UPSB v3

Tricks & Combos / [topic][1.6.1] Definition Of Smoothness

1. sketching

   Date: Fri, Jun 15 2007 16:48:10

   This project will attempt to define smoothness in a formal, quantitative way.
   
   After releasing the article on interrupted tricks, I am now certain there exists a concrete way to define smoothness, or at least offer a reasonable heuristic to evaluate it.
   
   I am willing to investigate the validity of the following thesis:
   
   Zombo's Theory of Smoothness (ZToS):
   
   Smoothness is inversely proportional to the number of pushes and catches in the execution of a combo, over the number of tricks it takes to execute the combo.
   
   Formula:
   
   Smoothness = 1 / [(Number of pushes + Number of catches) / Number of tricks]
   
   Push and Catch, which are two out of three (2/3) of the components of a trick, shall be referred to as NON-SPIN COMPONENTS (NSC).
   
   The units to represent smoothness is therefore "# Tricks / # NSCs", which I shall refer to as Zombo (Z).
   
   Reasoning:
   
   Now that we have the interrupted article released, it is possible that two persons who previously thought they did the same combo are not doing the same thing. This is because the breakdown now more accurately reflect the execution of the combo, so it's possible that one is doing one more catch than the other or whatever. In this sense, looking at the breakdown alone should give more sense of the smoothness of the combo itself.
   
   My reasoning stems from a study case, the Kam's 4 Loop Combo (K4LC). It was previously being broken down as:
   
   

   CODE
   A: Sonic 34-23 > Twisted Sonic 23-12 [Twist of Hand] > ThumbSpin 1.5
   
   
   I believe the K4LC can now being broken down differently, depending on the execution of the combo. To me, the smoothness of this combo comes from the facts that:
   
   1) It is uni-directional.
   2) The tricks are "cut down" and "pipelined" into each other. In other words, no trick is done complete.
   
   The optimal breakdown of the combo should be:
   
   
   CODE
   B: Sonic 34-23 [p 34][s 1.0] ~ Twisted Sonic 23-12 [s 1.5] ~ ThumbSpin [s 1.5][c T1]
   
   
   As you can see this version is much more streamlined, and represents more accurately the REAL execution of the tricks.
   
   If we look at breakdown A, we would conclude that the combo has 3 pushes and 3 catches and if we look at breakdown B, we have 1 push and 1 catch!
   
   Now if we use our fomula, A would amount to 0.5 Z (1 / [(3+3) / 3]), while B amounts to 1.5 Z (1 / [(1+1) / 3]).
   
   I believe this accurately represents reality: do a K4LC where you push and catch each trick invididually, then do it the regular way. You will see that the latter is much smoother, which leads me to believe the number of NSCs reflects the smoothness of the combo.
   
   It doesn't really make sense for now to say that B is 3 times smoother than A, other than the fact that it has 3 times less NSCs, but still it gives you a number to play with.
   
   If you had someone do the following K4LC:
   
   
   CODE
   C: Sonic 34-23 [p 34][s 1.0] ~ Twisted Sonic 23-12 [s 1.5][c 12] > ThumbSpin [p][s 1.5][c T1]
   
   
   Which has a smoothness of 0.75 Z, you can effectively say that B > C > A in terms of smoothness, which is correct.
   
   Now, when you compare different executions of the "same" combo (same in the sense that the same tricks are presented in the same order, only the fact that they are interrupted or not differs), just looking at the number of pushes and catches suffice. But if you want to compare different combos, you need to introduce a unit to balance the combos.
   
   Lemma #1:
   
   As the number of tricks increase, the number of NSCs increases.
   
   This lemma should be fairly intuitive. The bigger your combo is, the more likely you have to include pushes or catches.
   
   It can also be derived formally. The more tricks you have, the more you have to change direction, where direction refers to going up or down the fingers. This is because you eventually run out of space if you keep going in the same direction. Since changing direction takes a push, we can claim that the number of tricks rise with the number of pushes, hence NSCs.
   
   That's why I included the "Number of tricks". If you want to evaluate your combo vis-à-vis a shorter combo, you need to standardize your combo so that it is on the same scale as the other combo. It would not make sense to say that your combo is not as smooth as the other one because it has more tricks and thus more NSCs.
   
   This is the reason why we include the numer of tricks as a balance in the formula. If both combos are the "same", then it is constant and a non-factor. Otherwise, it has an impact in the smoothness.
   
   Issues
   
   Now, there are some issues with the formula above.
   
   First, using the number of tricks as the scale might not always be appropriate. One issue is that the definition of trick is very loose. For example, fingerpass is composed of 4 passes. A very bad fingerpass has in fact 8 NSCs, this happens when you stall at every pass to catch and push again.
   
   When we look at such trick, should it count for 4 tricks or 1? Normally each trick can have up to 1 push and 1 catch, but this one clearly does not. It therefore does not make sense to count fingerpass as one trick only.
   
   The other issue with using the number of tricks as the scale is that the time of execution of each trick varies, and THAT has an impact on the smoothness. If you consider a very fast combo, it is very likely to "spam" a lot of tricks. In our formula, this would mean we could have a lot of NSCs and the combo would still be smooth, which is untrue. This is because due to the speed of execution of the tricks, having a lot of NSCs in a short period of time ALSO looks not smooth.
   
   It seems as though the time should also be a factor in the formula, but it isn't always the case. Because, if we include it there, it means the exact same combo (same interrupt breakdown) is smoother if executed faster, which is not true! When we evaluate combo, we often differentiate smoothness from speed. A faster combo is not a smoother combo.
   
   The solution is to think of a new variable, which we will call "complexity". Complexity is not properly defined for now, but it should relate to the number of tricks, time of execution, difficulty of tricks, etc... We can argue that the number of NSCs increase as the combo becomes increasingly complex. In this sense the rectified formula would look like:
   
   Smoothness = 1 / [(Number of pushes + Number of catches) / Complexity]
   
   Second, using the number of pushes and number of catches as a measure of smoothness is not always appropriate. That's because the way the push or catch is executed is also a factor of smoothness. In our K4LC example, if the push of the Sonic 34-23 is done with the thumb, you would say that this push is of inferior quality as compared to the thumbless version, and thus the smoothness of the combo suffers from it. In this sense, not only the number of NSCs matters, but the QUALITY of the NSCs also matter. How this "quality" is determined is not known for now. But I imagine you would have to take the overall average quality of the NSCs in the combo, multiplied by the number of NSCs.
   
   Because of these two points, I now offer an extended version of my theory. It is much more abstract, because it relies on the concepts of complexity and quality which are not yet defined, but should those two concepts be defined, this one would supercede the aforementionned version of the theory, which is concrete and usable now.
   
   Zombo's Generalized Theory of Smoothness (ZGToS):
   
   Smoothness is inversely proportional to the number of pushes and catches in the execution of a combo and the quality of such pushes and catches, over the complexity of the executed combo.
   
   Formula:
   
   Smoothness = 1 / [(# NSCs / AQoN) / Complexity]
   
   AQoN = Average Quality of NSCs
   
   Additional Notes:
   
   - The quality of the spin also affects the smoothness of the combo. For example, in the K4LC version B, if you stall the spin of the twisted sonic such that the pen is "sandwiched" with the 2 above and 13 below, you would say such a combo is not smooth. However, this could be easily rectified in the breakdown as follow:
   
   
   CODE
   D: Sonic 34-23 [p 34][s 1.0] ~ Twisted Sonic 23-12 [s 0.75][c 13][p 13][s 0.75] ~ ThumbSpin [s 1.5][c T1]
   
   
   D would have a smoothness of 0.75 Z, which is less than 1.5 Z of B and is thus correct. It does have the same smoothness than C which seems wrong. However, in the ZGToS, you would also deduct more if the stall is too long, since it would reduce the complexity of the trick and thus make it even less smooth.
   
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   Feel free to comment on my theory, it is still very rough and untested. In fact, I need people to apply the formula on various combos to see how it compares with their innate sense of smoothness. You must first breakdown the combo in the correct "interrupted" way then apply the formula. Report all results here, thanks.
   
   
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