Addendum to Natural Topology on Hawk-Eye

In Natural Topology the line-calling decision-support system Hawk-Eye (which is used in professional tennis) is discussed. It is shown that an algorithm which only yields IN and OUT must lead to a bias in results, since there is no surjective morphism from the natural real numbers to a two-point space. We recommend a LET-feature, and an appropriate visualization for this feature. But we left out an explicit answer to a possible objection, namely that this shifts the bias to the borders between IN and LET and between LET and OUT.

The answer to this objection lies in the correct choosing of these new biases, which is done we think in our recommendation. Let us call the new features IN*, OUT* and LET, to compare them to the existing features IN, OUT.

On the border between IN* and LET, the bias favours IN* but only in such a way that any ball-trajectory calculation which receives IN*, is indeed IN outside the margin of error of the system currently in use.  Symmetrically, on the border between OUT* and LET, the bias favours OUT* but only in such a way that any ball-trajectory calculation which receives OUT*, is indeed OUT outside the margin of error of the system currently in use.

These new biases are unobjectionable, I believe, since they reflect what we want Hawk-Eye to do.

In the new system, if a ball is called IN*, we can be very very confident that it was indeed in. The same m.m. for OUT* and out. In all other situations, the system calls a LET, and the point should be replayed, reflecting that the decision between in and out could not be made accurately given the inherent measurement uncertainty.

 

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About fwaaldijk

mathematician (foundations & topology in constructive mathematics) and visual artist
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