Matpic
mathematics, in pictures
Concept

Reflection

A rigid motion of the plane that flips it about a line, leaving that line fixed and swapping the two sides. It preserves distances and tangency and reverses orientation, which is why reflecting a graph in the diagonal gives the inverse function.

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

13 rays leave the left focus in 13 directions and every one of them arrives at the rightfocuseach path is the same total length — 2.90 drawing units — which is the property thatdefines the ellipse and the reason the reflection works

Every ray comes back to the other focus

An ellipse has two foci and one property everybody remembers: the distances to them add to a constant. What that property forces is stranger and more useful — a mirror shaped like an ellipse sends every ray leaving one focus, in every direction, through the other.

geometry · conic sections -101234-101234xyslope 1.73slope 0.58eˣits inverseeˣ and its inverse, which is the same curve reflected in the dashed diagonal; the inverseis computed here by bisection, so its slope is measured rather than assumedat x = 0.55 the slope is 1.733 and the mirrored slope is 0.577; at x = 0.89 the slope is2.438 and the mirrored slope is 0.410; at x = 1.23 the slope is 3.430 and the mirroredslope is 0.292 — each pair multiplies to one

The slope of the mirror image

Undoing a function is reflecting its graph in the diagonal, and a reflection turns a slope into its reciprocal. That single observation supplies the derivative of every inverse — the logarithm, the roots, the inverse trigonometric functions — without differentiating any of them.

analysis · the derivative

Named alongside it

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

TangencyApproximationConicContinuityDandelin spheresDerivativeDirectrixEccentricityEllipseFocusInverseLimit

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