[postgis-tickets] r16479 - Add ChaikinSmoothing #4050
nicklas.aven at jordogskog.no
nicklas.aven at jordogskog.no
Tue Mar 20 06:11:17 PDT 2018
Author: nicklas
Date: 2018-03-20 06:11:17 -0700 (Tue, 20 Mar 2018)
New Revision: 16479
Added:
trunk/liblwgeom/cunit/cu_chaikin.c
trunk/liblwgeom/lwchaikins.c
trunk/regress/chaikin.sql
trunk/regress/chaikin_expected
Modified:
trunk/NEWS
trunk/doc/reference_processing.xml
trunk/liblwgeom/Makefile.in
trunk/liblwgeom/cunit/Makefile.in
trunk/liblwgeom/cunit/cu_tester.c
trunk/liblwgeom/liblwgeom.h.in
trunk/postgis/lwgeom_functions_analytic.c
trunk/postgis/postgis.sql.in
trunk/regress/Makefile.in
Log:
Add ChaikinSmoothing #4050
Modified: trunk/NEWS
===================================================================
--- trunk/NEWS 2018-03-20 09:37:03 UTC (rev 16478)
+++ trunk/NEWS 2018-03-20 13:11:17 UTC (rev 16479)
@@ -2,6 +2,7 @@
2018/xx/xx
* New Features *
+ - #4050, ST_ChaikinSmoothing (Nicklas Avén)
- #3989, ST_Buffer single sided option (Stephen Knox)
- #3876, ST_Angle function (Rémi Cura)
- #3564, ST_LineInterpolatePoints (Dan Baston)
Modified: trunk/doc/reference_processing.xml
===================================================================
--- trunk/doc/reference_processing.xml 2018-03-20 09:37:03 UTC (rev 16478)
+++ trunk/doc/reference_processing.xml 2018-03-20 13:11:17 UTC (rev 16479)
@@ -3081,7 +3081,7 @@
</refsection>
</refentry>
-<refentry id="ST_SimplifyVW">
+ <refentry id="ST_SimplifyVW">
<refnamediv>
<refname>ST_SimplifyVW</refname>
<refpurpose>Returns a "simplified" version of the given geometry using the Visvalingam-Whyatt algorithm</refpurpose>
@@ -3129,8 +3129,64 @@
<para><xref linkend="ST_SetEffectiveArea" />, <xref linkend="ST_Simplify" />, <xref linkend="ST_SimplifyPreserveTopology" />, Topology <xref linkend="TP_ST_Simplify"/></para>
</refsection>
</refentry>
- <refentry id="ST_SetEffectiveArea">
+
+ <refentry id="ST_ChaikinSmoothing">
<refnamediv>
+ <refname>ST_ChaikinSmoothing</refname>
+ <refpurpose>Returns a "smoothed" version of the given geometry using the Chaikin algorithm</refpurpose>
+ </refnamediv>
+
+ <refsynopsisdiv>
+ <funcsynopsis>
+ <funcprototype>
+ <funcdef>geometry <function>ST_ChaikinSmoothing</function></funcdef>
+ <paramdef><type>geometry</type> <parameter>geomA</parameter></paramdef>
+ <paramdef><type>integer</type> <parameter>nIterations</parameter></paramdef>
+ <paramdef><type>boolean</type> <parameter>preserveEndPoints</parameter></paramdef>
+ </funcprototype>
+ </funcsynopsis>
+ </refsynopsisdiv>
+
+ <refsection>
+ <title>Description</title>
+ <para> Returns a "smoothed" version of the given geometry using the Chaikin algorithm.
+ See <ulink url="http://www.idav.ucdavis.edu/education/CAGDNotes/Chaikins-Algorithm/Chaikins-Algorithm.html">Chaikins-Algorithm</ulink> for an explanation of the process.
+ For each iteration the number of vertex points will double.
+ The function puts new vertex points at 1/4 of the line before and after each point and removes the original point.
+ To reduce the number of points use one of the simplification functions on the result.
+ The new points gets interpolated values for all included dimensions, also z and m.</para>
+
+ <note><para>Note that returned geometry will get more points than the original.
+ To reduce the number of points again use one of the simplification functions on the result.
+ (see <xref linkend="ST_Simplify" /> and <xref linkend="ST_SimplifyVW" />)</para></note>
+ <note><para>Second argument, number of iterations is limited to max 5 iterations</para></note>
+ <note><para>Note third argument is only valid for polygons, and will be ignored for linestrings</para></note>
+ <note><para>This function handles 3D and the third dimension will affect the result.</para></note>
+ <para>Availability: 2.5.0</para>
+ </refsection>
+
+ <refsection>
+ <title>Examples</title>
+ <para>A triangle is smoothed</para>
+ <programlisting>
+
+select ST_AsText(ST_ChaikinSmoothing(geom)) smoothed
+FROM (SELECT 'LINESTRING(0 0, 8 8, 0 16)'::geometry geom) As foo;
+-result
+ smoothed
+-----------+-------------------+
+LINESTRING(0 0,6 6,6 10,0 16)
+
+ </programlisting>
+ </refsection>
+ <refsection>
+ <title>See Also</title>
+ <para><xref linkend="ST_Simplify" />, <xref linkend="ST_SimplifyVW" /></para>
+ </refsection>
+ </refentry>
+
+ <refentry id="ST_SetEffectiveArea">
+ <refnamediv>
<refname>ST_SetEffectiveArea</refname>
<refpurpose>
Sets the effective area for each vertex, storing the value in the M ordinate. A simplified geometry can then be generated by filtering on the M ordinate.
Modified: trunk/liblwgeom/Makefile.in
===================================================================
--- trunk/liblwgeom/Makefile.in 2018-03-20 09:37:03 UTC (rev 16478)
+++ trunk/liblwgeom/Makefile.in 2018-03-20 13:11:17 UTC (rev 16479)
@@ -114,6 +114,7 @@
lwgeom_wrapx.o \
lwunionfind.o \
effectivearea.o \
+ lwchaikins.o \
lwkmeans.o \
varint.o
Modified: trunk/liblwgeom/cunit/Makefile.in
===================================================================
--- trunk/liblwgeom/cunit/Makefile.in 2018-03-20 09:37:03 UTC (rev 16478)
+++ trunk/liblwgeom/cunit/Makefile.in 2018-03-20 13:11:17 UTC (rev 16479)
@@ -36,6 +36,7 @@
cu_tree.o \
cu_measures.o \
cu_effectivearea.o \
+ cu_chaikin.o \
cu_node.o \
cu_clip_by_rect.o \
cu_libgeom.o \
Added: trunk/liblwgeom/cunit/cu_chaikin.c
===================================================================
--- trunk/liblwgeom/cunit/cu_chaikin.c (rev 0)
+++ trunk/liblwgeom/cunit/cu_chaikin.c 2018-03-20 13:11:17 UTC (rev 16479)
@@ -0,0 +1,66 @@
+/**********************************************************************
+ *
+ * PostGIS - Spatial Types for PostgreSQL
+ * http://postgis.net
+ *
+ * Copyright 2012 Sandro Santilli <strk at kbt.io>
+ *
+ * This is free software; you can redistribute and/or modify it under
+ * the terms of the GNU General Public Licence. See the COPYING file.
+ *
+ **********************************************************************/
+
+#include <stdio.h>
+#include <stdlib.h>
+#include <string.h>
+#include "CUnit/Basic.h"
+
+#include "liblwgeom_internal.h"
+#include "cu_tester.h"
+
+static void do_test_chaikin(char *geom_txt,char *expected, int n_iterations, int preserve_end_points)
+{
+ LWGEOM *geom_in, *geom_out;
+ char *out_txt;
+ geom_in = lwgeom_from_wkt(geom_txt, LW_PARSER_CHECK_NONE);
+ geom_out = lwgeom_chaikin(geom_in, n_iterations, preserve_end_points);
+ out_txt = lwgeom_to_wkt(geom_out, WKT_EXTENDED, 3, NULL);
+ if(strcmp(expected, out_txt))
+ printf("%s is not equal to %s\n", expected, out_txt);
+ CU_ASSERT_STRING_EQUAL(expected, out_txt)
+ lwfree(out_txt);
+ lwgeom_free(geom_in);
+ lwgeom_free(geom_out);
+ return;
+}
+
+static void do_test_chaikin_lines(void)
+{
+ /*Simpliest test*/
+ do_test_chaikin("LINESTRING(0 0,8 8,16 0)","LINESTRING(0 0,6 6,10 6,16 0)", 1, 1);
+ /*2 iterations*/
+ do_test_chaikin("LINESTRING(0 0,8 8,16 0)","LINESTRING(0 0,4.5 4.5,7 6,9 6,11.5 4.5,16 0)", 2, 1);
+ /*check so it really ignores preserve_end_points set to off for linestrings*/
+ do_test_chaikin("LINESTRING(0 0,8 8,16 0)","LINESTRING(0 0,4.5 4.5,7 6,9 6,11.5 4.5,16 0)", 2, 0);
+ return;
+}
+
+static void do_test_chaikin_polygons(void)
+{
+ /*Simpliest test*/
+ do_test_chaikin("POLYGON((0 0,8 8,16 0,0 0))","POLYGON((0 0,6 6,10 6,14 2,12 0,0 0))", 1, 1);
+ /*2 iterations*/
+ do_test_chaikin("POLYGON((0 0,8 8,16 0,0 0))","POLYGON((0 0,4.5 4.5,7 6,9 6,11 5,13 3,13.5 1.5,12.5 0.5,9 0,0 0))", 2, 1);
+ /*2 iterations without preserving end points*/
+ do_test_chaikin("POLYGON((0 0,8 8,16 0,0 0))","POLYGON((3 3,5 5,7 6,9 6,11 5,13 3,13.5 1.5,12.5 0.5,10 0,6 0,3.5 0.5,2.5 1.5,3 3))", 2, 0);
+ return;
+}
+
+
+void chaikin_suite_setup(void);
+void chaikin_suite_setup(void)
+{
+ CU_pSuite suite = CU_add_suite("chaikin",NULL,NULL);
+ PG_ADD_TEST(suite, do_test_chaikin_lines);
+ PG_ADD_TEST(suite, do_test_chaikin_polygons);
+}
Modified: trunk/liblwgeom/cunit/cu_tester.c
===================================================================
--- trunk/liblwgeom/cunit/cu_tester.c 2018-03-20 09:37:03 UTC (rev 16478)
+++ trunk/liblwgeom/cunit/cu_tester.c 2018-03-20 13:11:17 UTC (rev 16479)
@@ -47,6 +47,7 @@
extern void lwstroke_suite_setup(void);
extern void measures_suite_setup(void);
extern void effectivearea_suite_setup(void);
+extern void chaikin_suite_setup(void);
extern void minimum_bounding_circle_suite_setup(void);
extern void misc_suite_setup(void);
extern void node_suite_setup(void);
@@ -97,6 +98,7 @@
lwstroke_suite_setup,
measures_suite_setup,
effectivearea_suite_setup,
+ chaikin_suite_setup,
minimum_bounding_circle_suite_setup,
misc_suite_setup,
node_suite_setup,
Modified: trunk/liblwgeom/liblwgeom.h.in
===================================================================
--- trunk/liblwgeom/liblwgeom.h.in 2018-03-20 09:37:03 UTC (rev 16478)
+++ trunk/liblwgeom/liblwgeom.h.in 2018-03-20 13:11:17 UTC (rev 16479)
@@ -990,6 +990,7 @@
extern LWGEOM* lwgeom_force_4d(const LWGEOM *geom);
extern LWGEOM* lwgeom_set_effective_area(const LWGEOM *igeom, int set_area, double area);
+extern LWGEOM* lwgeom_chaikin(const LWGEOM *igeom, int n_iterations, int preserve_endpoint);
/*
* Force to use SFS 1.1 geometry type
Added: trunk/liblwgeom/lwchaikins.c
===================================================================
--- trunk/liblwgeom/lwchaikins.c (rev 0)
+++ trunk/liblwgeom/lwchaikins.c 2018-03-20 13:11:17 UTC (rev 16479)
@@ -0,0 +1,202 @@
+
+/**********************************************************************
+ *
+ * PostGIS - Spatial Types for PostgreSQL
+ * http://postgis.net
+ *
+ * PostGIS is free software: you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation, either version 2 of the License, or
+ * (at your option) any later version.
+ *
+ * PostGIS is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with PostGIS. If not, see <http://www.gnu.org/licenses/>.
+ *
+ **********************************************************************
+ *
+ * Copyright 2018 Nicklas Avén
+ *
+ **********************************************************************/
+
+
+ #include "liblwgeom_internal.h"
+
+
+
+static POINTARRAY * ptarray_chaikin(POINTARRAY *inpts, int preserve_endpoint, int isclosed)
+{
+ uint32_t p, i, n_out_points=0, p1_set=0, p2_set=0;
+ POINT4D p1, p2;
+ POINTARRAY *opts;
+ double *dlist;
+ double deltaval, quarter_delta, val1, val2;
+ int ndims = 2 + ptarray_has_z(inpts) + ptarray_has_m(inpts);
+ int new_npoints = inpts->npoints * 2;
+ opts = ptarray_construct_empty(FLAGS_GET_Z(inpts->flags), FLAGS_GET_M(inpts->flags), new_npoints);
+
+ dlist = (double*)(opts->serialized_pointlist);
+
+ p1 = getPoint4d(inpts, 0);
+ /*if first point*/
+ if(preserve_endpoint)
+ {
+ ptarray_append_point(opts, &p1, LW_TRUE);
+ n_out_points++;
+ }
+
+ for (p=1;p<inpts->npoints;p++)
+ {
+ memcpy(&p2, &p1, sizeof(POINT4D));
+ p1 = getPoint4d(inpts, p);
+ if(p>0)
+ {
+ p1_set = p2_set = 0;
+ for (i=0;i<ndims;i++)
+ {
+ val1 = ((double*) &(p1))[i];
+ val2 = ((double*) &(p2))[i];
+ deltaval = val1 - val2;
+ quarter_delta = deltaval * 0.25;
+ if(!preserve_endpoint || p > 1)
+ {
+ dlist[n_out_points * ndims + i] = val2 + quarter_delta;
+ p1_set = 1;
+ }
+ if(!preserve_endpoint || p < inpts->npoints - 1)
+ {
+ dlist[(n_out_points + p1_set) * ndims + i] = val1 - quarter_delta;
+ p2_set = 1;
+ }
+ }
+ n_out_points+=(p1_set + p2_set);
+ }
+ }
+
+ /*if last point*/
+ if(preserve_endpoint)
+ {
+ opts->npoints = n_out_points;
+ ptarray_append_point(opts, &p1, LW_TRUE);
+ n_out_points++;
+ }
+
+ if(isclosed && !preserve_endpoint)
+ {
+ opts->npoints = n_out_points;
+ POINT4D first_point = getPoint4d(opts, 0);
+ ptarray_append_point(opts, &first_point, LW_TRUE);
+ n_out_points++;
+ }
+ opts->npoints = n_out_points;
+
+ return opts;
+
+}
+
+static LWLINE* lwline_chaikin(const LWLINE *iline, int n_iterations)
+{
+ POINTARRAY *pa, *pa_new;
+ int j;
+
+ if( lwline_is_empty(iline))
+ return lwline_clone(iline);
+
+
+ LWLINE *oline = lwline_construct_empty(iline->srid, FLAGS_GET_Z(iline->flags), FLAGS_GET_M(iline->flags));
+ pa = iline->points;
+ for (j=0;j<n_iterations;j++)
+ {
+ pa_new = ptarray_chaikin(pa,1,LW_FALSE);
+ if(j>0)
+ ptarray_free(pa);
+ pa=pa_new;
+ }
+ oline = lwline_construct(iline->srid, NULL, pa);
+
+ oline->type = iline->type;
+ return oline;
+
+}
+
+
+static LWPOLY* lwpoly_chaikin(const LWPOLY *ipoly, int n_iterations, int preserve_endpoint)
+{
+ uint32_t i, j;
+ POINTARRAY *pa, *pa_new;
+ LWPOLY *opoly = lwpoly_construct_empty(ipoly->srid, FLAGS_GET_Z(ipoly->flags), FLAGS_GET_M(ipoly->flags));
+
+ if( lwpoly_is_empty(ipoly) )
+ return opoly;
+ for (i = 0; i < ipoly->nrings; i++)
+ {
+ pa = ipoly->rings[i];
+ for(j=0;j<n_iterations;j++)
+ {
+ pa_new = ptarray_chaikin(pa,preserve_endpoint,LW_TRUE);
+ if(j>0)
+ ptarray_free(pa);
+ pa=pa_new;
+ }
+ if(pa->npoints>=4)
+ {
+ if( lwpoly_add_ring(opoly,pa ) == LW_FAILURE )
+ return NULL;
+ }
+ }
+
+ opoly->type = ipoly->type;
+
+ if( lwpoly_is_empty(opoly) )
+ return NULL;
+
+ return opoly;
+
+}
+
+
+static LWCOLLECTION* lwcollection_chaikin(const LWCOLLECTION *igeom, int n_iterations, int preserve_endpoint)
+{
+ LWDEBUG(2, "Entered lwcollection_set_effective_area");
+ uint32_t i;
+
+ LWCOLLECTION *out = lwcollection_construct_empty(igeom->type, igeom->srid, FLAGS_GET_Z(igeom->flags), FLAGS_GET_M(igeom->flags));
+
+ if( lwcollection_is_empty(igeom) )
+ return out; /* should we return NULL instead ? */
+
+ for( i = 0; i < igeom->ngeoms; i++ )
+ {
+ LWGEOM *ngeom = lwgeom_chaikin(igeom->geoms[i],n_iterations,preserve_endpoint);
+ if ( ngeom ) out = lwcollection_add_lwgeom(out, ngeom);
+ }
+
+ return out;
+}
+
+
+LWGEOM* lwgeom_chaikin(const LWGEOM *igeom, int n_iterations, int preserve_endpoint)
+{
+ LWDEBUG(2, "Entered lwgeom_set_effective_area");
+ switch (igeom->type)
+ {
+ case POINTTYPE:
+ case MULTIPOINTTYPE:
+ return lwgeom_clone(igeom);
+ case LINETYPE:
+ return (LWGEOM*)lwline_chaikin((LWLINE*)igeom, n_iterations);
+ case POLYGONTYPE:
+ return (LWGEOM*)lwpoly_chaikin((LWPOLY*)igeom,n_iterations,preserve_endpoint);
+ case MULTILINETYPE:
+ case MULTIPOLYGONTYPE:
+ case COLLECTIONTYPE:
+ return (LWGEOM*)lwcollection_chaikin((LWCOLLECTION *)igeom,n_iterations,preserve_endpoint);
+ default:
+ lwerror("lwgeom_chaikin: unsupported geometry type: %s",lwtype_name(igeom->type));
+ }
+ return NULL;
+}
Modified: trunk/postgis/lwgeom_functions_analytic.c
===================================================================
--- trunk/postgis/lwgeom_functions_analytic.c 2018-03-20 09:37:03 UTC (rev 16478)
+++ trunk/postgis/lwgeom_functions_analytic.c 2018-03-20 13:11:17 UTC (rev 16479)
@@ -127,7 +127,51 @@
PG_RETURN_POINTER(result);
}
+PG_FUNCTION_INFO_V1(LWGEOM_ChaikinSmoothing);
+Datum LWGEOM_ChaikinSmoothing(PG_FUNCTION_ARGS)
+{
+ GSERIALIZED *geom = PG_GETARG_GSERIALIZED_P(0);
+ GSERIALIZED *result;
+ int type = gserialized_get_type(geom);
+ LWGEOM *in;
+ LWGEOM *out;
+ int preserve_endpoints=1;
+ int n_iterations=1;
+ if ( type == POINTTYPE || type == MULTIPOINTTYPE )
+ PG_RETURN_POINTER(geom);
+
+ if ( (PG_NARGS()>1) && (!PG_ARGISNULL(1)) )
+ n_iterations = PG_GETARG_INT32(1);
+
+ if (n_iterations>5)
+ elog(ERROR,"Not more than 5 iterations please");
+ if (n_iterations< 1)
+ elog(ERROR,"Number of iterations must be between 1 and 5");
+
+ if ( (PG_NARGS()>2) && (!PG_ARGISNULL(2)) )
+ {
+ if(PG_GETARG_BOOL(2))
+ preserve_endpoints = 1;
+ else
+ preserve_endpoints = 0;
+ }
+
+ in = lwgeom_from_gserialized(geom);
+
+ out = lwgeom_chaikin(in, n_iterations, preserve_endpoints);
+ if ( ! out ) PG_RETURN_NULL();
+
+ /* COMPUTE_BBOX TAINTING */
+ if ( in->bbox ) lwgeom_add_bbox(out);
+
+ result = geometry_serialize(out);
+ lwgeom_free(out);
+ PG_FREE_IF_COPY(geom, 0);
+ PG_RETURN_POINTER(result);
+}
+
+
/***********************************************************************
* --strk at kbt.io;
***********************************************************************/
Modified: trunk/postgis/postgis.sql.in
===================================================================
--- trunk/postgis/postgis.sql.in 2018-03-20 09:37:03 UTC (rev 16478)
+++ trunk/postgis/postgis.sql.in 2018-03-20 13:11:17 UTC (rev 16479)
@@ -3144,7 +3144,14 @@
AS 'MODULE_PATHNAME', 'LWGEOM_SetEffectiveArea'
LANGUAGE 'c' IMMUTABLE STRICT _PARALLEL
COST 1; -- reset cost, see #3675
-
+
+-- Availability: 2.5.0
+CREATE OR REPLACE FUNCTION ST_ChaikinSmoothing(geometry, integer default 1, boolean default true)
+ RETURNS geometry
+ AS 'MODULE_PATHNAME', 'LWGEOM_ChaikinSmoothing'
+ LANGUAGE 'c' IMMUTABLE STRICT _PARALLEL
+ COST 1; -- reset cost, see #3675
+
-- ST_SnapToGrid(input, xoff, yoff, xsize, ysize)
-- Availability: 1.2.2
CREATE OR REPLACE FUNCTION ST_SnapToGrid(geometry, float8, float8, float8, float8)
Modified: trunk/regress/Makefile.in
===================================================================
--- trunk/regress/Makefile.in 2018-03-20 09:37:03 UTC (rev 16478)
+++ trunk/regress/Makefile.in 2018-03-20 13:11:17 UTC (rev 16479)
@@ -80,6 +80,7 @@
bestsrid \
binary \
boundary \
+ chaikin \
cluster \
concave_hull\
ctors \
Added: trunk/regress/chaikin.sql
===================================================================
--- trunk/regress/chaikin.sql (rev 0)
+++ trunk/regress/chaikin.sql 2018-03-20 13:11:17 UTC (rev 16479)
@@ -0,0 +1,6 @@
+SELECT '1', ST_astext(ST_ChaikinSmoothing('LINESTRING(0 0, 8 8, 0 16)'));
+SELECT '2', ST_astext(ST_ChaikinSmoothing('LINESTRING(0 0, 8 8, 0 16)',10));
+SELECT '3', ST_astext(ST_ChaikinSmoothing('LINESTRING(0 0, 8 8, 0 16)',0));
+SELECT '4', ST_astext(ST_ChaikinSmoothing('LINESTRING(0 0, 8 8, 0 16)',2));
+SELECT '5', ST_astext(ST_ChaikinSmoothing('POINT(0 0)'));
+SELECT '6', ST_astext(ST_ChaikinSmoothing('GEOMETRYCOLLECTION(POINT(1 1), LINESTRING(1 1, 1 3, 1 5), POLYGON((5 5, 5 10, 10 10, 10 5, 5 5), (6 6, 6 7, 7 7, 7 6, 6 6 )))', 2, 't'));
Added: trunk/regress/chaikin_expected
===================================================================
--- trunk/regress/chaikin_expected (rev 0)
+++ trunk/regress/chaikin_expected 2018-03-20 13:11:17 UTC (rev 16479)
@@ -0,0 +1,6 @@
+1|LINESTRING(0 0,6 6,6 10,0 16)
+ERROR: Not more than 5 iterations please
+ERROR: Number of iterations must be between 1 and 5
+4|LINESTRING(0 0,4.5 4.5,6 7,6 9,4.5 11.5,0 16)
+5|POINT(0 0)
+6|GEOMETRYCOLLECTION(POINT(1 1),LINESTRING(1 1,1 2.125,1 2.75,1 3.25,1 3.875,1 5),POLYGON((5 5,5 7.8125,5.3125 9.0625,5.9375 9.6875,6.875 10,8.125 10,9.0625 9.6875,9.6875 9.0625,10 8.125,10 6.875,9.6875 5.9375,9.0625 5.3125,7.8125 5,5 5),(6 6,6 6.5625,6.0625 6.8125,6.1875 6.9375,6.375 7,6.625 7,6.8125 6.9375,6.9375 6.8125,7 6.625,7 6.375,6.9375 6.1875,6.8125 6.0625,6.5625 6,6 6)))
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