Concept

Coordinates

Numbers that locate a point by its position relative to a chosen frame, such as distances along perpendicular axes or a radius and an angle. Different frames suit different problems: confocal ellipses and hyperbolas give coordinates whose natural origin is a pair of foci.

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

Two families of conics, crossing at right angles. 4 ellipses and 3 hyperbolas with the same pair of foci. Every ellipse meets every hyperbola at a right angle, checked at all 12 crossings.

Two families that cross at right angles

Fix two points and draw every ellipse with those foci, then every hyperbola with the same two. Each curve of the first family meets each of the second at a right angle, so between them they are a coordinate system — and the reason is one sentence about angle bisectors.

geometry · Conic sections

Two confocal ellipses, two confocal hyperbolas, and equal diagonals of 1.399. Two ellipses and two hyperbolas with the same foci cut out a four-sided region with curved sides. Its two diagonals, drawn as straight segments, both measure 1.3987.

Equal diagonals in a curved quadrilateral

Two ellipses and two hyperbolas sharing the same foci cut out a four-sided region with curved sides and no symmetry to speak of. Its two diagonals are nevertheless exactly equal. The reason is a stretch that carries one ellipse onto the other and moves every pair of points so that crossed distances match — a property only confocal curves have.

geometry · Conic sections

Named alongside it

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

Conic Ellipse Focus Hyperbola Orthogonality Reflection Tangency

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