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# Circular Interpolation for Computer Control of Machine Tools

IP.com Disclosure Number: IPCOM000090717D
Original Publication Date: 1969-Jun-01
Included in the Prior Art Database: 2005-Mar-05
Document File: 2 page(s) / 34K

IBM

## Related People

Wortzman, D: AUTHOR

## Abstract

A circle is approximated by an interpolation method in which a path is generated which comprises a many-sided polygon. An orthonormal matrix for a circle is used to generate a neighboring point on a circle. In this manner, circular interpolation of the locus to be traversed in approximating a circle can be accomplished with a low amount of computer time. The two points X(n), Y(n), and X(n-1), Y(n-1) chosen have to be sufficiently close so that the chord, of length delta S between them deviates insignificantly from the required circular path. Formulas for X(n) and Y(n) based upon recursion to the preceding point X(n-1), Y(n-1) are shown based upon trigonometric formulas.

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Circular Interpolation for Computer Control of Machine Tools

A circle is approximated by an interpolation method in which a path is generated which comprises a many-sided polygon. An orthonormal matrix for a circle is used to generate a neighboring point on a circle. In this manner, circular interpolation of the locus to be traversed in approximating a circle can be accomplished with a low amount of computer time. The two points X(n), Y(n), and X(n-1), Y(n-1) chosen have to be sufficiently close so that the chord, of length delta S between them deviates insignificantly from the required circular path. Formulas for X(n) and Y(n) based upon recursion to the preceding point X(n-1), Y(n-1) are shown based upon trigonometric formulas.

Preferably, the post-processor supplies the starting and ending points of the arc as displacements from the center of the circle. Since circular interpolation is performed in two dimensions only, the post-processor indicates the plane of cutting. Further, the post-processor indicates whether the motion is clockwise or counterclockwise or if acceleration or deceleration occurs during an arc.

In deriving the points on the circle, the two recursion formulas used are X(n) = aX(n-1) + bY(n-1) and Y(n) = aY(n-1) - bX(n-1). The numbers a and b are supplied by the post-processor. The numbers a and b are derived from a/2/ + b/2/ = 1. This is necessary for the first two equations to be true, that is, for X/2/(n) + Y/2/(n) = X/2/(n-1) + Y/2/(n-1) which places both points on the circle. A second necessary condition for a and b is derived in this manner. Consider the square of the average chord length delta...