Curvilinear — the series
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The arcs a curvilinear drawing uses
The taught way to draw a very wide view by hand is to run every straight edge of the world as a circular arc. That recipe has been repeated for sixty years without a surface attached to it, and it turns out to name one exactly — fitting a general conic to the image of a straight line returns a circle to nine decimal places under stereographic projection and returns nothing like a circle under any of the other standard picture surfaces.
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The arcs the five-point construction actually draws
The taught five-point construction draws circular arcs between five vanishing points and instructs a draughtsman to graduate the radius evenly. Read that way, the arcs miss a straight line's true image by up to 3.75 pixels on a 300-pixel disc. Read at the stereographic scale instead, the same arcs are exact to 4.3e-13 pixels — the construction was always drawing one projection, and the taught scale was never it.