A Mathematical Rule Defining the Curved Path of a Lawn Bowl

Ever since I started playing lawn bowls seven years ago I have tried to figure out a mathematical rule or formula that defines the path of the bowl from mat to jack when the bowl is running on a completely flat (henceforth theoretical) surface. I could find nothing online except one complex treatment that was too difficult for me to follow. I do not think answering this question will help me or anyone else bowl better but as a retired scientist, the question nagged me.  I believe that I have finally succeeded.

Everyone who bowls knows the approximate curve of a lawn bowl perfectly delivered onto the jack. It starts out almost on a straight path and then curves progressively as it slows down and eventually comes to a stop.  We are also taught, and discover to be true, that no matter what length the jack is, the correct angle of delivery is constant. The mathematical rule I have discovered can be stated as follows:

For each particular point on a bowl’s path, the angle subtended by (i) the line connecting the jack (target) and that particular point and (ii) the tangent to the curve at that particular point is equal to the angle subtended by (iii) the proper line of initial delivery for the particular bowl/surface and (iv) the centerline of the rink.

In the diagram above, B and B’ are any bowl positions on the perfect path between mat and jack. BT  and B’T’ are the tangents to the points B and B’ respectively.  JB and JB’ are the respective lines between points B and B’ and the jack J. ⍺ is the angles JBT and JB’T’ and these are equal to the original angle, JMX, the bias angle taken on the mat, which is dependent on the particular bowl’s bias and the friction of the green. 

What this rule guarantees is that a bowl released from any point along this path that has the same initial speed as a perfectly weighted bowl delivered from the full-length mat would have at that particular point, so long as it is directed at the same angle to the line to the jack as the full-length jack would have been given, will continue to follow the same path and arrive exactly at the jack. 

How can I convince you that this is true?

Rotate the line JB around the point so that JB lies on top of JM. Line BT will take up the position of T’V. This indicates that in practice if we move the mat up the green the delivery angle should be unaltered. This is what our mathematical rule would predict. Our mathematical rule, therefore, has correctly predicted something we know to be true from experience!