How interpolation impacts spatial calculations

Interpolation is the process of using known points in a geometry to approximate unknown points.

Several spatial methods and predicates use interpolation when the calculations involve circular arcs. Interpolation turns a circular arc into a sequence of straight lines. For example, a circularstring representing a quarter arc might be interpolated as a linestring with 11 control points.

Example

In Interactive SQL, connect to the sample database execute the following statement to create a variable called arc in which you will store a circularstring:
```
CREATE VARIABLE arc ST_CircularString;
```
Execute the following statement to create a circularstring and store it in the arc variable:
```
SET arc = NEW ST_CircularString( 'CircularString( -1 0, -0.707107 0.707107, 0 1 )' );
```
Execute the following statement to temporarily set the relative tolerance to 1% using the st_geometry_interpolation option.
```
SET TEMPORARY OPTION st_geometry_interpolation = 'relative-tolerance-percent=1';
```
Setting relative tolerance to 1% is optional, but makes the effects of interpolation more visible for the purposes of this example.
Open the Spatial Viewer (in Interactive SQL, click Tools Spatial Viewer ) and execute the following query to view the circularstring:
```
SELECT arc
  UNION ALL SELECT arc.ST_CurveToLine()
  UNION ALL SELECT arc.ST_CurveToLine().ST_PointN( row_num )
  FROM RowGenerator WHERE row_num <= arc.ST_CurveToLine().ST_NumPoints();
```
Notice how the arc is broken into a sequence of linestring. Since relative tolerance was set to 1%, each line segment shows up as a line that bows in from the true arc. The maximum distance between the interpolated line string and the true arc is 1% of the radius of the arc.