Anatomy of Bounce Juggling

In Numbers Juggling, we saw how toss jugglers tend to strike upon a combination of increased throw rate and increased throw height to toss juggle a greater number of objects. In particular, there is no tangible “ceiling” limiting the throw height. In contrast, in bounce juggling, the ground off which the balls are bouncing presents a very real barrier to the “depth” of the throws. As illustrated in Figure 1, not only are “deeper” throws impossible, bounce jugglers can’t even make their throws “shallower” as the balls must go down all the way to the ground in order to bounce.

Unlike the toss juggler then, it would seem that the bounce juggler has only the throw rate which can be manipulated for numbers juggling.

Not quite. The bounce juggler can indeed create an effect equivalent to the variation of throw height by changing the height of the throwing (and catching) hands above the ground. One way of doing this is shown in Figure 2 where the juggler is juggling the 3 and 5 ball cascades at the same throw rate. The required difference in throw heights is achieved simply by standing on platforms at different heights above the ground.

However, this method would mean hopping onto a different platform every time the throw height needs to change.

Another method of achieving different throw heights is illustrated in Figure 3.

Here, the juggler is standing in the same place but has extended his hands upwards to create the required difference in throw heights. Though this method is used to some limited extent by real world jugglers, the scenario depicted in Figure 3 is highly exaggerated and impractical.

There are however, other bounce techniques that recognize and exploit the fact that what we want to achieve by controlling the throw height is actually to control the flight time. Since throw height is synonymous with flight time for our purposes, we will use these terms interchangeably for the rest of this blog. Also, to simplify our study of these alternative techniques for controlling the flight time, we will assume the following Juggling Lab simulator defaults which are a fairly close model for most real life situations^[1]:

1. The juggler’s hands remain at a constant height H above ground for both catching and throwing and H=1m.
2. Acceleration due to gravity, g=9.8m/s², which is its value on Earth.
3. The bounce is not perfectly elastic^[2]. Fraction of kinetic energy retained after the bounce, r = 0.9.

The results and curves presented in what follows assumes the above unless otherwise stated.

Bounce Techniques

Four basic types of bounce juggling techniques are conventionally recognized^[3] and could be used by jugglers to trade-off the flight times against the throw rates required to bounce juggle a particular siteswap sequence. Each technique allows us to achieve a certain range of flight times by varying the vertical speed of the throw.

Hyperforce

The bounce juggling technique shown in Figure 2 and Figure 3 is called the hyperforce technique. In this method, the throw is made by pushing the object downward (force throw) and the catch is made while the object is on its way up after bouncing on the ground (force catch). Figure 4 plots the height of an object juggled in this way from the time it is thrown to the time it is caught.

Since the bounce will never be perfectly elastic, there is a certain minimum vertical speed with which the object must be thrown downward so that it can rise back again up to the height of the hand. This minimum speed corresponds to the highest flight time achievable by this technique. Lower flight times can be achieved by increasing the speed of the downward throw. Figure 5 shows a plot of achievable flight times and the corresponding throw speeds required for a hyperforce throw under the Juggling Lab default conditions.

In theory, using the hyperforce technique, we can achieve as low a flight time as we want by increasing the vertical throw speed, but a flight time higher than tmax is not achievable because decreasing the throw speed below vmin would mean that the object will not rise up to the catching hand at all. Higher flight times however, can be achieved using the other three bounce techniques.

Force

In the force bounce technique, we make a downward (force) throw similar to the hyperforce technique, but the catching is different. As shown in Figure 6, instead of intercepting the object on its way up after bouncing on the ground, the object is allowed to rise to its maximum height (above the hand level) and then start falling downward again before it is caught (lift catch).

The variation of height of the object with time for a force bounced pattern is shown in Figure 7.

Again, there is a minimum vertical throw speed (vmin) that is required to get the object to bounce up higher than the hand level, but for the force bounce technique, this sets the lower limit (tmin) for the flight time achievable under the Juggling Lab default conditions as indicated in Figure 8.

In theory, for the Juggling Lab default conditions, force bounce can achieve any flight time higher than tmin by increasing the vertical throw speed.

Lift

The lift bounce technique is shown in Figure 9.

Unlike in the hyperforce and force techniques, the object is now thrown upward (lift throw). A force catch is then made similar to the hyperforce case, i.e., the object is intercepted on its way up after the bounce. The variation of height of the object with time for a lift bounced pattern is shown in Figure 10.

Once again, there is a minimum vertical throw speed (vmin) required to get the object to bounce up to the catching hand height. Interestingly however, the relationship of the achievable flight time with the vertical throw speed under the Juggling Lab default conditions is not a monotonic one for lift bounce as seen in Figure 11.

Increasing the vertical throw speed initially leads to a reduction in the flight time so that the minimum flight time tmin is achieved at a speed v > vmin. Eventually though, the flight time starts increasing with throw speed and in theory, any flight time higher than tmin can be achieved by suitably increasing the vertical throw speed.

Hyperlift

The hyperlift bounce technique shown in Figure 12 combines the lift throw with the lift catch.

Figure 13 shows the variation of height of the object with time for a hyperlift bounced pattern.

The hyperlift technique allows us to achieve very high flight times with relatively low throw speeds, thus making it easier to throw high siteswap numbers. Similar to the force and lift techniques, under the Juggling Lab default conditions, hyperlift too achieves a minimum flight time tmin at the lowest vertical speed vmin required to make such a throw possible. Figure 14 then suggests that in theory, any flight time higher than tmin can be achieved by increasing the vertical throw speed.

The Choices

If we put together all the information we have so far, Figure 15 shows us all the bounce techniques available to us to achieve a particular flight time under the Juggling Lab default conditions. As we can see, for the lower range of flight times (below the solid black horizontal line in Figure 15) there is only one option that can be used. This lone option is hyperforce for extremely low flight times (red curve) and force for slightly higher flight times (green curve below solid black line). In real life, jugglers often prefer to make a quick toss throw and eliminate the bounce completely when very low flight times are required.

As the flight time increases above the solid black line, three options become available. There is a small interesting region where two of these three options consist of the lift technique (blue curve below top horizontal dashed line) while the third is force (green curve above the solid black line). Figure 16 shows an example plot of object height vs time where two different vertical speeds lead to the same flight time in lift bouncing.

For still larger flight times, the three options are made up of lift, force and hyperlift (brown curve in Figure 15) techniques. Another technique that jugglers resort to for very large flight times is to allow the objects to bounce two or more times.

Making The Choice

As previously mentioned, below the horizontal solid black line in Figure 15, there is only one technique possible for a bounce juggler and there is really no choice. Above this line though, the juggler has to choose between lift and force, and higher up above this line, the choice of hyperlift also becomes available.

For most jugglers, the lift bouncing technique seems to feel like the most natural one. Indeed, Juggling Lab defaults to using lift bounce whenever possible. I will now make some observations (still assuming Juggling Lab default conditions) regarding the different techniques which may or may not be relevant in determining the choice that a juggler intuitively makes among lift, force and hyperlift, but may serve as a starting point for understanding this choice.

Lift vs Force

It is a common perception that force bouncing is faster than lift bouncing in terms of throw rate and hence, more difficult. As can be seen from Figure 15 however, we can always increase the throw speed in the force technique to achieve the same flight time as that of the lift technique and so slow the pattern down to the same throw rate in both cases. Yet, jugglers seem to prefer lift bouncing. A possible reason is that it is easier to maintain a juggling pattern involving throws with low speeds. As Claude Shannon remarks in “Scientific Aspects of Juggling”, not only do low throw speeds imply lower energy requirements, “it also probably implies less dispersion in both time and direction, since these tend to be proportional to energy requirements.”

As opposed to toss juggling, a complication in bounce juggling (except hyperforce) is caused by the object changing its vertical direction of motion (when not in contact with the hands) both at the top of its flight and at the bottom of its flight (where it bounces). I suspect that jugglers feel the need to track both these transition points. This becomes difficult if the highest point of the object’s trajectory is above the eye level as the juggler then has to look both up and down. Under these assumptions, Figure 17 suggests another reason for jugglers picking lift bounce over force bounce.

Figure 17 indicates that to get the same flight time under the Juggling Lab default conditions, the force bounced object rises a distance h higher than the lift bounced one, thus increasing the vertical distance over which the juggler has to keep track of the object and making force bouncing harder than lift bouncing.

Lift vs Hyperlift

The same arguments that worked in favour of lift when compared to force, would now seem to work in favour of hyperlift when compared to lift: the speed required for a hyperlift throw is much less than that for a lift throw for the same flight time. Also, as can be seen from Figure 18, there is a higher chance of keeping all the action below eye level when using hyperlift rather than lift.

There is however, another factor indicated in Figure 18. For the hyperlift bounce, the last change in vertical direction happens when the object peaks for a second time at t2. In contrast, the last change in vertical direction for a lift bounce throw happens when the object bounces at t1. If the juggler needs to keep track of this last change in vertical direction in order to catch reliably, then we see that the time available between the last change in direction and the catch is less in the hyperlift than in the lift bounce technique. That may still make hyperlift feel more uncomfortable than lift in spite of its other advantages and also make it preferable to deal with very large flight times using multiple bounces instead of hyperlift.

Note that this argument regarding the timing of the last change in vertical direction, when applied to lift vs force (Figure 17) would make force bouncing more convenient! Clearly, a lot more analysis needs to be done to nail down what factors eventually decide the actual choice made by jugglers in real life.

Now that we have an understanding of the tools available to a bounce juggler for achieving different throw “heights” (i.e., flight times), we can explore how these tools may be used by a bounce juggler for numbers juggling and for juggling siteswap sequences. We will do this in the next few blogs where we will also compare some default simulator behaviours against real world bounce juggling.

Footnotes

1. The generalized case would allow any combination of H>0, g>0 and 0<r<1.
2. “Good” bounce balls bounce back to 90% or more of the height from which they’re dropped. A perfectly elastic bounce would mean that the ball returns to the same height from which it was dropped.
3. See this discussion on rec.juggling. See “Adding throw modifiers” section of Juggling Lab documentation for the notation used to represent these techniques.