Crokinole Angles:
A Photographic Investigation

Jeremy Bagai

1. Introduction
2. Method
3. Results
4. Resin Discs
5. Conclusions
6. Acknowledgements
7. Appendix: Camera Settings and Photoshop Workflow

Introduction

A Crokinole shot consists of flicking a wooden disc over a smooth wooden surface toward a target. Often that target is another disc resting somewhere on the wooden surface. Skillful play involves predicting the path the shooting disc will take after it strikes the target disc.

Hit that target disc head-on at slow speed, and the initial disc will come to a complete stop at the point of collision. Hit that same shot at greater speed, and the initial disc will “follow through” the point of contact to occupy the now-vacant space once held by the target disc (or beyond).

1a: Slow speed; red disc comes to a complete stop.	1b: Faster speed; red disc follows-through after contact.

Depending on your background with Crokinole, other carom games, and classical physics, you might find this perfectly natural or deeply unsettling. As a former competitive pool-player, I tend toward the latter.

But crokinole discs, being discs, can’t rotate forward or backward as they slide. They are pool balls in Flatland, destined always to be hit at center. And yet, when flicked with sufficient speed, they follow through. Why?

Because wooden discs are softer than synthetic polymer pool balls. Wooden discs compress when struck together. Not all of their energy is absorbed upon head-on collision. Their collisions are not perfectly elastic. How does this inelasticity affect other crokinole shots, at different angles, at different speeds? That’s the question this paper investigates.

Method

I filmed 250+ crokinole shots, varying speed and aim. Each shot segment was converted to a still image composed of successive video frames superimposed via Photoshop. (Details on camera settings and Photoshop workflow in the appendix.)

Exported images were coded for three variables: Speed, Aim, and Carom.

• Speed is measured by the distance between successive disc images, as close as possible before the point of impact. Videos were shot at a constant 60 frames per second, so a greater distance between disc images implies greater disc speed.
• Aim is the angle between the initial path of the shooter disc and the outcome path of the target disc, subtracted from 180. A smaller number implies a “thicker” hit (closer to head-on), while a greater number implies a “thinner” hit.
• Carom is the angle between the after-contact paths of the shooter and target discs.

2a: Thick hit; medium speed.	2b: Thin hit; slow speed.

In 2a the thick hit means the black target disc absorbs most of the energy of contact, so the red shooter disc slows down. In 2b the shooter disc barely loses speed after it grazes the target disc.

Speed and aim are, theoretically, under the player’s control. The carom angle is what we want to predict, both over the board by feel, and within this paper by way of data.

Results

Michel Cloutier makes a strong claim about the carom angle in his excellent Crokinole: The Illustrated guide. “The 90-degree rule states that when your disc comes in contact with an opponent’s disc it will always rebound at a 90-degree angle.” He provides the following illustration:

This is an effective way to introduce carom angle to a beginner.

Cloutier continues, “The great thing about the 90-degree rule is that the rebound will always follow the tangent line regardless of the shooter's angle and the opponent’s disc will always move at a 90-degree angle.” This is an oversimplification. Imagine this were true, and imagine we collected data in the manner described above, and plotted carom angle against aim:

A graphic illustration of the proposed 90-degree angle rule. Regardless of aim – from direct hit (aim = 0) to glancing contact (aim = 90) – the carom angle is always 90 degrees.

In the real world, things are different. We have already seen so in the images above (2a;2b). Here is my complete data set, plotted the same way:   

With very thin hits (aim greater than 50, on the right side of the chart), the carom angle approaches but never reaches 90 degrees. As aim decreases to 20 degrees, carom angle slowly decreases with it. With aim below 20 degrees, carom angle drops quickly. Example images below:

Thick hitThin hit
3a-d. Four shots at constant speed. Thicker hits (lower aim angles) produce more follow (lower carom angles).

The term “follow” is sometimes used to mean only head-on follow-through, or “force-follow.” But we see here that “follow” (the shrinking of the carom angle) affects all shots. Most dramatically when aim is less than 10, but the effect is always there.

Now let’s separate the data by speed:

While speed does affect carom angle, it is not as strong a predictor as is aim. The examples below are arranged horizontally from thick to thin and vertically from slow to fast:

Slow / ThickSlow / Thin
Fast / ThickFast / Thin
4a-l. With the thinner hits on the right side of the table, there is very little effect of speed. Carom angle is always around 85 degrees. When the hit is much thicker (far left), greater speed produces greater follow.

Note that carom angle could not be measured in the two images on the upper-left. The shooter disc did not travel far enough after impact to be able to determine its path. This is why there are no yellow-colored (slow) data points when aim is less than 10 in Chart 3.

Thicker hits produce more follow, but also have more of their energy absorbed by contact. Making the shooter disc travel several inches beyond the point of contact for a head-on shot tests your ability to shoot extremely hard while maintaining perfect aim. An easier route may be to give yourself a slight angle, so that less of your initial speed is absorbed by contact:

5a. Aim = 1. Shooter travels ½ disc length after contact.	5b. Aim = 3. Shooter travels 2 disc lengths after contact.	5c. Aim = 4. Shooter travels 2.5 disc lengths after contact.

Resin Discs

A recent development within the Crokinole scene is the availability of discs made of hard resin instead of soft wood. The discs are pretty: pretty to look at; pretty expensive; and pretty darn resistant to follow-through.

6a. Wood. Speed = 69. Follow extends halfway through the target disc.	6b. Resin. Speed = 77. No follow at all.

If they don’t follow through, maybe resin discs follow the 90-degree rule? Let’s find out:

Pretty close! Most of the data shows a horizontal line around 85 degrees, regardless of aim. (You might call it the 85-Degree Resin rule.) The examples below illustrate the different effects of aim on carom angle in wood and resin discs:

Wood / ThickWood / Thin
Resin / ThickResin / Thin
7a-h. In the top row (wood), carom angle decreases with aim. In the bottom row (resin), carom angle is constant over aim, until aim is very low (bottom left).

Conclusions

Expert players likely already know this material, consciously or subconsciously. But for the aspiring intermediate (like me), there is much to absorb and apply:

• Carom angles are never a perfect 90 degrees; there is always some amount of follow.
• Follow increases with the thickness of the hit; the effect is continuous over the aiming range, but is greatest when aim < 10.
• Follow increases with the speed of the shot; the effect is negligible with thinner hits, moderate with thicker hits, and is greatest when aim < 10.
• Resin discs are much closer to a hypothetical elastic ideal, but they still show some follow effects when aim is between 4 and 10. They don’t show any force-follow when hit head-on.

Factors not investigated here:

• Disc side (concave or convex). I didn’t think to control for this – I should have re-read Cloutier’s Crokinole: The Illustrated guide, where this factor is discussed, one more time. I suspect the convex side will travel farther along the same vector as the concave side, but I don’t know this for sure.
• Wax. I decided not to use any for the recorded shots, rather than try to enforce a similar amount for all shots. I suspect that waxed discs will travel farther along the same vector as unwaxed discs, but I don’t know this for sure.
• Board speed. I shot the videos a few months after receiving a board for Christmas (thanks Chris!). I waxed the board shortly thereafter, and play sped up considerably. I suspect that with a freshly waxed board, discs will travel farther along the same vector as they would on less freshly waxed board, but I don’t know this for sure.

Acknowledgements

Thanks to Michel Cloutier for kind permission to use his illustration and quote from his wonderful book, Crokinole: The Illustrated guide.

Thanks to David Levy, Benjamin James, and Simon Woodhead for helpful comments earlier drafts.

Thanks to Karen Schwartz for tripod help and early photographic experimentation, to Chris Berkman for converting what would have been two tedious hours of solo disc-flicking into one pleasantly social hour by replacing target discs as I shot at them, to Bryan Alberstat for significant Photoshop help, and to Benjamin James for introducing me to this wonderful game in the first place – inside a travel trailer, during a heavy dust storm, at Burning Man.

Appendix: Camera Settings and Photoshop Workflow

Crokinole videos were taken with a Galaxy S21 phone on a tripod, at 60 frames per second and a shutter speed of 1/350. This 350/60 ratio of shutter speed to FPS is far higher than the standard practice of 2/1 (e.g. shutter speed of 1/120 for 60 FPS), but I found that at slower shutter speeds the disc images were blurry and hard to measure. Higher shutter speed meant less light, so I raised the ISO to 1600 to compensate. Most photographers dislike the image noise associated with high ISO, but that wasn’t an issue for the purposes of this paper.

In Photoshop, I split the videos into singe-shot video segments using the Timeline panel. I used Import Video Frames To Layers to create a single multi-layer .psd file for each shot, then selected all layers, and then selected the Darken blending preset. (Different disc colors may warrant a different blending preset.) I used built-in PS measurement tools to capture speed, aim, and carom values. I made extensive use of Photoshop actions to automate repetitive tasks (but it still took forever).

Jeremy Paul Bagai
jeremy@fortuitouspress.com