HAWK-EYE FOOTPRINTS

Rod Cross Feb 2014

A footprint of a tennis ball is used by Hawk-Eye to determine whether the ball was in or out. However, the footprint itself is not measured by Hawk-Eye. It is the ball trajectory through the air that is measured. Hawk-Eye can therefore predict where the ball landed, but that still doesn’t determine whether the ball was in or out. Consider a hypothetical case where the ball lands vertically on the court near the baseline, as shown below.

The ball first contacts the court at the point shown by the red dot. If the red dot is beyond the baseline, then it looks like the ball was out. But the ball squashes and makes a circular footprint on the court, extending to the blue dot. If the blue dot were to touch the line, then the ball is in. Actually, the ball could still be in even if the blue dot doesn’t touch the line, since cloth fibres extending out from the ball surface might touch the line.

The same problems occur if the ball is incident at an oblique angle on the court. The ball first contacts the court over a small patch at the bottom of the ball. The ball then slides forward but the ball also squashes, so the back edge of the ball can either move backwards or forwards depending on how fast the ball squashes and how fast the centre of the ball moves forwards.

What does a ball footprint look like, and how big is it? Two examples are shown below. The first shows a tennis ball incident obliquely from the left at 22 m/s on a horizontal blackboard covered in chalk dust. The ball was incident at 17.7 degrees to the horizontal. The two white lines on the blackboard are 100 mm apart.

The second is the footprint of a smooth rubber ball, 60 mm diameter, incident from the left at 23 m/s and 18.5 degrees to the horizontal.

The smooth rubber ball leaves a smooth footprint and is pointy at the left end. The tennis ball is pointy at the right end and shows trailing marks at the left end due to fibres extending out of the cloth cover. If those fibres touch the line, then the ball is technically in. The tennis ball first contacted the blackboard close to the left white line on the blackboard and then slid forward. But the ball squashed rapidly, expanding backwards faster than it slid forward, at least during the first 0.1 ms of the bounce.

The footprint of the rubber ball shows that the ball slid forward and then gripped the blackboard during the latter half of the bounce. While the ball was sliding it swept the chalk dust forward. When the ball gripped, the bottom of the ball came to rest, but the whole ball still moved forward, rolling across the chalk. The video below shows what happened, at 600 frames/s. You need to advance one frame at a time to see the details. Near the end of the bounce, the near vertical line does not slide forward. Rather, the bottom of the line remains fixed to the blackboard, meaning that the ball rolls forward.

600 fps QuickTime video

PAINTED BALL

The footprint of a tennis ball can also be measured by painting the ball and impacting the ball on white paper. Here is a footprint I obtained on 8^th Jan 2014, and a graph of footprint diameter vs ball speed. The ball was incident vertically on a sheet of paper.

By an amazing coincidence, Rod Laver did the same thing on exactly the same day. He even used blue acrylic paint. A brief part of his efforts is shown in the following movie. His signed footprint was auctioned for the Roger Federer Foundation. A longer movie version can be seen at http://youtu.be/gQloLkYDUZw

Rod Laver hitting a ball dipped in paint