Concave vs. Convex Loop (Which is which/better?)

Because ML elbows are at a fixed angle when he is loop driving.  So the forward rotation is done with the shoulders.  They are taught to use the shoulder as major pivot point of their swing, not the elbow.

I've had a few different coaches that span many different points in my life.  PRC coaches teaches the basic FH that way.  Mostly, shoulder movement.  They don't say lock the elbow, but it just happens, to give more control over the ball, if you pivot with mostly shoulder movement.

Edited by power7 - 06/24/2012 at 2:59am

@Zeio

Edited by hobbes203 - 06/24/2012 at 6:03am

V-Griper wrote: zeio wrote: For those who have chosen to jump to conclusions, I invite them to answer the following question. Given two balls released from the top at the same time, which one reaches the bottom in the shortest time, Red or Green? Why? Sliding or rolling? Friction? Off the cuff I would say green(if they are rolling, or sliding with no friction). Larger initial acceleration(linear and rotational) in the beginning and greater momentum and inertia at the end to maintain velocity. The green ball is accelerating/moving/rolling a lot faster. You will have to explain the relevance. I don't get it.

Doesn't matter whether there is friction or not.  The end result is the same.

You are correct in picking the green, but you may be surprised to know that the acceleration of the ball down the path is actually constant.  The mass and thus inertia of the ball also does not have any effect on its speed, as evidenced by the conservation of energy.  It just so happens that the curve produces the best speed from the acceleration due to gravity.

The curve seen in the image is referred as the Brachistochrone curve, which is Greek for "the shortest time."  The brachistochrone curve happens to be a segment of an inverted cycloid.  Before the brachistochrone was solved, the closely related Tautochrone curve, meaning "same time," was found to be part of a cycloid also. 









For those who are still reading:

The brachistochrone problem was first posed by Johann Bernoulli, in which a point is to start from rest at point a and, solely under the effect of gravity, follow down to point b along a curve that is to be covered in the least time.

The brachistochrone problem was posed by Johann Bernoulli in Acta Eruditorum in June 1696. He introduced the problem as follows:-     I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise. The problem he posed was the following:-     Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time. ... Johann Bernoulli was not the first to consider the brachistochrone problem. Galileo in 1638 had studied the problem in his famous work Discourse on two new sciences. His version of the problem was first to find the straight line from a point A to the point on a vertical line which it would reach the quickest. He correctly calculated that such a line from A to the vertical line would be at an angle of 45 reaching the required vertical line at B say. He calculated the time taken for the point to move from A to B in a straight line, then he showed that the point would reach B more quickly if it travelled along the two line segments AC followed by CB where C is a point on an arc of a circle. Although Galileo was perfectly correct in this, he then made an error when he next argued that the path of quickest descent from A to B would be an arc of a circle - an incorrect deduction.

The brachistochrone problem was one of the earliest problems posed in the calculus of variations. Newton was challenged to solve the problem in 1696, and did so the very next day (Boyer and Merzbach 1991, p. 405). In fact, the solution, which is a segment of a cycloid, was found by Leibniz, L'Hospital, Newton, and the two Bernoullis. Johann Bernoulli solved the problem using the analogous one of considering the path of light refracted by transparent layers of varying density (Mach 1893, Gardner 1984, Courant and Robbins 1996). Actually, Johann Bernoulli had originally found an incorrect proof that the curve is a cycloid, and challenged his brother Jakob to find the required curve. When Jakob correctly did so, Johann tried to substitute the proof for his own (Boyer and Merzbach 1991, p. 417).

Follow the second quoted passage to see the maths behind it.  Equations (16) and beyond consider the condition where friction is present.  The curve is slightly different as a result of that.


Edited by zeio - 06/24/2012 at 7:01am

What amazes me is those mathematicians from the past didn't have Mathematica.   How did they know there was a symbolic solution?   It is amazing what people can do when their mind is "stupidfied" with TV.  I would imagine this is common knowledge for those that design roller coasters. Zeio, where else would you apply this?   I don't see what this has to do with TT.

mikepong wrote: can someone correlate these math equations or whatever it is to TT?

zeio wrote: And for that to happen the best path to follow during a swing is a curve that most closely resembles the brachistochrone.

The brachistochrone works because the brachistochrone allows the ball to drop or accelerate faster initially.  This isn't the case with a swing that opposes gravity.

I know where you are going with this.  You are thinking of the brachistochone with a upwards swing where the swing starts out shallow to get moving and then starts in the upwards direction.  Good one, but if the paddle attitude is always changing the resulting timing errors will result in poor control.  However, if the paddle attitude can be kept constant during the up swing it may work.   The big question is will this make any practical difference?   Most of us can swing far faster than what is necessary anyway.

The reason why a follow through matters is because a swing that results in a proper follow throw begins before you hit the ball, and therefore affects the way you hit the ball.

zeio wrote: I am getting there. Forget the maths and the gravity.  The keyword here is time.  It is time, not distance, that really matters in producing the best swing.  And for that to happen the best path to follow during a swing is a curve that most closely resembles the brachistochrone.  Concave or convex, positive or negative, however you name it.  It may sound counter-intuitive, but the longer path does produce the least time and we see signs of it in videos of the pros.  And one thing is for sure, that path has never been and probably never will be a straight line as those who are in denial of any curvature in swing mechanics may have imagined.  They could have been doing it all along without even realizing. The answer is out there, that the way we live, the tools we use, and the many things we come across during our lives are largely inspired by Nature.  As it turns out, we humans tend to ignore it and overcomplicate things.  As Johann Bernoulli himself has put it - "Nature always tends to act in the simplest way, and so it here lets one curve serve two different functions, while under any other hypothesis we should need two curves..."

I love things that challenge my intuitive assumptions about how things work. Even if this is not substantially relevant to TT it is still cool.

Edited by V-Griper - 06/25/2012 at 9:47am

To keep things straight, the energy source behind the two balls is actually the potential energy.  Potential energy can exist in quite a few forms, and each of them comes from a particular type of force, namely the conservative force.  In the brachistochrone problem, the force of gravity used to initiate a change of state from rest to motion is a type of conservative force and it gives rise to gravitational potential energy.  As the balls trace down the paths, the potential energy within them is converted into kinetic energy.  By the conservation of energy, in a closed system where friction is neglected, both potential and kinetic energy are conserved.  In layman's terms, the balls are powered by an engine that is 100% efficient with no loss.  Theoretically speaking, they start off at the same level of potential energy and should end up with the same final velocities at the end of the paths.  In an experiment run in Germany where kinetic energy is not conserved due to friction, the final velocity of the ball along the brachistochrone curve has been found to be greater than that of the ball along the straight line, as cited by this source.  So it looks like the brachistochrone is also more efficient in aspects other than time.

For table tennis, we draw on a number of potential energy.  When one winds up during a backswing, elastic potential energy is stored in parts of the body, waiting to be released.   Chemical potential energy from food is transformed into kinetic energy when we do work to swing the paddle back and forth.  When players lower their stances, they also gain gravitational potential energy.  Instead of using the gravitational potential energy gained from the force of gravity alone to start motion as in the brachistochrone problem, the goal here is to make the most out of the additional potential energy available to us to overcome gravity in a three dimensional swing.

The timing problem is not really a problem as we have all witnessed.  This is what training is for.  One may be a fluke, two could be coincidence, but three is a trend.  Every pro may approach it differently, but the basic rule still holds.  Just because the human nature within most of us tends to think only the pros can do it well should not deter the rest from pondering, discussing, and mimicking, for this is how one gains knowledge.

Of all western sources I've managed to dig up, only those in skiing and surfing is the brachistochrone curve ever directly mentioned and applied for performance gain.  Not to worry, here is a thesis just about its application in table tennis, titled "Kinetics Analysis of the Kinetic Chain of the Execution of Strokes in Succession while in Motion in Table Tennis" by 賈鵬(Jia Peng) and published in 2007.  I do not have access to the full paper, but the abstract will suffice.  Below is an excerpt about the benefits of integrating the brachistochrone curve into stroke mechanics.

(3)人体在移动后动力再链接包含了“引拍、挥拍、还原”的周期过程,尤其在连续击球时,“引拍、挥拍、还原”的线路轨迹、周期长短、频率高低以及击球前的准备充分程度都直接影响击球的效果。在研究过程中把“引拍、挥拍、还原”的轨迹、周期、频率和单摆、锥摆的轨迹、周期、频率运用力学的方法做了详尽的比较分析,也利用“最速降线”对“还原、引拍”进行了力学上的比较分析,得出人体在持拍连续击球的时候,利用近似椭圆的轨迹特点,利于发挥速度、频率优势,在节奏快的激烈对抗中获取时空上的利益,以期能够获得技战术上的优势。
(3) The kinetic chain of the human body after some displacement involves a cycle of "backswing, swing, and recovery".  The path, period, and frequency of the cycle, as well as the level of readiness before each stroke all have a direct effect on its result.  During research, the path, period, and frequency of the "backswing, swing, and recovery" and of the simple pendulum and the conical pendulum are given extensive comparative analysis using kinetics.  The same analysis is also performed using the "brachistochrone curve" for the "recovery and backswing."  The results derived suggest that when the human body executes shots in succession, use of the ellipse-like path facilitates the play of speed and frequency to get the upper hand in space-time during a fast-paced and fierce confrontation in hopes of gaining further technical and tactical advantages.

At this stage, we see the potential advantage of applying the brachistochrone curve for the recovery and backswing before and after a swing.

Other than the thesis above, there is another user by the name "jlw" posting about the brachistochrone curve(1), (2).  The poor dude got practically zero constructive feedback.

Edited by zeio - 06/25/2012 at 12:02pm

pnachtwey wrote: I find it interesting but we don't swing using gravity alone.   It is would very nice to put a  3D accelerometer on the back of a blade to see what is really happening.  It doesn't surprise me that some of the authors didn't get any attention without hard data.

power7 wrote: http://www.google.com/products/catalog?q=accelerometer&hl=en&rlz=1C1CHKZ_enUS437US437&prmd=imvnsr&bav=on.2,or.r_gc.r_pw.r_cp.r_qf.,cf.osb&biw=1024&bih=673&um=1&ie=UTF-8&tbm=shop&cid=14940118607995515195&sa=X&ei=rb3oT__EF6HY0QG5iPSMCg&ved=0CJQBEPMCMAA $25 USD...cheaper than some popular rubbers.

Edited by V-Griper - 06/25/2012 at 3:44pm

power7 wrote: zeio wrote: I am getting there. Forget the maths and the gravity.  The keyword here is time.  It is time, not distance, that really matters in producing the best swing.  And for that to happen the best path to follow during a swing is a curve that most closely resembles the brachistochrone.  Concave or convex, positive or negative, however you name it.  It may sound counter-intuitive, but the longer path does produce the least time and we see signs of it in videos of the pros.  And one thing is for sure, that path has never been and probably never will be a straight line as those who are in denial of any curvature in swing mechanics may have imagined.  They could have been doing it all along without even realizing. The answer is out there, that the way we live, the tools we use, and the many things we come across during our lives are largely inspired by Nature.  As it turns out, we humans tend to ignore it and overcomplicate things.  As Johann Bernoulli himself has put it - "Nature always tends to act in the simplest way, and so it here lets one curve serve two different functions, while under any other hypothesis we should need two curves..." This isn't non-Euclidean geometry, so the linear path takes the least amount of time.  Nor is this a gravity power problem with no friction. The modelling is incorrect.   The only thing applicable is Bernoulli's principle on the ball spinning creating low pressure systems thus make the ball "curve" in flight...

The brachistochrone problem was solved and proven in the Euclidean space in the 17th century.  The classical mechanics was developed during this period as well.  Non-Euclidean space did not emerge until the early 19th century.  What matters though is that the brachistochrone was first experimentally observed under the hypothesis of Galileo in the physical space, which, by Albert Einstein's theory of general relativity, is curved.  As the balls are traveling at far below the speed of light, the classical mechanics used to solve the problem is extremely accurate.  Unless you are hallucinating or something, I don't see how this model can be labeled wrong.

Yes, the reason a table tennis ball curves can be explained by the Bernoulli's principle.  But the Bernoulli's principle alone is insufficient in describing the sequence of events leading up to the pressure difference.  The boundary layer separation is necessary to fill the void.