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Pythagorean Theorem Square Root Circuit

IP.com Disclosure Number: IPCOM000081981D
Original Publication Date: 1974-Sep-01
Included in the Prior Art Database: 2005-Feb-28
Document File: 3 page(s) / 58K

Publishing Venue

IBM

Related People

Childress, LS: AUTHOR

Abstract

A circuit arrangement based on the graphic approach to determining the square root of a number by simulation of a compass, straightedge and precision measuring device, rather than the conventional computer iterative process, operates as follows.

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Pythagorean Theorem Square Root Circuit

A circuit arrangement based on the graphic approach to determining the square root of a number by simulation of a compass, straightedge and precision measuring device, rather than the conventional computer iterative process, operates as follows.

Referring to Fig. 1, the quantity for which the square root is sought "a" and a unit segment "1" are plotted end-to-end to form AB, and then bisected to establish midpoint M. A semicircle is drawn with a radius equal to a+1 with the center at M, a perpendicular drawn from C to intersect the semicircle at a point D, the length of CD being equal to square root of a. For the angle alpha illustrated in Fig. 1;

(Image Omitted)

This may also be expressed as a+1 over 2 sin alpha = a. (Equation 2)

Referring to Fig. 2, the logic for generating the square root of a number includes a read-only memory (ROM) which contains the sin alpha for different values of alpha. Only the cos alpha can be computed without resorting to another square-root function, i.e., sin alpha = square root of1 - cos alpha. A simple solution to establishing the proper addressing of the ROM is to use "a" as the ROM address of the sin alpha required, because the solution of Equation 2 above is unique for each value of alpha.

More sophisticated approaches involving additional circuitry can produce a method consistent with standard "off-the-shelf" trigonometric function generators (ROM). The block diagram of the instant method is shown in Fig. 2 in its simplest form, i.e., for values of "a" equal to or greater than 1.

Referring back to Fig. 1, it is noted that Equation 1 is valid only for values of 1 equal to or greater than 1, for at a = 1, alpha equals 90 degrees and a+1 over -1 = x = 0. For values of "a" less than 1, alpha exceeds 90 degrees and, therefore, sin alpha = the sin (180 degrees -B) = cos B. Equation 1 is then modified by cos B = x over c = 1-a over 1+a where x = a+1 over 2, then a = a-1 over and y = a+1 over 2 sin B = square of root of a.

For values of "a" less then 1, Fig. 3 depicts the situation for circuit logic manipulations. Alpha has exceeded 90 degrees, therefore sin alpha = sin (180 degrees -B) = sin B. Cos B = x over c = 1-a over 1+a, where x = a+1 over 2 -a = 1-a over 2 and y = a+1 over 2 sin B = square root of a.

An alternate method of handling values of "a" less than 1, shown g...