I don't give a rat's left cortex what definition of line-holder you're using. I am asking you to prove your claim that they exist and that they mean what you claim they mean by SHOWING us the line-holders.
It's obvious that those are actually either a figment of your imagination, or something you simply made up. An honest researcher would've shown us, already.
Considering that I already posted a number of times on how lines may be extrapolated, once to you specifically, too (page 30 of this thread), who is the honest guy here, and who the great pretender? Hmm?
Line Logic
To Belz - There is a line passing through any two points. We could think of those two points as 'line-holders'.
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What line logic is there behind an engraved line? It has an infinity of points, and infinity of implied lines. It has
only two endpoints, however. These are easy points on the line to identify. It would be harder to determine the mid-point (to know the mid-point, we have to know the end-points first), or the two quarter-points, and so on.
To translate apparently free-hand lines into exact lines, we have to come up with some method. The concept of 'forced lines' is part of this method. Out of the infinity of all the possible lines only those lines, which are special, are usable. A line is special, when it complies to an idea, such as having the category of end-points. The forced nature of the line comes from having to comply to an idea.
Endpoints. Good example of the concept of end-points working well is the 'Frame'. Sometimes, however extrapolating from endpoints does not look good, because the engraved line has its thickness, and the endpoints themselves are not distinct but appear as half-circles (under magnification). Such endpoints of an engraved line do not readily force a single line as translation. We have a choice of different lines to make, which all look equally good (bad). What other categories of linemaking could we use instead?
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Line confined between edges of an engraved line
An engraved line looks fairly straight, but we find that there is only one line we can draw through it, which bounces between the edges, touching them on both sides without protrusion, or it travels with an edge of the engraved line, in effect smoothing it out.
Example: line 'b', 'd', 'e', 'g' in the image shown
The color lines are generated by CAD, but are visually selfsame with the lines I once drew manually over a copy of the image on paper at the magnification of about 4 times lifesize.
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Tangential Points. on
bowed lines.
An irregularly bowed line forces a line, which rests against the bow.. The forced line will be held by two
line-holder points of contact between the bow and the line.
Example: line 'c' in the image shown The example shows that in this case the forced line runs with the edge of the engraved line on one side, instead of meeting a single point.
Hybrid lines
A line starts out as one type, but ends as another type.
Example line 'a'. It starts out from the bottom as the type 'bouncing between edges' in the first marked segment, and ends as a tangent in the other segment of the engraved line. Of course, this line has an alternative based purely on the "bow" technique. It isn't the lone forced line, it is one of two possible forced lines. The two lines dilate at about 1/10th of a degree from the looks of it.
Hopefully someone will know what I mean by bewilderment, when an experiment works out like a dream, and then works itself into the pre-existent overall system. Seeing this happen in essence automatically, as if following instructions, is extremely convincing to the researcher involved (like me).
In this particular experiment lines a-b-c-d create a small but perfect system in conjunction with the x,y axes. The lines 'a' and 'b' had worked out with the y-axis to a visually exact triangle, found on a regular 5-pointed star.
That extrapolated star then worked itself right into the already extensive system of the Cone & Square formation.
Long straight line edges also force extrapolation of a single line.
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Tangential Lines
A line can be tangential to two or more arcs occurring on an engraved line, or separately. This is another way of forcing a line. (line-holders, Belz?)
Enough for now, Belz?
I wrote a reply for JSF, where I went into other details of forcing lines. I think it is on page 33 of this thread. Check it out.
