Visual Plane
every picture states the point it is correct from
Concept

Discriminant — where it appears

The quantity B² − 4AC of a conic's coefficients, whose sign names the conic as an ellipse, a parabola or a hyperbola. It gives the type without solving anything, which is why a figure can report the kind of curve a shadow is while it is being drawn.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

horizonthe eye stands 8.78 m from the centrecorrect from 26 cm, at 160 mm wideellipse · nearest point +3.96 m

The conic a circle becomes

A circle photographed is an ellipse, or a parabola, or a hyperbola, and which one is decided by a single incidence: whether the circle reaches the plane through the eye parallel to the picture. Not the lens, not the tilt, not how far away it is. The discriminant of the image agrees with that one test at every point of a sweep, and at the crossing it is zero to 1e-13.

foundations · Imagedcircle horizon25withheldfive marks fitted, one withheldcorrect from 26 cm, at 160 mm widethe withheld mark is 2e-13 px off the fitted conic

Five marks and the sixth

Five points determine a conic exactly — five coefficients up to scale, five equations, nothing left over — so a fit through five marks on a photograph is not a fit at all. The sixth mark, withheld, lands on the curve to 1.9e-13 px. And the moment a sixth mark is used, the arithmetic changes character completely: it becomes a least-squares problem, and the residual starts telling you something the five could never say.

foundations · Fiveconic parabola · clearance +0.000 mcorrect from 9 cm, at 160 mm wideparabola · B² − 4AC = 0.00e+0

Three conics are one conic and a choice of horizon

Ellipse, parabola and hyperbola are one curve and three answers to which line is infinitely far away. Pitching one camera over one 6 metre circle walks through all three, and the crossing sits at 30.465545° by two instruments with different units — but the line that decides is not the horizon, and the popular name for the choice names the wrong one of the two lines a picture of a plane has.

foundations · Imagedcircle horizonboth familiescorrect from 16 cm, at 160 mm wide40 generators · straight to 2.3e-13 px

A curved surface made of straight lines

A hyperboloid of one sheet carries two families of exactly straight lines, so a photograph of a cooling tower is full of straight lines bounding nothing flat — 28 of them here, drawn to 2.3e-13 px of straightness against a parallel of the same surface that bows 120 pixels. Both families' directions satisfy one asymptotic equation, so their vanishing points lie on one conic in the picture and not on two, to 1.6e-12 px with no fitting anywhere.

foundations · Quadric

Named alongside it

The objects these essays reach for when they reach for this one.

Conicpoint at infinityProjective mapCircleEllipseHorizonHyperbolaParabolaRuled surfaceVanishing linecentre of projectionContour generator

All concepts