas and algorithms that intrigue me:

- logarithmic disection of the plane, and its similarity transform (self-similarity).
- simple TM, and complement with rotation (related to pï/2 rotatlon [*i = e^(i*pi) transform])
- scientific illustration style
- insect reproduction and mutualism theme
- visual aspects:

1) Bistable percept and motion blur: with slow rotation, I percieve a wholistic pattern rotating. But with fast rotation half the contours appear fixed, and the other half rotating (with no shrinking).

2007-10-01 Ted used a "2-D in-plane" or "3-D tunnel" spacial cue to switch between percepts.

2) Checkerboard flashing and motion, influences by fMRI retinotopic mapping and MT stimulii.
3) Smooth angular accelaratlon ramps, continuous angular velocity.

- continuous loop, toroidal in space-time

4) Half of the logarithmic spiral contours are parallel to, and half perpendicular to, the motion. Those that are parallel form a continuous band when motion is fastest, while those perpendicular can't be seen. The visible set of contours switches when the zoom direction switches, even though the direction of rotation doesn't change.

Paremeters

    72x4 = 288 frames, 640 x 640 px., constant acceleration and deceleration and scaling reversal but not rotation reversal. 

    Motion blur: maximum of 21 frame blends with gaussian profile, sigma = .27. 

    24 frames/s x 288 frames = 12s.

Programs and code

    Basis image: Orthologia_merged_1.tif

Matlab program:

    The basis image filename is a hardcoded parameter of the main function:

    This function generates individual frames of the animation, where the number of frames desired is an input parameter. Each of four segments of the animation is generated separately, by changing the hardcoded parameters signScaleDirection and signAccelerateDirection for each segment. See this file's header for more details on how it is used.

The main function (Orthologia_animate.m) calls these functions:

AffineTransformImage_fa.m (scale and rotate image)
interp2_fa.m (2-D interpolation)
interp1_fa.m (vector "flat aperture" interpolation)
removenan.m (utility, replaces NaN values with 0's)

Also available is a related image scaling function:
ScaleImage_fa.m (not used)

Animation assembly

    The four sets of individual frames generated by Orthologia_animate.m were composited into AVI format using Space Software, and was converted to .swf Flash using CinemaForge.

Possible improvements

    A good artist could improve the basis image using stipling and other texture techniques. They would need to be used with discretion so as not to break up the hard contrast boundaries. More and better hair. 3-D cues that don't introduce an implied lighting direction.

    A few more hard contrast boundaries, and particularly intersections, along spiral contours. The smoothed boundaries at corners should have at least one inflection point.

    The basis image (Orthologia spiralis) alone would look nice on a digitally "aged" book page.