Given that Google Wave will be shutting its doors by the end of the year, I wanted to post a document that's long been gestating on it. I wrote up my theory of hard, soft, and mixed transitions in a rough draft form months ago and had shared it with a whole mess of people whose opinion I respect for feedback and clarification. It is still not finished by a longshot and there are more than a few things in this document that I've found since to be inaccurate, but given that I frequently get questions from people asking about this theory, I thought an incomplete document would be better than none at all. Enjoy!
I spent the holidays mapping out every combination of circle size, transition type, and destination circle using Jon's concept of loops and arcs as a foundation. I did a couple videos of the results, but think it would also be a good idea to put this stuff out there for discussion.
Arcs and loops: Jon came up with this framework for describing the transitions between unit circle hybrids and their equivalents in larger circles. BTW, please correct if I'm getting any of this wrong, Jon. This is your concept and I don't want to be abusing it if I don't completely grok it (which granted is redundant ;). Anys, I've come to think of loops and arcs as regions that retain direction of rotation between figures such that when doing soft transitions rotation continues in its original direction, only converted into a different region. I had made the mistake before of thinking of these as distinct sections of movement, as though flowers were being built out of tinker-toys when instead they describe the moments when the relationship of direction and momentum between poi and hand shift. Put simply, loops are antispin (or inspin) petals or isolations. Though it's an imperfect description, we can think of these regions as being those in which the path of the poi and hand intersect each other, either when the poi crosses over the hand path in antispin or the path of the poi and hand permanently intersected in a loop. Arcs represent a relationship between these two elements in which hand and poi move parallel but do not intersect--these are the areas between flower petals or the full length of an extension.
Hard and soft transitions: my own concept, though I'm sure it's similar to concepts dreamed up by others in the past. In simplest terms, hard and soft transitions represent either the complete conservation of rotation or the complete reversal of rotation There are two points of movement in epitrochoid moves: the hand and the poi. When both retain direction of rotation but switch into either a nested or neighboring figure, this can be said to be a soft transition. The switch from an extension to either horizontal or vertical cateye is one such example. When both hand and poi completely reverse direction, it can look like a stall or a float, but I'm referring to it here as a hard transition. In between the two are transitions in which the hand reverses direction but the poi doesn't (this comes out looking like the standard C-CAP) or when the hand continues direction of rotation but the poi reverses (I'm arguing that these are still a type of CAP, though the diversity of them is limited by gravity). I don't have good terms for these types yet.
IT and ET: All transitions begin with the root figure and then move into either a nested/concentric figure which shares a point of intersection or an entirely external figure which shares a point of intersection. I'm considering the nested/concentric version to be intratangent (IT) and the external figure to be extratangent (ET).
Pure figure and mixed figure: being as how antispin flowers have both loop and arc elements, in this nomenclature I'm considering flowers to be examples of mixed figures whereas isolations are pure loops and extensions are pure arcs.
Charting all resulting figures from the perspective of soft transitions, more than a few patterns begin to emerge that we can verify between figures of different sizes. For one, arc regions remain arc regions whether switching the figure in IT or ET. With IT figures, antispin flowers change size/number of petals, but soft transitions between arcs of said figures retain arc definition and thus a 4-petal flower soft transposed to triquetra will drop the two petals most distant to the transition arc and share the nearest 2 petals. The same is true of transitioning from cateye to any other antispin flower and back. If the root figure is extension, the overall size of the extension hand circle will shift accordingly, but the integrity of the character of the move remains exactly the same (extension of different base sizes). Isolations are a special case because they require the poi path to switch to arc to exit into IT--thus there is no soft root to IT transition for them.
Soft transpositions to the ET figure result in a shift in figure from either a mixed figure (flower) to a pure figure (isolation or extension) depending upon the point of departure. All soft transitions via arc to the ET retain character just as the transitions to the IT do, as is the case with all transitions via loop. Any soft transition via loop must inevitably be restricted to unit circle when going to pure figure while arc pure figures may retain any size from unit circle extension to full-arm extension depending upon the tastes of the spinner.
Transitions that are both and hand poi hard have the effect of reversing the figures created by soft transitions in direction, ie, a hard transition to IE via antispin flower arc to unit circle will result in a cateye, just as in soft transitions, but result in a figure created in the opposite direction. Likewise, a transition from 4-petal flower via IT hard arc to triquetra will share the same arc segment and nearest 2 flower petals just as in soft transitions but will return to the most recent petal before continuing to the farthest petal rather than travelling first through the next petal as in soft transitions. Hard transitions can be seen as analogous to stalls/floats that reverse direction after completely arresting momentum.
This is where things get very interesting, because whether one does the hard transition with the poi or the hand, as long as the other point is in a soft transition it results is a reversal of region characteristics. One such example: when going from 4-petal antispin via arc to an IT 4-petal or 3-petal with a mixed transition, the outcome will be the poi and hand moving in parallel rather than in opposite directions and thus we know we are headed for a pure arc figure, yet the orientation of the poi is facing into the figure rather than outside. To exit the hand circle the poi must cross over the hand path, thus becoming a loop. At this point we can treat the resulting pattern as either an inspin flower or a unique cardioid-shape figure I would consider to represent a type of CAP.
When making the same transition to ET, however, we similarly find a point at which the poi must cross over the hand path to follow the hand into ET. As with the IT, this shifts an arc into a loop, but given that the loop is still on the outside of the resulting circle, we have an antispin flower once again--one that is in diamond rather than box mode. When switching to unit circle, this results in a horizontal cateye.
It turns out that any mixed transition whether poi hard/hand soft or poi soft/hand hard will reverse a region's character from arc to loop or vice-versa. This is useful to know not just because it allows mode transformations from diamond to square orientation in antispin flowers, but also because it allows transitions in and out of any mixed figure to other mixed figures. When using soft or hard transitions, any ET transition must be from mixed to pure or vice-versa.