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The Geopoly Interface To The SQLite R*Tree Module

# 1. Overview

The Geopoly module is an alternative interface to the R-Tree extension that uses the GeoJSON notation (RFC-7946) to describe two-dimensional polygons. Geopoly includes functions for detecting when one polygon is contained within or overlaps with another, for computing the area enclosed by a polygon, for doing linear transformations of polygons, for rendering polygons as SVG, and other similar operations.

The source code for Geopoly is included in the amalgamation but is not included in the library unless the -DSQLITE_ENABLE_GEOPOLY compile-time option is used.

Geopoly operates on "simple" polygons - that is, polygons for which the boundary does not intersect itself. Geopoly thus extends the capabilities of the R-Tree extension which can only deal with rectangular areas. On the other hand, the R-Tree extension is able to handle between 1 and 5 coordinate dimensions, whereas Geopoly is restricted to 2-dimensional shapes only.

Each polygon in the Geopoly module can be associated with an arbitrary number of auxiliary data fields.

## 1.1. GeoJSON

The GeoJSON standard is syntax for exchanging geospatial information using JSON. GeoJSON is a rich standard that can describe nearly any kind of geospatial content.

The Geopoly module only understands a small subset of GeoJSON, but a critical subset. In particular, GeoJSON understands the JSON array of vertexes that describes a simple polygon.

A polygon is defined by its vertexes. Each vertex is a JSON array of two numeric values which are the X and Y coordinates of the vertex. A polygon is a JSON array of at least four of these vertexes, and hence is an array of arrays. The first and last vertex in the array must be the same. The polygon follows the right-hand rule: When tracing a line from one vertex to the next, the area to the right of the line is outside of the polygon and the area to the left is inside the polygon. In other words, the net rotation of the vertexes is counter-clockwise.

For example, the following JSON describes an isosceles triangle, sitting on the X axis and with an area of 0.5:

    [[0,0],[1,0],[0.5,1],[0,0]]

A triangle has three vertexes, but the GeoJSON description of the triangle has 4 vertexes because the first and last vertex are duplicates.

## 1.2. Binary storage format

Internally, Geopoly stores polygons in a binary format - an SQL BLOB. Details of the binary format are given below. All of the Geopoly interfaces are able to accept polygons in either the GeoJSON format or in the binary format.

# 2. Using The Geopoly Extension

A geopoly table is created as follows:

    CREATE VIRTUAL TABLE newtab USING geopoly(a,b,c);

The statement above creates a new geopoly table named "newtab". Every geopoly table contains a built-in integer "rowid" column and a "_shape" column that contains the polygon associated with that row of the table. The example above also defines three auxiliary data columns named "a", "b", and "c" that can store whatever additional information the application needs to associate with each polygon. If there is no need to store auxiliary information, the list of auxiliary columns can be omitted.

Store new polygons in the table using ordinary INSERT statements:

    INSERT INTO newtab(_shape) VALUES('[[0,0],[1,0],[0.5,1],[0,0]]');

UPDATE and DELETE statements work similarly.

## 2.1. Queries

To query the geopoly table using an indexed geospatial search, use one of the functions geopoly_overlap() or geopoly_within() as a boolean function in the WHERE clause, with the "_shape" column as the first argument to the function. For example:

    SELECT * FROM newtab WHERE geopoly_overlap(_shape, $query_polygon);

The previous example will return every row for which the _shape overlaps the polygon in the $query_polygon parameter. The geopoly_within() function works similarly, but only returns rows for which the _shape is completely contained within $query_polygon.

Queries (and also DELETE and UPDATE statements) in which the WHERE clause contains a bare geopoly_overlap() or geopoly_within() function make use of the underlying R*Tree data structures for a fast lookup that only has to examine a subset of the rows in the table. The number of rows examines depends, of course, on the size of the $query_polygon. Large $query_polygons will normally need to look at more rows than small ones.

Queries against the rowid of a geopoly table are also very quick, even for tables with a vast number of rows. However, none of the auxiliary data columns are indexes, and so queries against the auxiliary data columns will involve a full table scan.

# 3. Special Functions

The geopoly module defines several new SQL functions that are useful for dealing with polygons. All polygon arguments to these functions can be either the GeoJSON format or the internal binary format.

## 3.1. The geopoly_overlap(P1,P2) Function

If P1 and P2 are both polygons, then the geopoly_overlap(P1,P2) function returns a non-zero integer if there is any overlap between P1 and P2, or it returns zero if P1 and P2 completely disjoint. If either P1 or P2 is not a polygon, this routine returns NULL.

The geopoly_overlap(P1,P2) function is special in that the geopoly virtual table knows how to use R*Tree indexes to optimize queries in which the WHERE clause uses geopoly_overlap() as a boolean function. Only the geopoly_overlap(P1,P2) and geopoly_within(P1,P2) functions have this capability.

## 3.2. The geopoly_within(P1,P2) Function

If P1 and P2 are both polygons, then the geopoly_within(P1,P2) function returns a non-zero integer if P2 is completely contained within P1, or it returns zero if any part of P2 is outside of P1. If P1 and P2 are the same polygon, this routine returns non-zero. If either P1 or P2 is not a polygon, this routine returns NULL.

The geopoly_within(P1,P2) function is special in that the geopoly virtual table knows how to use R*Tree indexes to optimize queries in which the WHERE clause uses geopoly_within() as a boolean function. Only the geopoly_within(P1,P2) and geopoly_overlap(P1,P2) functions have this capability.

## 3.3. The geopoly_area(P) Function

If P is a polygon, then geopoly_area(P) returns the area enclosed by that polygon. If P is not a polygon, geopoly_area(P) returns NULL.

## 3.4. The geopoly_blob(P) Function

If P is a polygon, then geopoly_blob(P) returns the binary encoding of that polygon as a BLOB. If P is not a polygon, geopoly_blob(P) returns NULL.

## 3.5. The geopoly_json(P) Function

If P is a polygon, then geopoly_json(P) returns the GeoJSON representation of that polygon as a TEXT string. If P is not a polygon, geopoly_json(P) returns NULL.

## 3.6. The geopoly_svg(P,...) Function

If P is a polygon, then geopoly_svg(P,...) returns a text string which is a Scalable Vector Graphics (SVG) representation of that polygon. If there is more one argument, then second and subsequent arguments are added as attributes to each SVG glyph. For example:

    SELECT geopoly_svg($polygon,'class="poly"','style="fill:blue;"');

If P is not a polygon, geopoly_svg(P,...) returns NULL.

Note that geopoly uses a traditional right-handed cartesian coordinate system with the origin at the lower left, whereas SVG uses a left-handed coordinate system with the origin at the upper left. The geopoly_svg() routine makes no attempt to transform the coordinate system, so the displayed images are shown in mirror image and rotated. If that is undesirable, the geopoly_xform() routine can be used to transform the output from cartesian to SVG coordinates prior to passing the polygons into geopoly_svg().

## 3.7. The geopoly_bbox(P) and geopoly_group_bbox(P) Functions

If P is a polygon, then geopoly_bbox(P) returns a new polygon that is the smallest (axis-aligned) rectangle completely enclosing P. If P is not a polygon, geopoly_bbox(P) returns NULL.

The geopoly_group_bbox(P) function is an aggregate version of geopoly_bbox(P). The geopoly_group_bbox(P) function returns the smallest rectangle that will enclose all P values seen during aggregation.

## 3.8. The geopoly_contains_point(P,X,Y) Function

If P is a polygon, then geopoly_contains_point(P,X,Y) returns a non-zero integer if and only if the coordinate X,Y is inside or on the boundary of the polygon P. If P is not a polygon, geopoly_contains_point(P,X,Y) returns NULL.

## 3.9. The geopoly_xform(P,A,B,C,D,E,F) Function

The geopoly_xform(P,A,B,C,D,E,F) function returns a new polygon that is an affine transformation of the polygon P and where the transformation is defined by values A,B,C,D,E,F. If P is not a valid polygon, this routine returns NULL.

The transformation converts each vertex of the polygon according to the following formula:

    x1 = A*x0 + B*y0 + E
    y1 = C*x0 + D*y0 + F

So, for example, to move a polygon by some amount DX, DY without changing its shape, use:

    geopoly_xform($polygon, 1, 0, 0, 1, $DX, $DY)

To rotate a polygon by R radians around the point 0, 0:

    geopoly_xform($polygon, cos($R), sin($R), -sin($R), cos($R), 0, 0)

Note that a transformation that flips the polygon might cause the order of vertexes to be reversed. In other words, the transformation might cause the vertexes to circulate in clockwise order instead of counter-clockwise. This can be corrected by sending the result through the geopoly_ccw() function after transformation.

## 3.10. The geopoly_regular(X,Y,R,N) Function

The geopoly_regular(X,Y,R,N) function returns a convex, simple, regular, equilateral, equiangular polygon with N sides, centered at X,Y, and with a circumradius of R. Or, if R is negative or if N is less than 3, the function returns NULL. The N value is capped at 1000 so that the routine will never render a polygon with more than 1000 sides even if the N value is larger than 1000.

As an example, the following graphic:

Was generated by this script:

    SELECT '<svg width="600" height="300">';
    WITH t1(x,y,n,color) AS (VALUES
       (100,100,3,'red'),
       (200,100,4,'orange'),
       (300,100,5,'green'),
       (400,100,6,'blue'),
       (500,100,7,'purple'),
       (100,200,8,'red'),
       (200,200,10,'orange'),
       (300,200,12,'green'),
       (400,200,16,'blue'),
       (500,200,20,'purple')
    )
    SELECT
       geopoly_svg(geopoly_regular(x,y,40,n),
            printf('style="fill:none;stroke:%s;stroke-width:2"',color))
       || printf(' <text x="%d" y="%d" alignment-baseline="central" text-anchor="middle">%d</text>',x,y+6,n)
      FROM t1;
    SELECT '</svg>';

## 3.11. The geopoly_ccw(J) Function

The geopoly_ccw(J) function returns the polygon J with counter-clockwise (CCW) rotation.

RFC-7946 requires that polygons use CCW rotation. But the spec also observes that many legacy GeoJSON files do not following the spec and contain polygons with clockwise (CW) rotation. The geopoly_ccw() function is useful for applications that are reading legacy GeoJSON scripts. If the input to geopoly_ccw() is a correctly-formatted polygon, then no changes are made. However, if the circulation of the input polygon is backwards, then geopoly_ccw() reverses the circulation order so that it conforms to the spec and so that it will work correctly with the Geopoly module.

# 4. Implementation Details

The geopoly module is an extension to the R-Tree extension. Geopoly uses the same underlying logic and shadow tables as the R-Tree extension. Geopoly merely presents a different interface, and provides some extra logic to compute polygon decoding, overlap, and containment.

## 4.1. Binary Encoding of Polygons

Geopoly stores all polygons internally using a binary format. A binary polygon consists of a 4-byte header following by an array of coordinate pairs in which each dimension of each coordinate is a 32-bit floating point number.

The first byte of the header is a flag byte. The least significant bit of the flag byte determines whether the coordinate pairs that follow the header are stored big-endian or little-endian. A value of 0 for the least significant bit means big-endian and a value of 1 means little endian. Other bits of the first byte in the header are reserved for future expansion.

The next three bytes in the header record the number of vertexes in the polygon as a big-endian integer. Thus there is an upper bound of about 16 million vertexes per polygon.

Following the header is the array of coordinate pairs. Each coordinate is a 32-bit floating point number. The use of 32-bit floating point values for coordinates means that any point on the earth's surface can be mapped with a resolution of approximately 2.5 meters. Higher resolutions are of course possible if the map is restricted to a single continent or country. Note that the resolution of coordinates in the geopoly module is similar in magnitude to daily movement of points on the earth's surface due to tidal forces.

The list of coordinates in the binary format contains no redundancy. The last coordinate is not a repeat of the first as it is with GeoJSON. Hence, there is always one fewer coordinate pair in the binary representation of a polygon compared to the GeoJSON representation.

## 4.2. Shadow Tables

The geopoly module is built on top of the R-Tree extension and uses the same underlying shadow tables and algorithms. For indexing purposes, each polygon is represented in the shadow tables as a rectangular bounding box. The underlying R-Tree implementation uses bounding boxes to limit the search space. Then the geoploy_overlap() and/or geopoly_within() routines further refine the search to the exact answer.