I am not very good at understanding much of what my teachers told me when I was young. Firstly because I was young and inexperienced in language and life. Secondly because I was subjectively pragmatic and saw no current utility for the advice or information.
Ah the pragmatism of youth! Stuffing myself with the advice of my tutors and the subjective interests of my own I unconsciously prepared for adolescents in which most of my knowledge would undergo a review and pruning process. We did not know it at the time but our brains developmental roadmap includes a time when it savagely prunes connections that are not used to prepare for the adult stage. Thus we find ourselves evaluating the survival, social, emotional, communication,, imagination,and philosophical and religious utility of the synaptic connections within our neurology, and thus the access to stored information within our system.
Does this happen only at adolescents? Or does it occur when we are old and possibly syphilitic, or at bqest infected by some viral genes with their own reforming agenda?
In any case I had no pragmatic use for the terms collinearity and coincidence in their geometrical sense ,as I was not developing a subjectiv description of my experiential continuum, but now I am, I find the terms collinear and coincident very useful.
First if you stand between twowo parallel mirrors and shift slightly you will see all the coincident points reflected in amaze around a substantial reflection. Parallax in view enables them to be distinguished, otherwise they are coincident and not distinguishable. The firs rule is coincidence is not identity.
Now if you look through a polished pane of glass you will see the collinearity of the reflected form with the forms on the other side of the glass pane. Such collinearity again is subjective, as the reflection is an optical illusion but the collinearity of reflection of a reflected triangle in a perpendicular mirror with a triangle rotated by pi radians is a concept ot hold onto . The rule of this again is collinearity is not identity.
It is sensible to use the collinear geometry in a effected mirror as a model of a negative space, but care must be taken to deal with the rotational symmetry breaking involved.
The point of coincidence is that a centre of rotation may ge coincident with an infinent number of such centres without one affecting any of the others. The collapsing of the mathematics to the same iterative formulas does not change that. Where a centre of rotation is coincident the number of rotating centres may be contributory to the point mass, at the centre of symmetry, and indeed point mass may be a direct measure of the coincidence of centres of rotation. Thus the greater the point mass the greater the number of coincident centres of rotation.
Objects which share a centre of rotation but not by coincidence must have a connecting geometry which passes influence in the form of strain or pressure or tension between the objects,but which does not effect coincident ocentres of rotation for other objects or an object. Such linked objects with this kind of transference geometry could be noted as a couple, dipole, triple, tripole etc.
Now collinearity is the same, and the mass of a collinearly structured line element may reflect or correspond to the amount of collinearity. If collinear lines do in fact meet or join then the resulting geometry again is able to transfer influence to all linked collinear lines or objects, but not to those which are merely collinear
There is a continuity implication to collinearity if and only if a line is defined in terms of a continuous fine structured region. If it is defined as a aggregation of fine regions of a rotational nature, then only contiguity may be implied, and again only if explicitly defined
Thus we see that continuity and discreteness are defined attributes and apply to a region in space. I deal with regions in space not abstract points or lines, and thus Amy subjective definitions define whether I am dealing with a continuous region or a discrete one. I get to choose, but the density function in space decides on the fractal density deposition and spatial configuration and disposition.
In short collinearity enables me to elain the mirror world as an optical illusion that has collinear aspects with reality, but rotational differences, while reality exhibits collinearity in the join or meet of lines with the same spatial orientation and coincident points of reference but which may have different attributes of influence on surrounding space.