Page:Calligraphy for computers (Hershey, 1967) (IA DTIC AD0662398).djvu/18

for the slope of its tangent. In these equations a and b are principal radii of the ellipse. Solution leads to the equation

${\frac {y}{x}}={\frac {-1+{\sqrt {1+{\frac {a^{2}}{b^{2}}}\left({\frac {dy}{dx}}\right)^{2}}}}{{\frac {a^{2}}{b^{2}}}{\frac {dy}{dx}}}}$

(5)

Along a side of the polygon, x and y are related linearly, and the slope dy/dx is constant. The point of tangency between ellipse and polygon may be found by the solution of two simultaneous linear equations in x and y. A number of solutions have been obtained, but only the solutions in the following table are within reasonable bounds.

Side of Polygon	Height of Ellipse
$y={\frac {1}{4}}\left(x-2\right)$	$2a=22.0{\textrm {for}}{\frac {b}{a}}={\frac {2}{3}}$
$y={\frac {1}{3}}\left(x-{\frac {3}{2}}\right)$	$2a=18.5{\textrm {for}}a=b$
$y={\frac {1}{2}}\left(x-1\right)$	$2b=18.5{\textrm {for}}{\frac {a}{b}}={\frac {2}{3}}$

The height for polygonalization is not well defined but seems to range from 18 to 22 raster units.

Professional printers measure the size of type in points such that one inch equals 72 points. The point size of type is the normal distance from the base line of one line of type to the base line of the next line of type. The design of character within a character block depends upon the amount of white space which is to be provided between lines of type. Printers often increase the white space to more than normal with additional leading between lines of type. The normal distance from one line to the next is one em, which is sub-divided further into printers units such that one em equals 18 units.

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