Illustrated Physics
physics, drawn
Concept

Conic section — where it appears

A curve cut from a cone — ellipse, parabola or hyperbola — and the shape every inverse-square orbit and every perfect focusing mirror takes. Which of the three appears is decided by a single number, the eccentricity, and by whether the total energy of the orbit is negative, zero or positive.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

A sphere does not have a focus. Parallel rays reflected off a spherical mirror. Rays striking further from the axis cross it nearer the mirror, so there is no single point where all of them meet — the blur is spherical aberration, and a perfect sphere has it inherently.

The mirror that cannot focus, and the shape that can

A perfect sphere does not bring parallel light to a point. The blur is not a manufacturing defect — it is what the shape does, and the shape is used anyway, for a reason worth knowing.

optics · Imaging The surface a spin decides. The free surface of a liquid in a dish of radius 0.5 m turning at 10, 20, 40 revolutions a minute. In the rotating frame the surface is a level set of gz − ½ω²r², so it is a paraboloid exactly and not to some approximation, and nothing about the liquid appears in its shape: the same curve is got with mercury, water or oil. The rim stands 14.0 mm, 55.9 mm, 223.6 mm above the centre at those rates. A parabola z = r²/4f has focal length f, so these surfaces are mirrors of focal length g/2ω² — 4.47 m at 10 rpm, 1.12 m at 20 rpm, 0.28 m at 40 rpm. That is checked here on the drawn curves rather than quoted: a vertical ray reflected off the surface at a quarter, a half, three quarters and the whole of the radius crosses the axis at the same height to 0.00%, which is what a mirror with no spherical aberration means. Doubling the spin quarters the focal length, and there is no other adjustment: the dish can only ever look straight up.

The surface a spin decides

Spin a dish of liquid and its surface settles into a paraboloid — exactly, with nothing about the liquid in the shape. A parabola of that form has a focal length of g over twice the spin rate squared, so a bucket of mercury turning at twenty revolutions a minute is a telescope mirror figured by a clock instead of by grinding.

fluids · Hydrostatics The surface that focuses 1 into 1.52 with no error at all. A Cartesian oval: the locus of points for which 1 times the distance from the object plus 1.52 times the distance to the image is a constant. Every ray drawn takes exactly the same optical path, so every one arrives at the image point — not nearly, and not for small angles, but exactly, for rays at any angle the surface reaches. There is no spherical aberration because there is no approximation: this is what Fermat's principle asks for, solved rather than expanded. The surface is not a sphere, not a conic in general, and not anything a grinding machine makes easily, which is most of why lenses are spherical and aberrated instead.

The surface that images one point exactly

Fermat's principle says every ray from an object to its image must take the same time. Written as an equation, that is a curve Descartes found — a surface with no aberration at all, at any angle. It exists, it is easy to compute, and it images exactly one pair of points and nothing else.

optics · Fermat Every slope answered by one curve. The boundary of everywhere one throwing speed can reach, drawn about the hand, with straight lines from the hand at −30°, 0°, 20°, 45° running out to it. Distances are in units of v²/g. Because the boundary is a parabola with its focus at the hand, the distance to it along any direction is r = (v²/g)/(1 + sin α), and each drawn length was found separately — by searching every launch angle for the one that lands farthest along that line — and agrees with the formula to ten decimal places. At −30° the greatest reach is 2.000 v²/g, launched at 30.0°; at 0° the greatest reach is 1.000 v²/g, launched at 45.0°; at 20° the greatest reach is 0.745 v²/g, launched at 55.0°; at 45° the greatest reach is 0.586 v²/g, launched at 67.5°. Uphill the reach shrinks and downhill it grows without limit as the line approaches straight down, and the launch that achieves it always bisects the angle between the line and the vertical. The small dots are the foci of those best throws: every trajectory's focus lies on a circle of radius v²/2g about the hand, and the farthest throw along a line is the one whose focus lies on that line.

One curve answers every slope

A throw up a hillside, down one, off a height and into a basket look like four problems with four answers. They are one problem. The edge of everywhere a throw can reach is a parabola with its focus at the hand, and written about that focus it gives the farthest reach in any direction in one line — along with the reason the shot that needs the least effort is the one whose aim matters least.

mechanics · Projectile

Named alongside it

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

Spherical aberrationFocal lengthStationary pointAberrationApparent weightAspheric surfaceCausticCentrifugal forceEnvelopeEquipotentialFermat's principleFree surface

All concepts