I have no clue if the following is of any value. But as I began writing this article I realized it might be more significant than even I realized.
I think all players realize that pool balls are spheres, and a sphere's circumference is nothing more than a circle made up of 360 degrees. This fact can be of specific value in certain situations (see shot at right), and of general value in others. For example, try to see the object ball's contact point as a flat plane (or visualize the entire ball as a cube.) The "plane" of the hit is the tangent line.
Above is a ball, the circumference of which is divided into just 8 sections (of 360), each section consisting of 45 degrees. (Just divide 360 degrees by the number of sections.)
Below I have reduced the number of sections (planes) on the object ball to 4, creating a square. Now the tangent line is much easier to visualize. Notice that the tangent line is at a right angle to the aiming line.
Now, please do not run to your nearest table and start trying to see the object ball as a cube as an alternative to proven aiming techniques. This view is offered only to stimulate the imagination, and to submit another way of seeing the tangent line. At the illustrated distance, the ball would still have to be hit to an accuracy of 15 degrees or so. Visualizing the ball as a cube for aiming purposes would make it more difficult to achieve that kind of aiming accuracy. "seeing" the ball for what it is is still best when playing the game.
The shot below is an ideal example of how a frozen ball can be used like a rail to kick the object ball using an interesting visualization of the tangent line.
Using The Object Ball As A Rail
You have been left with the 9-ball in the jaws of the corner pocket. The 8-ball is frozen to the opposite rail, too far from the corner to cut easily, and too straight to provide for an reasonable bank shot. An easy safety is all but impossible. Call a carom-kick off the 8-ball for the win.
A frozen object ball can be used as a rail extension. Thinking of the planes previously discussed, simply visualize a line on the face of the object ball that is parallel with the rail. Hit the line where it touches the object ball, and the cue ball will rebound much the same way it would have rebounded off the rail itself.
Practice this shot at lag speed with a cue ball hit 1/2 tip below center. Set up other angles on a frozen ball and "discover" what happens with the same speed and hit.
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