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11: Alexander I. Bobenko

… ► Matveev), published by Springer in 1994, Painlevé Equations in the Differential Geometry of Surfaces (with U. …

12: About Color Map

… ►Surface visualizations in the DLMF represent functions of the form $z=f(x,y)$ by the height $z$ or the magnitude, $|z|$, for complex functions, over the $x\times y$ plane. … ►To provide an easily interpreted encoding of surface heights, a rainbow-like mapping of height to color is used. … ►By painting the surfaces with a color that encodes the phase, $\mathrm{ph}f$, both the magnitude and phase of complex valued functions can be displayed. …

13: 4.3 Graphics

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§4.3(iii) Complex Arguments: Surfaces

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14: 5.19 Mathematical Applications

… ►The volume $V$ and surface area $S$ of the $n$-dimensional sphere of radius $r$ are given by …

15: 4.15 Graphics

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§4.15(iii) Complex Arguments: Surfaces

… ►The corresponding surfaces for $\cos \left(x+\mathrm{i}y\right)$, $\cot \left(x+\mathrm{i}y\right)$, and $\sec \left(x+\mathrm{i}y\right)$ are similar. …they can be obtained by translating the surfaces shown in Figures 4.15.8, 4.15.10, 4.15.12 by $-\frac{1}{2}\pi$ parallel to the $x$-axis, and adjusting the phase coloring in the case of Figure 4.15.10. ►The corresponding surfaces for $\arccos \left(x+\mathrm{i}y\right)$, $\mathrm{arccot}\left(x+\mathrm{i}y\right)$, $\mathrm{arcsec}\left(x+\mathrm{i}y\right)$ can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).

16: 19.33 Triaxial Ellipsoids

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§19.33(i) Surface Area

►The surface area of an ellipsoid with semiaxes $a,b,c$, and volume $V=4\pi abc/3$ is given by … ►If a conducting ellipsoid with semiaxes $a,b,c$ bears an electric charge $Q$, then the equipotential surfaces in the exterior region are confocal ellipsoids: …

17: 21.1 Special Notation

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$g,h$	positive integers.
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$a\circ b$	intersection index of $a$ and $b$, two cycles lying on a closed surface. $a\circ b=0$ if $a$ and $b$ do not intersect. Otherwise $a\circ b$ gets an additive contribution from every intersection point. This contribution is $1$ if the basis of the tangent vectors of the $a$ and $b$ cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is $-1$.
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18: Viewing DLMF Interactive 3D Graphics

… ►Users can render a 3D scene and interactively rotate, scale, and otherwise explore a function surface. …

19: 36.14 Other Physical Applications

… ►Applications include scattering of elementary particles, atoms and molecules from particles and surfaces, and chemical reactions. …

20: 4.29 Graphics

… ►The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …