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

Tangent — where it appears

A line meeting a conic twice at the same point, which is the polar of the point it touches and survives every projection as a tangent. It is the polar of its own point of contact, which is how a tangency is checked on a drawn conic without differentiating anything.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

six tangents across 3 of the four arcs1.69% of the width

Six tangents and the point nobody drew

Brianchon's theorem is a test a reader can run on a finished drawing with nothing but a straightedge — six tangents, three diagonals, and a question about whether they meet. Pointed at the drawing office's four-centre ellipse it rejects the curve by 1.7 per cent of the figure's own width, 546 times the instrument's own floor, with no true ellipse to compare against.

wrong · Conic 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 the pointone point, one conicconstructed and computed agree to 2e-13

The polar with a straightedge

Two secants through a point cut a conic at four places; the complete quadrangle they make has two more diagonal points; the line through those is the polar. Not one length, angle or midpoint is used, so the whole construction survives the projection that made the picture — and three unrelated pairs of secants land on the same line to 4.3e-13, while moving the point moves it by fifteen orders of magnitude more.

foundations · Poleandpolar fitted on five lines, tested on seven points7.4e-10 px

Five tangents name the same conic

Fitting a conic to five lines is the same six-coefficient nullspace problem as fitting one to five points, with the roles exchanged, and it lands on the same curve to 2.6 × 10⁻¹¹ pixels. What duality does not promise is that the two are equally good evidence — and measured at equal marking precision they are, to within seventeen per cent, because a drawn tangent is made of points.

foundations · Fiveconic

Named alongside it

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

ConicDualityDual conicleast squaresProjective mapsingular valuescentre of projectionCircleComplete quadrangleConditioningConic fitCross-ratio

All concepts