COGO: Computing Arc To Chord (Arc2Chord) Separation

Computing the arc to chord (Arc2Chord) separation between an arc on the circumference of a circle and the chord subtended by that arc is considered to be a vital measure for determining how well a vertex defined polygon matches its related circle (or arc in a circular arc).

In cadastral applications, an arc to chord separation value of 3mm is considered to be a good value for approximating a polygon to a circular arc.

This value is relatively easy to calculate and is give here for those who need it.

Note that I have a schema call cogo in which I create functions like this. You can use anything you like.

 /* ----------------------------------------------------------------------------------------
 * @function   : ArcToChordSeparation
 * @precis     : Returns the separation between an arc on the circumference of a circle
 *               and the chord it subtends at the centre of the arc.
 * @version    : 1.0
 * @usage      : FUNCTION ArcToChordSeparation( @p_dRadius Float,
 *                                              @p_dAngle Float)
 *                RETURNS FLOAT;
 *               eg select dbo.ArcToChordSeparation(10,90.0) as Arc2ChordSeparation;
 * @param      : p_dRadius    : Radius of Circle
 * @paramtype  : p_dRadius    : FLOAT
 * @param      : p_dAngle     : Angle subtended at centre for which chord length is to be calculated
 * @paramtype  : p_dAngle     : FLOAT
 * @return     : Arc2ChordSep : The chord subtended by the angle
 * @rtnType    : Arc2ChordSep : FLOAT
 * @note       : Does not throw exceptions
 * @note       : Assumes planar projection eg UTM.
 * @history    : Simon Greener - Feb 2005 - Original coding.
 * @history    : Simon Greener - May 2011 - Converted to SQL Server 2008 TSQL
 * @copyright  : Licensed under a Creative Commons Attribution-Share Alike 2.5 Australia License. (
 Create Function cogo.ArcToChordSeparation(@p_dRadius Float,
                                             @p_dAngle  Float)
 Returns Float
       @dAngleRad              Float,
       @dCentreToChordMidPoint Float,
       @dArcChordSeparation    Float;
       SET @dAngleRad              = @p_dAngle  * PI() / 180;
       SET @dCentreToChordMidPoint = @p_dRadius * COS(@dAngleRad/2);
       SET @dArcChordSeparation    = @p_dRadius - @dCentreToChordMidPoint;
       Return @dArcChordSeparation;

An example of how to use this function is as follows:

 select cogo.ArcToChordSeparation(10,90.0) as Arc2ChordSep;

|_. Arc2ChordSep| |2.92893218813452|

Hope this helps someone.