Test Unicode Math and Symbol Display

(from Spherical Trigonometry section)

68. The Oblique Spherical Triangle. Let a, b, c denote the sides and α, β, γ the corresponding opposite angles of the spherical triangle, Δ its area, E its spherical excess, R the radius of the sphere upon which the triangle lies, and α ′, β ′, γ ′, a, b, c the corresponding parts of the polar triangle.

0° < a + b + c < 360°,          180° < α + β + γ < 540°.

α = 180° - a ′,       β = 180° - b ′,       γ = 180° - c ′,

a = 180° - α ′,       b = 180° - β ′,       c = 180° - γ ′.

sin α

sin a

sin β

sin b

sin γ

sin c

. (Law of Sines).

cos a = cos b cos c + sin b sin c cos α .

cos α = -cos β cos γ + sin β sin γ cos a. (Law of Cosines)

tan

√

sin (s - b ) · sin (s - c )

sin s · sin (s - a )

, where s = ½(a + b + c ).

tan

√

- cos σ · cos (σ - α )

cos (σ - β ) · cos (σ - γ )

, where σ = ½(α + β + γ ).

sin ½(α + β )

sin ½(α - β )

tan ½c

tan ½(a - b )

cos ½(α + β )

cos ½(α - β )

tan ½c

tan ½(a + b )

sin ½(a + b )

sin ½(a - b )

ctn ½γ

tan ½(α - β )

cos ½(a + b )

cos ½(a - b )

ctn ½γ

tan ½(α + β )

sin ½(α + β ) cos ½c = cos ½(a - b ) cos ½γ.

cos ½(α + β ) cos ½c = cos ½(a + b ) sin ½γ.

sin ½(α - β ) sin ½c = sin ½(a - b ) cos ½γ.

cos ½(α - β ) sin ½c = sin ½(a + b ) sin ½γ.

tan

√

tan ½s · tan ½(s - a ) · tan ½(s - b ) · tan ½(s - c ).

Δ =

πR²E

180

, α + β + γ - 180° = E .

The information on this page is from HANDBOOK OF MATHEMATICAL TABLES and FORMULAS compiled by Richard Stevens Burington, Ph. D., published by Handbook Publishers, Inc. of Sandusky, Ohio. 1953 edition. (pages 22 and 23)

Using Unicode and HTML to display math formulas on a web page is fairly easy, as long as the formulas are linear (in a typographic sense). When square roots, fractions, and other unusual features are required, some kind of special processing/rendering software is necessary in order to avoid using the HTML table format with kludges as in parts of the example test above.

There is a Mathematical Mark-up Language. It's an XML application for treating mathematic typographic requirements. To learn more about this, please visit: http://www.w3.org/TR/REC-MathML/

For this page, the "thin space" character is used between the italic letter and the prime symbol or the right parenthesis. This is done because without it, "β )" looks like "β)". (On this system, without the thin space, the italic beta impacts the prime symbol or the right parenthesis.) Unfortunately, not every good Unicode symbol font has the special spacing characters, like the thin space, so your system may be showing a few null boxes. The fraction bars and the top bars of the square root symbols are formed with <HR> tags within table cells.

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