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Overview</a></div> +<div class="fancy-toc2"><a href="#geojson">1.1. GeoJSON</a></div> +<div class="fancy-toc2"><a href="#binary_storage_format">1.2. Binary storage format</a></div> +<div class="fancy-toc1"><a href="#using_the_geopoly_extension">2. Using The Geopoly Extension</a></div> +<div class="fancy-toc2"><a href="#queries">2.1. Queries</a></div> +<div class="fancy-toc1"><a href="#special_functions">3. Special Functions</a></div> +<div class="fancy-toc2"><a href="#the_geopoly_overlap_p1_p2_function">3.1. The geopoly_overlap(P1,P2) Function</a></div> +<div class="fancy-toc2"><a href="#the_geopoly_within_p1_p2_function">3.2. The geopoly_within(P1,P2) Function</a></div> +<div class="fancy-toc2"><a href="#the_geopoly_area_p_function">3.3. The geopoly_area(P) Function</a></div> +<div class="fancy-toc2"><a href="#the_geopoly_blob_p_function">3.4. The geopoly_blob(P) Function</a></div> +<div class="fancy-toc2"><a href="#the_geopoly_json_p_function">3.5. The geopoly_json(P) Function</a></div> +<div class="fancy-toc2"><a href="#the_geopoly_svg_p_function">3.6. The geopoly_svg(P,...) Function</a></div> +<div class="fancy-toc2"><a href="#the_geopoly_bbox_p_and_geopoly_group_bbox_p_functions">3.7. The geopoly_bbox(P) and geopoly_group_bbox(P) Functions</a></div> +<div class="fancy-toc2"><a href="#the_geopoly_contains_point_p_x_y_function">3.8. The geopoly_contains_point(P,X,Y) Function</a></div> +<div class="fancy-toc2"><a href="#the_geopoly_xform_p_a_b_c_d_e_f_function">3.9. The geopoly_xform(P,A,B,C,D,E,F) Function</a></div> +<div class="fancy-toc2"><a href="#the_geopoly_regular_x_y_r_n_function">3.10. The geopoly_regular(X,Y,R,N) Function</a></div> +<div class="fancy-toc2"><a href="#the_geopoly_ccw_j_function">3.11. The geopoly_ccw(J) Function</a></div> +<div class="fancy-toc1"><a href="#implementation_details">4. Implementation Details</a></div> +<div class="fancy-toc2"><a href="#binary_encoding_of_polygons">4.1. Binary Encoding of Polygons</a></div> +<div class="fancy-toc2"><a href="#shadow_tables">4.2. Shadow Tables</a></div> +</div> +</div> +<script> +function toggle_toc(){ +var sub = document.getElementById("toc_sub") +var mk = document.getElementById("toc_mk") +if( sub.style.display!="block" ){ +sub.style.display = "block"; +mk.innerHTML = "▼"; +} else { +sub.style.display = "none"; +mk.innerHTML = "►"; +} +} +</script> +</div> + + + + +<h1 id="overview"><span>1. </span>Overview</h1> + +<p> +The Geopoly module is an alternative interface to the <a href="rtree.html">R-Tree extension</a> that uses +the <a href="http://geojson.org">GeoJSON</a> notation +(<a href="https://tools.ietf.org/html/rfc7946">RFC-7946</a>) 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 +<a href="https://en.wikipedia.org/wiki/Scalable_Vector_Graphics">SVG</a>, and other +similar operations. + +</p><p> +The source code for Geopoly is included in the <a href="amalgamation.html">amalgamation</a>. However, +depending on configuration options and the a particular version of SQLite +you are using, the Geopoly extension may or may not be enabled by default. +To ensure that Geopoly is enabled for your build, +add the <a href="compile.html#enable_geopoly">-DSQLITE_ENABLE_GEOPOLY=1</a> compile-time option. + +</p><p> +Geopoly operates on "simple" polygons - that is, polygons for which +the boundary does not intersect itself. Geopoly thus extends the capabilities +of the <a href="rtree.html">R-Tree extension</a> which can only deal with rectangular areas. +On the other hand, the <a href="rtree.html">R-Tree extension</a> is +able to handle between 1 and 5 coordinate dimensions, whereas Geopoly is restricted +to 2-dimensional shapes only. + +</p><p> +Each polygon in the Geopoly module can be associated with an arbitrary +number of auxiliary data fields. + +</p><h2 id="geojson"><span>1.1. </span>GeoJSON</h2> + +<p>The <a href="https://tools.ietf.org/html/rfc7946">GeoJSON standard</a> is syntax for +exchanging geospatial information using JSON. GeoJSON is a rich standard +that can describe nearly any kind of geospatial content. + +</p><p>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. + +</p><p>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. + +</p><p> +For example, the following JSON describes an isosceles triangle, sitting +on the X axis and with an area of 0.5: + +</p><div class="codeblock"><pre>[[0,0],[1,0],[0.5,1],[0,0]] +</pre></div> + +<p> +A triangle has three vertexes, but the GeoJSON description of the triangle +has 4 vertexes because the first and last vertex are duplicates. + +</p><h2 id="binary_storage_format"><span>1.2. </span>Binary storage format</h2> + +<p> +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. + +</p><h1 id="using_the_geopoly_extension"><span>2. </span>Using The Geopoly Extension</h1> + +<p> +A geopoly table is created as follows: + +</p><div class="codeblock"><pre>CREATE VIRTUAL TABLE newtab USING geopoly(a,b,c); +</pre></div> + +<p> +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. + +</p><p> +Store new polygons in the table using ordinary INSERT statements: + +</p><div class="codeblock"><pre>INSERT INTO newtab(_shape) VALUES('[[0,0],[1,0],[0.5,1],[0,0]]'); +</pre></div> + +<p> +UPDATE and DELETE statements work similarly. + +</p><h2 id="queries"><span>2.1. </span>Queries</h2> + +<p> +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: + +</p><div class="codeblock"><pre>SELECT * FROM newtab WHERE geopoly_overlap(_shape, $query_polygon); +</pre></div> + +<p> +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. + +</p><p> +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. + +</p><p> +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. + +</p><h1 id="special_functions"><span>3. </span>Special Functions</h1> + +<p> +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. + +<a name="goverlap"></a> + +</p><h2 id="the_geopoly_overlap_p1_p2_function"><span>3.1. </span>The geopoly_overlap(P1,P2) Function</h2> + +<p> +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. + +</p><p> +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. + +<a name="gwithin"></a> + +</p><h2 id="the_geopoly_within_p1_p2_function"><span>3.2. </span>The geopoly_within(P1,P2) Function</h2> + +<p> +If P1 and P2 are both polygons, then the geopoly_within(P1,P2) function returns +a non-zero integer if P1 is completely contained within P2, or it returns zero +if any part of P1 is outside of P2. 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. + +</p><p> +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. + +<a name="garea"></a> + +</p><h2 id="the_geopoly_area_p_function"><span>3.3. </span>The geopoly_area(P) Function</h2> + +<p> +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. + +<a name="gblob"></a> + +</p><h2 id="the_geopoly_blob_p_function"><span>3.4. </span>The geopoly_blob(P) Function</h2> + +<p> +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. + +<a name="gjson"></a> + +</p><h2 id="the_geopoly_json_p_function"><span>3.5. </span>The geopoly_json(P) Function</h2> + +<p> +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. + +<a name="gsvg"></a> + +</p><h2 id="the_geopoly_svg_p_function"><span>3.6. </span>The geopoly_svg(P,...) Function</h2> + +<p> +If P is a polygon, then geopoly_svg(P,...) returns a text string which is a +<a href="https://en.wikipedia.org/wiki/Scalable_Vector_Graphics">Scalable Vector Graphics (SVG)</a> +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: + +</p><div class="codeblock"><pre>SELECT geopoly_svg($polygon,'class="poly"','style="fill:blue;"'); +</pre></div> + +<p> +If P is not a polygon, geopoly_svg(P,...) returns NULL. + +</p><p> +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(). + +<a name="gbbox"></a> + +</p><h2 id="the_geopoly_bbox_p_and_geopoly_group_bbox_p_functions"><span>3.7. </span>The geopoly_bbox(P) and geopoly_group_bbox(P) Functions</h2> + +<p> +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. + +</p><p> +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. + +<a name="gpoint"></a> + +</p><h2 id="the_geopoly_contains_point_p_x_y_function"><span>3.8. </span>The geopoly_contains_point(P,X,Y) Function</h2> + +<p> +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. + +<a name="xform"></a> + +</p><h2 id="the_geopoly_xform_p_a_b_c_d_e_f_function"><span>3.9. </span>The geopoly_xform(P,A,B,C,D,E,F) Function</h2> + +<p> +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. + +</p><p> +The transformation converts each vertex of the polygon according to the +following formula: + +</p><div class="codeblock"><pre>x1 = A*x0 + B*y0 + E +y1 = C*x0 + D*y0 + F +</pre></div> + +<p> +So, for example, to move a polygon by some amount DX, DY without changing +its shape, use: + +</p><div class="codeblock"><pre>geopoly_xform($polygon, 1, 0, 0, 1, $DX, $DY) +</pre></div> + +<p> +To rotate a polygon by R radians around the point 0, 0: + +</p><div class="codeblock"><pre>geopoly_xform($polygon, cos($R), sin($R), -sin($R), cos($R), 0, 0) +</pre></div> + +<p> +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 <a href="geopoly.html#ccw">geopoly_ccw()</a> function after transformation. + + +<a name="regpoly"></a> + +</p><h2 id="the_geopoly_regular_x_y_r_n_function"><span>3.10. </span>The geopoly_regular(X,Y,R,N) Function</h2> + +<p> +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. + +</p><p> +As an example, the following graphic: + +</p><blockquote> +<svg width="600" height="300" style="border:1px solid black"> +<polyline points="140.003,100 80.0019,134.644 80.0019,65.3565 140.003,100" style="fill:none;stroke:red;stroke-width:2"></polyline> <text x="100" y="106" alignment-baseline="central" text-anchor="middle">3</text> +<polyline points="240.003,100 200,140.003 159.997,100 200,59.9973 240.003,100" style="fill:none;stroke:orange;stroke-width:2"></polyline> <text x="200" y="106" alignment-baseline="central" text-anchor="middle">4</text> +<polyline points="340.003,100 312.358,138.042 267.637,123.511 267.637,76.4893 312.358,61.9583 340.003,100" style="fill:none;stroke:green;stroke-width:2"></polyline> <text x="300" y="106" alignment-baseline="central" text-anchor="middle">5</text> +<polyline points="440.003,100 419.998,134.644 380.002,134.644 359.997,100 380.002,65.3565 419.998,65.3565 440.003,100" style="fill:none;stroke:blue;stroke-width:2"></polyline> <text x="400" y="106" alignment-baseline="central" text-anchor="middle">6</text> +<polyline points="540.003,100 524.94,131.276 491.101,138.995 463.959,117.353 463.959,82.6471 491.101,61.005 524.94,68.7243 540.003,100" style="fill:none;stroke:purple;stroke-width:2"></polyline> <text x="500" y="106" alignment-baseline="central" text-anchor="middle">7</text> +<polyline points="140.003,200 128.286,228.286 100,240.003 71.7143,228.286 59.9973,200 71.7143,171.714 100,159.997 128.286,171.714 140.003,200" style="fill:none;stroke:red;stroke-width:2"></polyline> <text x="100" y="206" alignment-baseline="central" text-anchor="middle">8</text> +<polyline points="240.003,200 232.363,223.511 212.358,238.042 187.642,238.042 167.637,223.511 159.997,200 167.637,176.489 187.642,161.958 212.358,161.958 232.363,176.489 240.003,200" style="fill:none;stroke:orange;stroke-width:2"></polyline> <text x="200" y="206" alignment-baseline="central" text-anchor="middle">10</text> +<polyline points="340.003,200 334.644,219.998 319.998,234.644 300,240.003 280.002,234.644 265.356,219.998 259.997,200 265.356,180.002 280.002,165.356 300,159.997 319.998,165.356 334.644,180.002 340.003,200" style="fill:none;stroke:green;stroke-width:2"></polyline> <text x="300" y="206" alignment-baseline="central" text-anchor="middle">12</text> +<polyline points="440.003,200 436.956,215.305 428.286,228.286 415.305,236.956 400,240.003 384.695,236.956 371.714,228.286 363.044,215.305 359.997,200 363.044,184.695 371.714,171.714 384.695,163.044 400,159.997 415.305,163.044 428.286,171.714 436.956,184.695 440.003,200" style="fill:none;stroke:blue;stroke-width:2"></polyline> <text x="400" y="206" alignment-baseline="central" text-anchor="middle">16</text> +<polyline points="540.003,200 538.042,212.358 532.363,223.511 523.511,232.363 512.358,238.042 500,240.003 487.642,238.042 476.489,232.363 467.637,223.511 461.958,212.358 459.997,200 461.958,187.642 467.637,176.489 476.489,167.637 487.642,161.958 500,159.997 512.358,161.958 523.511,167.637 532.363,176.489 538.042,187.642 540.003,200" style="fill:none;stroke:purple;stroke-width:2"></polyline> <text x="500" y="206" alignment-baseline="central" text-anchor="middle">20</text> +</svg> +</blockquote> + +<p>Was generated by this script: + +</p><div class="codeblock"><pre>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>'; +</pre></div> + +<a name="ccw"></a> + +<h2 id="the_geopoly_ccw_j_function"><span>3.11. </span>The geopoly_ccw(J) Function</h2> + +<p>The geopoly_ccw(J) function returns the polygon J with counter-clockwise (CCW) rotation. + +</p><p> +<a href="https://tools.ietf.org/html/rfc7946">RFC-7946</a> 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. + + + +</p><h1 id="implementation_details"><span>4. </span>Implementation Details</h1> + +<p>The geopoly module is an extension to the <a href="rtree.html">R-Tree extension</a>. Geopoly +uses the same underlying logic and shadow tables as the <a href="rtree.html">R-Tree extension</a>. +Geopoly merely presents a different interface, and provides some extra logic +to compute polygon decoding, overlap, and containment. + +</p><h2 id="binary_encoding_of_polygons"><span>4.1. </span>Binary Encoding of Polygons</h2> + +<p> +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. + +</p><p> +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. + +</p><p> +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. + +</p><p> +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. + +</p><p> +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. + +</p><h2 id="shadow_tables"><span>4.2. </span>Shadow Tables</h2> + +<p> +The geopoly module is built on top of the <a href="rtree.html">R-Tree extension</a> 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. +</p><p align="center"><small><i>This page last modified on <a href="https://sqlite.org/docsrc/honeypot" id="mtimelink" data-href="https://sqlite.org/docsrc/finfo/pages/geopoly.in?m=7b64a47e71">2023-12-05 14:43:20</a> UTC </small></i></p> + |