Spherical cap, torus, and ellipsoid

The same elementary table that listed the cube, box, ball, prism, cylinder, and solid annulus and then the pyramid, cone, frustum, obelisk, and wedge finishes with five more solids: a spherical cap, a spherical zone, a torus, a barrel of circular section, and an ellipsoid.

Drag a point on a board below. Turn the 3D figures. $O$ is the table's name for the surface when it splits one off. The barrel leaves $O$ blank — this page does not invent one. The ellipsoid sends $O$ to Legendre's formula (L), unpacked below.

This math handbook writes angles in radians unless a degree sign is shown.

Spherical cap

The piece of a ball cut off by one plane parallel to the equator (a parallel of latitude). Let $r$ be the radius of the ball and $h$ the height of the cap:

$V=\frac{\pi }{3}{h}^2(3r-h),O=2\pi rh.$

$O$ here is the curved top, not the cutting disk. Drag $A$ for $r$ and $K$ for the plane; $h$ is the distance from that plane to the pole.

Spherical zone

The slice of a ball between two parallels. Let $R$ and $\rho$ be the radii of those latitude circles, $h$ the height of the slice, and $r$ still the radius of the ball:

$V=\frac{\pi h}{6}(3R^2+3{\rho }^2+h^2),O=2\pi rh.$

$O$ is the middle band — the same $2\pi rh$ as the cap, independent of where the two planes sit. Drag $P$ and $Q$ for the parallels.

Torus

A doughnut: a disk of radius $\rho$ swept around a circle of radius $r$ in its plane. The table prints

$V=2\pi r^2\rho ,O=4{\pi }^{2}r\rho .$

The live $V$ is the volume of that torus. Drag $A$ for $r$ and $P$ for $\rho$. This is not the solid annulus (a cylinder with a cylindrical hole).

Barrel

A cask of circular section: equal ends of radius $r$, bulge diameter $D$, height $h$. The table gives an approximation, and leaves the surface blank:

$V=0.0873h(2D+2r{)}^2.$

Drag $R$ for $r$, $M$ for the bulge, and $B$ for $h$. The live $V$ is that approximate formula, not a mesh volume.

Ellipsoid

Three pairwise perpendicular semi-axes $a$, $b$, and $c$, ordered $c<b<a$:

$V=\frac{4}{3}\pi abc.$

The stretch that scales a ball along three axes fills volume by one product. Drag $A$, $B$, and $K$ for those axes. Keep $c<b<a$, matching the table's order. The live $V$ is that product. The live $O$ is Legendre's formula from the next section, not a mesh area.

Surface and Legendre's formula

The stretch that gave the volume does not give the skin. Volume fills the solid: scale the three axes and the whole ball scales by those three factors at once, so $V=\frac{4}{3}\pi abc$. Area cares about direction. A patch that sat along $a$ keeps more of its size than a patch that sat along $c$; a diagonal patch mixes the two. There is no single multiple of $4\pi a^2$ that is $O$.

So you add the changed patches as you go — an integral. That integral cannot be written from $a$, $b$, and $c$ by finitely many arithmetic operations and roots. The table names two pieces of it $F(k,\phi )$ and $E(k,\phi )$, and packages them as Legendre's formula (L):

$O=2\pi c^2+\frac{2\pi b}{\sqrt{a^2-c^2}}(c^{2}F(k,\phi )+(a^2-c^2)E(k,\phi )).$

$O$ is the surface. $F$ and $E$ are not areas — they are how those named integrals score the axes. The first term in (L) is elementary. The rest weights the two integrals. Both ellipsoid boards compute this $O$: drag $K$ and $B$, and the surface updates with $F$ and $E$.

The modulus $k$ and the stop $\phi$ are not a fourth and fifth axis. They are read off $a$, $b$, and $c$:

$k=\frac{a}{b}\frac{\sqrt{b^2-c^2}}{\sqrt{a^2-c^2}},\phi =\arcsin \frac{\sqrt{a^2-c^2}}{a}.$

Keep $c<b<a$ so that $0<k<1$ and $\phi$ sits in $(0,\pi /2)$. The stop $\phi$ uses only $a$ and $c$: it is how far the integral walks. $k$ uses all three — $b$ is the modulus.

The meridian through $a$ and $c$ is an ellipse, in the plane of $A$ and $K$ on the ellipsoid. Turn the figure to see that slice. Its outer circle has radius $a$; the inner dashed circle has radius $c$; the ellipse sits between them. From $Q$ on the outer circle, drop parallel to the $c$-axis to the ellipse at $R$. Then $R=(a\cos t,0,c\sin t)$ in that plane, where $t=\angle ACQ$ is the eccentric angle — the polar angle of $Q$, not of $R$. The ellipse circumference already used this machine in the plane, with the planar minor axis in place of $c$.

A step $dt$ moves $Q$ through an arc $adt$. The same step moves $R$ through

$ds=a\sqrt{1-{\epsilon }^2{\cos }^{2}t}dt$

along that meridian, with $\epsilon =\sqrt{1-c^2/a^2}$. Divide by $a$ and integrate from $A$ toward the top. Replace $t$ by $\pi /2-t$ and $\cos$ becomes $\sin$; the definite integral does not change, and that integrand is the second kind. The first kind puts the square root in the denominator. This page does not unwind the rest of the surface into more $ds$; the table does not. It names the same two integrals, reads a different modulus $k$ from all three axes, and stops at the $\phi$ above instead of walking the full quarter:

$E(k,\phi ):={\int }_0^{\phi }\sqrt{1-k^2{\sin }^{2}t}dt,F(k,\phi ):={\int }_0^{\phi }\frac{dt}{\sqrt{1-k^2{\sin }^{2}t}}.$

The limits $0$ to $\pi /2$ are what complete means on the circumference page: the full quarter, not a stop at a smaller $\phi$. Here the upper limit is the table's smaller $\phi$, so $F$ and $E$ are incomplete.

Drag $Q$ in the first quadrant of that meridian. Keep it near the outer circle in the plane of $A$ and $K$ — the drop uses the polar angle in that plane, so a yank in $y$ does not change $t$. $R$ traces the meridian; $\angle ACQ$ is the dummy $t$ of those integrals. Ray $CS$ is locked at the table's $\phi$ — that is where (L) stops, not at the top of the circle. Drag $K$ toward $C$ to flatten the meridian: the stop opens toward $\pi /2$. Drag $B$ to change $b$: $k$ moves, the stop does not. Keep $c<b<a$. Turn the figure.

When the three axes close toward one radius, $k$ and $\phi$ both degenerate; (L) asked for $c<b<a$ and does not take $a=b=c$ as a substitution. The sphere $O=4\pi r^2$ is that limit, not a plug-in. Flatten $c$ toward $0$ instead and the stop opens toward $\pi /2$, so the integrals become the complete ones from the circumference page.