Contact geometry gives you THE shoot line of THE shot aimed.
C.ounter T.hunk E.stimates give you AN aim line without specifying a shot.
WTF is WRONG with you? Contact geometry isn't even a term that is used or useful for pool playing. You keep pushing this nonsense as if you are actually saying something that is useful or practical.
From Wikipedia on the subject since you skipped over the last time I published the link.
"
contact point p is an (
n − 1)-dimensional
linear subspace of the
tangent space to
M at
p.
[1][2] A contact element can be given by the kernel of a linear function on the tangent space to
M at
p. However, if a subspace is given by the kernel of a linear function ω, then it will also be given by the zeros of λω where λ ≠ 0 is any nonzero real number. Thus, the kernels of { λω : λ ≠ 0 } all give the same contact element. It follows that the space of all contact elements of
M can be identified with a quotient of the
cotangent bundle T*
M (with the zero section {\displaystyle 0_{M}}
removed),
[1] namely:
{\displaystyle {\text{PT}}^{*}M=({\text{T}}^{*}M-{0_{M}})/\!\sim \ {\text{ where, for }}\omega _{i}\in {\text{T}}_{p}^{*}M,\ \ \omega _{1}\sim \omega _{2}\ \iff \ \exists \ \lambda \neq 0\ :\ \omega _{1}=\lambda \omega _{2}.}
A
contact structure on an odd dimensional manifold
M, of dimension 2
k+1, is a smooth
distribution of contact elements, denoted by ξ, which is generic at each point.
[1][2] The genericity condition is that ξ is
non-integrable.
Assume that we have a smooth distribution of contact elements, ξ, given locally by a
differential 1-form α; i.e. a smooth
section of the cotangent bundle. The non-integrability condition can be given explicitly as:
[1]
{\displaystyle \alpha \wedge ({\text{d}}\alpha )^{k}\neq 0\ {\text{where}}\ ({\text{d}}\alpha )^{k}=\underbrace {{\text{d}}\alpha \wedge \ldots \wedge {\text{d}}\alpha } _{k-{\text{times}}}.}
"
Maybe you can diagram that and explain to us how it gives the shooter "THE shoot (shot) line of THE shot aimed." Please explain how the player uses this information or ANY type of formula to establish the shot line. While I personally think you're just trolling I will play along since I think you're completely full of BS.
Now for the second thing you wrote in your pathetic continuing attempts to mock that which you don't know or understand,
"C.ounter T.hunk E.stimates give you AN aim line without specifying a shot."
Um, nope, the shot dictates the sight lines that the shooter uses to get to the shot line. Sight lines are aiming/alignment lines. Aiming encompasses the entire act of getting the body into position to make something go to a target. In the case of pocket billiards where the task is to send a cue ball into an object ball and have that object ball go into a hole the AIMING is done to find where to put the cue down. There is no estimating/sighting/aligning without a specific shot in mind. Center to Edge aiming gives the shooter a set of "keys" to use where one of those keys IS what brings them to the shot line. There is no estimating in CTE. There can be making the wrong choice when presented with two choices and one is the right one. It isn't a matter of saying well it could be A-inside but need a smidgen of adjustment to the right or left. It's, for example, a choice between A-Inside or A-outside, or A inside and B outside, sometimes, like with some locksets two keys can open the same lock and as with any building manager knowing which keys go to which locks is a matter of experience. Over time the manager gets to the point where he instantly recognizes the correct keys and quickly has it in hand ready to unlock the door.
You make this out to be much more complicated than it is by casting aspersions that are not true and which come from a place of ignorance. This nonsense about "contact geometry" is merely a trolling mechanism. You could insert any physics/geometry/math theory and claim it does whatever for aiming in pool and it would be equally ridiculous.
Center to Edge aiming is s set of specific instructions that train the shooter's visual acuity by directing them to use objective reference points and a specific focus technique in order to bring that shooter's eyes and body to the correct center ball shot line, the line which the cueball MUST travel down to make the object ball in the absence of applied spin. That's it. No math needed. No formulas. No rulers. No imaginary balls or estimation. Just an open mind and training to recognize the reference points and practice to know which perception/visual "key" to apply to the shot being faced.
