Dual quadric — where it appears
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
Also named here as polar plane — the same set of essays touches all of them, so they are one junction rather than several.
The ball at the edge of the frame
A ball photographed near the edge of a wide picture is not drawn as a circle. It is an ellipse, longer along the radius from the centre of the picture than across it, and the centre of that ellipse is not the image of the centre of the ball. Both are properties of the flat sheet the picture is on, and an exact pinhole produces both.
Every quadric has one outline
The curve where a solid turns away from the eye is the section of the solid by one plane — the eye's polar plane — and the outline is three matrix products with no sampling in them. It works for a ball, a dish and a hyperboloid, and it fails for a cone, whose dual outline collapses to a single point and forgets which two lines pass through it.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
ConicContour generatorPolar planeQuadricCamera matrixDemonstrationAdjugatecentre of projectionDegeneracyDevelopable surfaceDiscriminantDual conic