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Perhaps our thinking consciously "deludes itself" that this difference disappears com­plete­ly "to the limit"; but if this were the case, the curve would also dis­ap­pear, being no longer supported or justified by distances with a progressive in­cli­na­tion.

Going beyond the veracity of the possible cases would arrive at a rectilinear pace, interpreting the curvature by points only with a forcing.
The potential contradiction lies in the fact that there can be no polygon without a length of the sides, in the presence of which the curvature insists or there would be only polygons, which from the maximum approximation tables looks to tend to reduce it, thence no real π.

This is true also if we are dealing with lines that, yet from a polygon of only 64 sides, even with an advanced graphics computer, must be enlarged in order to be able to distinguish them, as can be seen from this high-precision figure that zooms the previous one as far as the page allows, where the side -ai- on the back­ground [angle 120°] is magnified and superimposed (thick lines – only half -ai- keeps visible, but you can view the full image).
By increasing aplenty the number of sides, the vertical distance between the side and arc midpoints becomes appreciable only with a very strong mag­ni­fi­ca­tion, o­ver­taking any optical and mental effect, including the possible re­ad­just­ment due to a stressed graphics.

On the other hand, I challenge the most experienced eye, or the most keen im­ag­i­na­tion (in physiological terms) to become aware of the hor­i­zon­tal dif­fer­ence of 95 hundredths of a thousandth in the size of an already microscopic chord.
It is true that this is work entrusted to formulas, but it is also true that this min­i­mum of un­re­li­a­bil­i­ty is the result of an a priori acceptance, in which thought can dissolve.

12-sided poligons
64-sided poligon (zoom hover)

I will repeat myself, but suppose for purely visual absurdity, we have max­i­mized the figure (the background curvature is only symbolic) until we have reached with -av- and -vc- the graphic rendering of two indivisible and ir­re­duc­i­ble el­e­men­ta­ry units (not points, which are abused since they are noth­ing more than a position by coordinates), represented in the figure by two ellipses.
Two contiguous elements, whose only distance is due to their shape.

Although obviously the curvature would be minimal, if not imperceptible at this magnitude, it would remain so that even if it wanted to identify with the linear dimension of the unit itself, that of each subsequent unit would be inclined with respect to the tangent at the center of the previous unit.

Such inclination will be added to the third unit with a fourth and so on… and, whenever expressing a numerical distance between three or more units, it will only reiterate the aforementioned gap.