Coefficient Values: Part Two

The essential paint stroke — or wave stroke — in spectral art comes down to choosing a chiral pair, q:v and setting the values of one or the other or both coefficients. We label these coefficients here as C and C', cee and cee-prime.

As observed in the previous section, under Fourier synthesis they both give rise to apparently identical waveforms, but the one produced by C exhibits counterclockwise circular polarization while that produced by C' is clockwise.

This is not a big distinction! Visually, it is just that the imaginary component of C swings high, then low, while the imaginary component of C' swings low, then high. Tweedledee. Tweedledum.

Artists come in all different styles and temperaments, ranging from the Jackson Pollock action painters to the Maxfield Parrish precisionists, sealing exact applications of trasparent oils between layers of varnish. The Jackson Pollocks, and maybe even the general wave-painter, won't find much to interest them here. Preferring some amount of chance, they set some coefficients in -tiletex and see what happens, or perhaps even use -tiletex as a can of coefficient spray paint. That is perfectly fine.

But this section is for the Maxfield Parrishes, who wish to set the coefficients in a chiral pair in such a way as to get a very particular wave form under Fourier synthesis.

To paint the waveform associated with the chiral pair q:v, we set each of its coefficients to complex numbers, C and C.' Under Fourier synthesis, this sets in train a pair of waveforms which interact both constructively and destructively, resulting in the composite waveform of the chiral pair, its visible manifestation that we see, in part, in the real and imaginary output images of -tiletex. To characterise this behavior, we plot these values on the complex plane, ℂ, as we have done in Figure 10.

C and C' form rays which radiate out from the origin of ℂ and have particular phase angles. Whatever these may be, there is an interior angle separating C and C' with measure 2α and which can be bisected, as shown by the dotted line. Let θ represent the angular displacement of this dotted line from the real axis. Then the measure of C from the real axis is the sum θ + α and the measure of C' from the real axis is the difference θ – α. Finally, Let ΔC = | C – C'|.

We can harness θ, α, and ΔC as parameters to precisely control the waveform produced by chiral pair q:v:

1. θ variously shifts the aggregate waveform produced by chiral pair q:v between the real and imaginary output images. Increasing ΔC diminishes this effect and the action of θ vanishes when either C or C' become zero. We think of θ as a 'balance control.'
2. α variously changes the phase of the aggregate waveform, shifting its position to and fro in its propagation direction. We think of α as a 'phase control.'
3. ΔC changes the polarization of the aggregate waveform from linear to eliptical to circular; the handedness of the polarization depends upon the larger of C or C'. There is no obvious visual analog of this action, but the effect is one that diminishes the actions of both θ and α. We think of ΔC as a 'damper control.'

Lets examine how these three control parameters behave. We will begin with θ which functions in a way like a balance control on a stereo amplifier. It apportions how much of the waveform appears in the real or imaginary output images, and whether the waveform will be normal or inverted.

The Bisector as Balance Control

Lets consider a setting where ΔC = 0; so that C has the same length as C'. We set out by twirling θ around a pivot at the origin, letting it assume various values. Figure 11 illustrates the most salient aspects of this.

Real	Imaginary	Real	Imaginary	Real	Imaginary
(a) θ = 0°		(b) θ = 90°		(c) θ arbitrary
Figure 11

Figure 11(a) corresponds to the bisector aligning with the real axis; it is ground upon which we have already trod, reprising Figure 9. C and C' are complex conjugates and both transform under Fourier synthesis to component waveforms, each with real and imaginary parts. The real parts of both components are in phase. Blue in the following graph, they overlap and cannot be distinguished from one another. But the imaginary parts, in red, are 180° out of phase.

The sums of these parts are drawn in bold, again, blue for real and red for imaginary. Being in phase, the two real parts, one from C, the other from C' reinforce one another. They directly add up to a waveform twice as large as the contributing parts. In contrast, the two imaginary parts cancel out.

The top part of Figure 11(a) illustrates the consequences: the composite waveform of the chiral pair has only a real part, its imaginary part vanished. Thus, θ = 0° corresponds to the "real only" balance setting.

In the prosaic world of -tiletex, this case arises when we set the angle pixel of both coefficients to black. That sets θ and α to zero and aligns the bisector with the real axis. Figure 11(a) prevails and under Fourier synthesis both coefficients morph to their component waveforms. In the vector sum of these, only the real part survives, the imaginary vanishing.

If we twirl this "balance control" angle θ until it aligns with 90°, we arrive at an almost identical scene but with the actors' roles reversed. Figure 11(b), corresponds to the bisector aligning with the imaginary axis and finds the two imaginary parts in phase, but the real parts are 180° out. Thus, θ = 90° corresponds to the "imaginary only" balance setting:

In general, when we are not neatly aligned with one or another cardinal points, waveforms of equal frequency combine as vectors under the Parallelogram Rule. Amplitudes correspond to the magnitude and phase angles correspond to the orientation of a vector, and we can add waveforms with different phase angles as we would any other vector. With this distinction in mind, we can take up the third case, Figure 11(c) for some arbitrary angle θ.

The animated graph charts the full sweep of θ from 0° to 360° illustrating how the component wavefrorms of C and C' combine under various values of θ. Their real and imaginary parts, the light-weight lines, each form vector sums, one for the imaginary parts, one for the real, resulting in the real and imaginary parts of the composite, these drawn in heavy-weight lines.

Increasing θ shifts counterclockwise the real and imaginary parts of waveforms derived from C, right to left, these drawn with indigo (real) and red-violet (imaginary) lines. Concurrently, the real and imaginary parts of coefficient C' shift clockwise, left to right, these drawn with blue-green (real) and amber (imaginary) lines. The real and imaginary parts of the chiral pair's composite waveform, what we see in output, are drawn with heavy-weight lines, blue for the real part, red for imaginary. These vector sums wax or wane as their supporting parts phase in or out, the composites reaching their maxima or minima at the cardinal points. 

In the prosaic world of -tiletex, the artist manipulates θ indirectly. She first picks a chiral pair for its nice frequency and orientation, establishing q:v, then locates the two coefficients, each being q horizontal and v vertical steps away from the Zero Point. From here, the artist deals entirely with the angle image. θ corresponds to the average intensity of the two angle pixels, so if the artist sets one to an intensity of 27 and the other to an intensity of 63, the averages of these intensities is (27 + 63) ÷ 2 = 45, the measure of θ. One should not take this measure to be in degrees, however, because in the typical eight bit, unsigned integer scheme, there are only 256 levels of gray. A change of pixel intensity of one is equivalent to about 360° ÷ 255 ≈ 1.411765°. In this example, an intensity of 45 represents 63.53°.

Without getting too far ahead with this example, the phase control, α, is one half of the difference between these intensities, (63 − 27) ÷ 2 = 18; 18 × 1.411765° ≈ 25.4118°. We will delve into the deeper ramifications of α in the next section.