From b1549d873a5b69ba7c6b4124c51488e0b9f2d412 Mon Sep 17 00:00:00 2001 From: Nail Sharipov Date: Tue, 24 Sep 2024 11:13:01 +0300 Subject: [PATCH] embedded star --- docs-gen/src/embedded/star.html | 171 ++++++++++++++++++ .../src/overlay/extract/define_contour.svg | 14 +- docs-gen/src/overlay/extract/define_shape.svg | 12 +- .../overlay/extract/extract_outer_contour.svg | 15 +- .../src/overlay/extract/matching_contours.svg | 12 +- .../overlay/extract/segment_under_point.svg | 10 +- .../extract/segment_under_segment_a.svg | 10 +- .../extract/segment_under_segment_ab.svg | 18 +- .../extract/segment_under_segment_b.svg | 10 +- .../overlay/overlay_graph/difference_ba.svg | 12 +- .../src/overlay/overlay_graph/example.svg | 16 +- .../src/overlay/overlay_graph/exclusion.svg | 12 +- .../overlay/overlay_graph/intersection.svg | 12 +- .../overlay/overlay_graph/overlay_graph.md | 6 +- docs-gen/src/overlay/overlay_graph/union.svg | 10 +- docs/embedded/star.html | 171 ++++++++++++++++++ docs/overlay/extract/define_contour.svg | 14 +- docs/overlay/extract/define_shape.svg | 12 +- .../overlay/extract/extract_outer_contour.svg | 15 +- docs/overlay/extract/matching_contours.svg | 12 +- docs/overlay/extract/segment_under_point.svg | 10 +- .../extract/segment_under_segment_a.svg | 10 +- .../extract/segment_under_segment_ab.svg | 18 +- .../extract/segment_under_segment_b.svg | 10 +- docs/overlay/overlay_graph/difference_ba.svg | 12 +- docs/overlay/overlay_graph/example.svg | 16 +- docs/overlay/overlay_graph/exclusion.svg | 12 +- docs/overlay/overlay_graph/intersection.svg | 12 +- docs/overlay/overlay_graph/overlay_graph.html | 18 +- docs/overlay/overlay_graph/union.svg | 10 +- docs/print.html | 18 +- docs/searchindex.js | 2 +- docs/searchindex.json | 2 +- 33 files changed, 535 insertions(+), 179 deletions(-) create mode 100644 docs-gen/src/embedded/star.html create mode 100644 docs/embedded/star.html diff --git a/docs-gen/src/embedded/star.html b/docs-gen/src/embedded/star.html new file mode 100644 index 0000000..f6fd63e --- /dev/null +++ b/docs-gen/src/embedded/star.html @@ -0,0 +1,171 @@ + + + + + Star + + + + +
+ +
+
+

Subject Star

+
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+

Clip Star

+
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+ + diff --git a/docs-gen/src/overlay/extract/define_contour.svg b/docs-gen/src/overlay/extract/define_contour.svg index 39d4ed1..e358ba6 100644 --- a/docs-gen/src/overlay/extract/define_contour.svg +++ b/docs-gen/src/overlay/extract/define_contour.svg @@ -10,8 +10,8 @@ inkscape:version="1.3.2 (091e20e, 2023-11-25)" sodipodi:docname="define_contour.svg" inkscape:export-filename="extract_contour.png" - inkscape:export-xdpi="96" - inkscape:export-ydpi="96" + inkscape:export-xdpi="192" + inkscape:export-ydpi="192" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd" xmlns="http://www.w3.org/2000/svg" @@ -27,13 +27,13 @@ inkscape:deskcolor="#d1d1d1" inkscape:document-units="px" showgrid="true" - inkscape:zoom="0.82174391" - inkscape:cx="555.52587" - inkscape:cy="279.28409" + inkscape:zoom="0.98330828" + inkscape:cx="700.18733" + inkscape:cy="363.56858" inkscape:window-width="1872" inkscape:window-height="1212" - inkscape:window-x="634" - inkscape:window-y="181" + inkscape:window-x="98" + inkscape:window-y="411" inkscape:window-maximized="0" inkscape:current-layer="layer1" showguides="false"> diff --git a/docs-gen/src/overlay/extract/define_shape.svg b/docs-gen/src/overlay/extract/define_shape.svg index b538934..a855c92 100644 --- a/docs-gen/src/overlay/extract/define_shape.svg +++ b/docs-gen/src/overlay/extract/define_shape.svg @@ -10,8 +10,8 @@ inkscape:version="1.3.2 (091e20e, 2023-11-25)" sodipodi:docname="define_shape.svg" inkscape:export-filename="define_shape.png" - inkscape:export-xdpi="96" - inkscape:export-ydpi="96" + inkscape:export-xdpi="192" + inkscape:export-ydpi="192" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd" xmlns="http://www.w3.org/2000/svg" @@ -27,13 +27,13 @@ inkscape:deskcolor="#d1d1d1" inkscape:document-units="px" showgrid="true" - inkscape:zoom="0.81786064" - inkscape:cx="88.645909" - inkscape:cy="319.12527" + inkscape:zoom="2.6516575" + inkscape:cx="88.058128" + inkscape:cy="319.04572" inkscape:window-width="2008" inkscape:window-height="1023" inkscape:window-x="221" - inkscape:window-y="225" + inkscape:window-y="197" inkscape:window-maximized="0" inkscape:current-layer="layer1"> + diff --git a/docs-gen/src/overlay/overlay_graph/example.svg b/docs-gen/src/overlay/overlay_graph/example.svg index 7e4e2da..7d65579 100644 --- a/docs-gen/src/overlay/overlay_graph/example.svg +++ b/docs-gen/src/overlay/overlay_graph/example.svg @@ -9,9 +9,9 @@ id="svg5" inkscape:version="1.3.2 (091e20e, 2023-11-25)" sodipodi:docname="example.svg" - inkscape:export-filename="A_subtract_B.png" - inkscape:export-xdpi="96" - inkscape:export-ydpi="96" + inkscape:export-filename="A_B_overlay.png" + inkscape:export-xdpi="192" + inkscape:export-ydpi="192" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd" xmlns="http://www.w3.org/2000/svg" @@ -27,13 +27,13 @@ inkscape:deskcolor="#d1d1d1" inkscape:document-units="px" showgrid="true" - inkscape:zoom="0.63511096" - inkscape:cx="1222.621" - inkscape:cy="876.22484" + inkscape:zoom="1.0000609" + inkscape:cx="1082.4341" + inkscape:cy="791.45184" inkscape:window-width="1872" inkscape:window-height="1212" - inkscape:window-x="571" - inkscape:window-y="172" + inkscape:window-x="98" + inkscape:window-y="200" inkscape:window-maximized="0" inkscape:current-layer="layer1" showguides="false"> diff --git a/docs-gen/src/overlay/overlay_graph/exclusion.svg b/docs-gen/src/overlay/overlay_graph/exclusion.svg index 943067b..f754cc6 100644 --- a/docs-gen/src/overlay/overlay_graph/exclusion.svg +++ b/docs-gen/src/overlay/overlay_graph/exclusion.svg @@ -9,7 +9,7 @@ id="svg5" inkscape:version="1.3.2 (091e20e, 2023-11-25)" sodipodi:docname="exclusion.svg" - inkscape:export-filename="A_subtract_B.png" + inkscape:export-filename="A_exclusion_B.png" inkscape:export-xdpi="96" inkscape:export-ydpi="96" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" @@ -27,13 +27,13 @@ inkscape:deskcolor="#d1d1d1" inkscape:document-units="px" showgrid="true" - inkscape:zoom="1.0227509" - inkscape:cx="561.72037" - inkscape:cy="304.57075" + inkscape:zoom="4.071028" + inkscape:cx="552.80878" + inkscape:cy="780.02411" inkscape:window-width="1872" inkscape:window-height="1212" - inkscape:window-x="655" - inkscape:window-y="76" + inkscape:window-x="136" + inkscape:window-y="611" inkscape:window-maximized="0" inkscape:current-layer="layer1" showguides="false"> diff --git a/docs-gen/src/overlay/overlay_graph/intersection.svg b/docs-gen/src/overlay/overlay_graph/intersection.svg index bc65ddb..f8a4553 100644 --- a/docs-gen/src/overlay/overlay_graph/intersection.svg +++ b/docs-gen/src/overlay/overlay_graph/intersection.svg @@ -9,7 +9,7 @@ id="svg5" inkscape:version="1.3.2 (091e20e, 2023-11-25)" sodipodi:docname="intersection.svg" - inkscape:export-filename="A_subtract_B.png" + inkscape:export-filename="A_intersection_B.png" inkscape:export-xdpi="96" inkscape:export-ydpi="96" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" @@ -27,13 +27,13 @@ inkscape:deskcolor="#d1d1d1" inkscape:document-units="px" showgrid="true" - inkscape:zoom="1.4208567" - inkscape:cx="597.17493" - inkscape:cy="341.69526" + inkscape:zoom="0.78873544" + inkscape:cx="771.48809" + inkscape:cy="500.16771" inkscape:window-width="1872" inkscape:window-height="1212" - inkscape:window-x="392" - inkscape:window-y="186" + inkscape:window-x="128" + inkscape:window-y="539" inkscape:window-maximized="0" inkscape:current-layer="layer1" showguides="false"> diff --git a/docs-gen/src/overlay/overlay_graph/overlay_graph.md b/docs-gen/src/overlay/overlay_graph/overlay_graph.md index 4778043..59b1655 100644 --- a/docs-gen/src/overlay/overlay_graph/overlay_graph.md +++ b/docs-gen/src/overlay/overlay_graph/overlay_graph.md @@ -24,16 +24,16 @@ The side associated solely with body **B** ![Example](difference_ba.svg) ### Union, C = A or B -![Example](union.svg) The resulting segments of **C** must belong to either body **A** or body **B**, or to both. The opposite side of each segment must not belong to anybody. The side associated with one of the bodies will represent the inner part of the resulting shape. +![Example](union.svg) ### Intersection, C = A and B -![Example](intersection.svg) The resulting segments of **C** must belong to both bodies **A** and **B**. The opposite side of each segment must not belong to both bodies simultaneously. The side associated with both bodies **A** and **B** will represent the inner part of the resulting shape. +![Example](intersection.svg) ### Exclusion, C = A xor B -![Example](exclusion.svg) The resulting segments of **C** must belong to either body **A** or body **B**, but not to both simultaneously. The opposite side of each segment must either belong to both bodies or to neither. The side associated with one of the bodies (**A** or **B**) will represent the inner part of the resulting shape. +![Example](exclusion.svg) \ No newline at end of file diff --git a/docs-gen/src/overlay/overlay_graph/union.svg b/docs-gen/src/overlay/overlay_graph/union.svg index 81c479f..3ed3bdf 100644 --- a/docs-gen/src/overlay/overlay_graph/union.svg +++ b/docs-gen/src/overlay/overlay_graph/union.svg @@ -9,7 +9,7 @@ id="svg5" inkscape:version="1.3.2 (091e20e, 2023-11-25)" sodipodi:docname="union.svg" - inkscape:export-filename="A_subtract_B.png" + inkscape:export-filename="A_union_B.png" inkscape:export-xdpi="96" inkscape:export-ydpi="96" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" @@ -27,13 +27,13 @@ inkscape:deskcolor="#d1d1d1" inkscape:document-units="px" showgrid="true" - inkscape:zoom="1.0442051" - inkscape:cx="554.96762" - inkscape:cy="258.09107" + inkscape:zoom="4.1564258" + inkscape:cx="555.40508" + inkscape:cy="258.51538" inkscape:window-width="1872" inkscape:window-height="1212" inkscape:window-x="656" - inkscape:window-y="76" + inkscape:window-y="48" inkscape:window-maximized="0" inkscape:current-layer="layer1" showguides="false"> diff --git a/docs/embedded/star.html b/docs/embedded/star.html new file mode 100644 index 0000000..f6fd63e --- /dev/null +++ b/docs/embedded/star.html @@ -0,0 +1,171 @@ + + + + + Star + + + + +
+ +
+
+

Subject Star

+
+ + +
+
+ + +
+
+ + +
+
+ + +
+
+
+

Clip Star

+
+ + +
+
+ + +
+
+ + +
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+ + +
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+ +
+ + diff --git a/docs/overlay/extract/define_contour.svg b/docs/overlay/extract/define_contour.svg index 39d4ed1..e358ba6 100644 --- a/docs/overlay/extract/define_contour.svg +++ b/docs/overlay/extract/define_contour.svg @@ -10,8 +10,8 @@ inkscape:version="1.3.2 (091e20e, 2023-11-25)" sodipodi:docname="define_contour.svg" inkscape:export-filename="extract_contour.png" - inkscape:export-xdpi="96" - inkscape:export-ydpi="96" + inkscape:export-xdpi="192" + inkscape:export-ydpi="192" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd" xmlns="http://www.w3.org/2000/svg" @@ -27,13 +27,13 @@ inkscape:deskcolor="#d1d1d1" inkscape:document-units="px" showgrid="true" - inkscape:zoom="0.82174391" - inkscape:cx="555.52587" - inkscape:cy="279.28409" + inkscape:zoom="0.98330828" + inkscape:cx="700.18733" + inkscape:cy="363.56858" inkscape:window-width="1872" inkscape:window-height="1212" - inkscape:window-x="634" - inkscape:window-y="181" + inkscape:window-x="98" + inkscape:window-y="411" inkscape:window-maximized="0" inkscape:current-layer="layer1" showguides="false"> diff --git a/docs/overlay/extract/define_shape.svg b/docs/overlay/extract/define_shape.svg index b538934..a855c92 100644 --- a/docs/overlay/extract/define_shape.svg +++ b/docs/overlay/extract/define_shape.svg @@ -10,8 +10,8 @@ inkscape:version="1.3.2 (091e20e, 2023-11-25)" sodipodi:docname="define_shape.svg" inkscape:export-filename="define_shape.png" - inkscape:export-xdpi="96" - inkscape:export-ydpi="96" + inkscape:export-xdpi="192" + inkscape:export-ydpi="192" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd" xmlns="http://www.w3.org/2000/svg" @@ -27,13 +27,13 @@ inkscape:deskcolor="#d1d1d1" inkscape:document-units="px" showgrid="true" - inkscape:zoom="0.81786064" - inkscape:cx="88.645909" - inkscape:cy="319.12527" + inkscape:zoom="2.6516575" + inkscape:cx="88.058128" + inkscape:cy="319.04572" inkscape:window-width="2008" inkscape:window-height="1023" inkscape:window-x="221" - inkscape:window-y="225" + inkscape:window-y="197" inkscape:window-maximized="0" inkscape:current-layer="layer1"> + diff --git a/docs/overlay/overlay_graph/example.svg b/docs/overlay/overlay_graph/example.svg index 7e4e2da..7d65579 100644 --- a/docs/overlay/overlay_graph/example.svg +++ b/docs/overlay/overlay_graph/example.svg @@ -9,9 +9,9 @@ id="svg5" inkscape:version="1.3.2 (091e20e, 2023-11-25)" sodipodi:docname="example.svg" - inkscape:export-filename="A_subtract_B.png" - inkscape:export-xdpi="96" - inkscape:export-ydpi="96" + inkscape:export-filename="A_B_overlay.png" + inkscape:export-xdpi="192" + inkscape:export-ydpi="192" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd" xmlns="http://www.w3.org/2000/svg" @@ -27,13 +27,13 @@ inkscape:deskcolor="#d1d1d1" inkscape:document-units="px" showgrid="true" - inkscape:zoom="0.63511096" - inkscape:cx="1222.621" - inkscape:cy="876.22484" + inkscape:zoom="1.0000609" + inkscape:cx="1082.4341" + inkscape:cy="791.45184" inkscape:window-width="1872" inkscape:window-height="1212" - inkscape:window-x="571" - inkscape:window-y="172" + inkscape:window-x="98" + inkscape:window-y="200" inkscape:window-maximized="0" inkscape:current-layer="layer1" showguides="false"> diff --git a/docs/overlay/overlay_graph/exclusion.svg b/docs/overlay/overlay_graph/exclusion.svg index 943067b..f754cc6 100644 --- a/docs/overlay/overlay_graph/exclusion.svg +++ b/docs/overlay/overlay_graph/exclusion.svg @@ -9,7 +9,7 @@ id="svg5" inkscape:version="1.3.2 (091e20e, 2023-11-25)" sodipodi:docname="exclusion.svg" - inkscape:export-filename="A_subtract_B.png" + inkscape:export-filename="A_exclusion_B.png" inkscape:export-xdpi="96" inkscape:export-ydpi="96" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" @@ -27,13 +27,13 @@ inkscape:deskcolor="#d1d1d1" inkscape:document-units="px" showgrid="true" - inkscape:zoom="1.0227509" - inkscape:cx="561.72037" - inkscape:cy="304.57075" + inkscape:zoom="4.071028" + inkscape:cx="552.80878" + inkscape:cy="780.02411" inkscape:window-width="1872" inkscape:window-height="1212" - inkscape:window-x="655" - inkscape:window-y="76" + inkscape:window-x="136" + inkscape:window-y="611" inkscape:window-maximized="0" inkscape:current-layer="layer1" showguides="false"> diff --git a/docs/overlay/overlay_graph/intersection.svg b/docs/overlay/overlay_graph/intersection.svg index bc65ddb..f8a4553 100644 --- a/docs/overlay/overlay_graph/intersection.svg +++ b/docs/overlay/overlay_graph/intersection.svg @@ -9,7 +9,7 @@ id="svg5" inkscape:version="1.3.2 (091e20e, 2023-11-25)" sodipodi:docname="intersection.svg" - inkscape:export-filename="A_subtract_B.png" + inkscape:export-filename="A_intersection_B.png" inkscape:export-xdpi="96" inkscape:export-ydpi="96" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" @@ -27,13 +27,13 @@ inkscape:deskcolor="#d1d1d1" inkscape:document-units="px" showgrid="true" - inkscape:zoom="1.4208567" - inkscape:cx="597.17493" - inkscape:cy="341.69526" + inkscape:zoom="0.78873544" + inkscape:cx="771.48809" + inkscape:cy="500.16771" inkscape:window-width="1872" inkscape:window-height="1212" - inkscape:window-x="392" - inkscape:window-y="186" + inkscape:window-x="128" + inkscape:window-y="539" inkscape:window-maximized="0" inkscape:current-layer="layer1" showguides="false"> diff --git a/docs/overlay/overlay_graph/overlay_graph.html b/docs/overlay/overlay_graph/overlay_graph.html index 4f5e7a4..07b51eb 100644 --- a/docs/overlay/overlay_graph/overlay_graph.html +++ b/docs/overlay/overlay_graph/overlay_graph.html @@ -188,17 +188,17 @@

Diffe The side associated solely with body B will represent the inner part of the resulting shape. Example

Union, C = A or B

-

Example -The resulting segments of C must belong to either body A or body B, or to both. The opposite side of each segment must not belong to anybody.
-The side associated with one of the bodies will represent the inner part of the resulting shape.

+

The resulting segments of C must belong to either body A or body B, or to both. The opposite side of each segment must not belong to anybody.
+The side associated with one of the bodies will represent the inner part of the resulting shape. +Example

Intersection, C = A and B

-

Example -The resulting segments of C must belong to both bodies A and B. The opposite side of each segment must not belong to both bodies simultaneously.
-The side associated with both bodies A and B will represent the inner part of the resulting shape.

+

The resulting segments of C must belong to both bodies A and B. The opposite side of each segment must not belong to both bodies simultaneously.
+The side associated with both bodies A and B will represent the inner part of the resulting shape. +Example

Exclusion, C = A xor B

-

Example -The resulting segments of C must belong to either body A or body B, but not to both simultaneously. The opposite side of each segment must either belong to both bodies or to neither.
-The side associated with one of the bodies (A or B) will represent the inner part of the resulting shape.

+

The resulting segments of C must belong to either body A or body B, but not to both simultaneously. The opposite side of each segment must either belong to both bodies or to neither.
+The side associated with one of the bodies (A or B) will represent the inner part of the resulting shape. +Example

diff --git a/docs/overlay/overlay_graph/union.svg b/docs/overlay/overlay_graph/union.svg index 81c479f..3ed3bdf 100644 --- a/docs/overlay/overlay_graph/union.svg +++ b/docs/overlay/overlay_graph/union.svg @@ -9,7 +9,7 @@ id="svg5" inkscape:version="1.3.2 (091e20e, 2023-11-25)" sodipodi:docname="union.svg" - inkscape:export-filename="A_subtract_B.png" + inkscape:export-filename="A_union_B.png" inkscape:export-xdpi="96" inkscape:export-ydpi="96" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" @@ -27,13 +27,13 @@ inkscape:deskcolor="#d1d1d1" inkscape:document-units="px" showgrid="true" - inkscape:zoom="1.0442051" - inkscape:cx="554.96762" - inkscape:cy="258.09107" + inkscape:zoom="4.1564258" + inkscape:cx="555.40508" + inkscape:cy="258.51538" inkscape:window-width="1872" inkscape:window-height="1212" inkscape:window-x="656" - inkscape:window-y="76" + inkscape:window-y="48" inkscape:window-maximized="0" inkscape:current-layer="layer1" showguides="false"> diff --git a/docs/print.html b/docs/print.html index b5b5c87..f8af7eb 100644 --- a/docs/print.html +++ b/docs/print.html @@ -1014,17 +1014,17 @@

Diffe The side associated solely with body B will represent the inner part of the resulting shape. Example

Union, C = A or B

-

Example -The resulting segments of C must belong to either body A or body B, or to both. The opposite side of each segment must not belong to anybody.
-The side associated with one of the bodies will represent the inner part of the resulting shape.

+

The resulting segments of C must belong to either body A or body B, or to both. The opposite side of each segment must not belong to anybody.
+The side associated with one of the bodies will represent the inner part of the resulting shape. +Example

Intersection, C = A and B

-

Example -The resulting segments of C must belong to both bodies A and B. The opposite side of each segment must not belong to both bodies simultaneously.
-The side associated with both bodies A and B will represent the inner part of the resulting shape.

+

The resulting segments of C must belong to both bodies A and B. The opposite side of each segment must not belong to both bodies simultaneously.
+The side associated with both bodies A and B will represent the inner part of the resulting shape. +Example

Exclusion, C = A xor B

-

Example -The resulting segments of C must belong to either body A or body B, but not to both simultaneously. The opposite side of each segment must either belong to both bodies or to neither.
-The side associated with one of the bodies (A or B) will represent the inner part of the resulting shape.

+

The resulting segments of C must belong to either body A or body B, but not to both simultaneously. The opposite side of each segment must either belong to both bodies or to neither.
+The side associated with one of the bodies (A or B) will represent the inner part of the resulting shape. +Example

Extract Shapes

Once we apply boolean filter to Overlay Graph, we can begin extract contours.

Build Contour

diff --git a/docs/searchindex.js b/docs/searchindex.js index ced5252..77e84dc 100644 --- a/docs/searchindex.js +++ b/docs/searchindex.js @@ -1 +1 @@ -Object.assign(window.search, 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» iOverlay","id":"0","title":"iOverlay"},"1":{"body":"The iOverlay is a poly-bool library that supports main operations such as union, intersection, difference, xor, and self-intersection by the even-odd rule.","breadcrumbs":"iOverlay » Introduction","id":"1","title":"Introduction"},"10":{"body":"Not Overlap Test Squares iOverlay(Swift) iOverlay(Rust) MT off iOverlay(Rust) MT on Clipper2(C++) 5 0.000009 0.000004 0.000004 0.000005 25 0.000041 0.000016 0.000016 0.000021 113 0.000204 0.000077 0.000076 0.000097 481 0.001052 0.000400 0.000406 0.000457 1985 0.004978 0.001965 0.001959 0.002114 8065 0.021336 0.009956 0.009912 0.010783 32513 0.089523 0.045412 0.044751 0.056281 130561 0.375594 0.194658 0.189546 0.369146 523265 1.663087 0.885775 0.860105 2.695334 2095105 6.947932 3.759691 3.627623 20.665812 8384513 28.777569 16.449567 15.293312 167.966801","breadcrumbs":"iOverlay » Performance Comparison » Not Overlap Test","id":"10","title":"Not Overlap Test"},"11":{"body":"Lines Net Test Lines iOverlay(Swift) iOverlay(Rust) MT off iOverlay(Rust) MT on Clipper2(C++) 4 0.000014 0.000006 0.000006 0.000004 8 0.000049 0.000018 0.000018 0.000012 16 0.000195 0.000063 0.000064 0.000043 32 0.001295 0.000258 0.000259 0.000176 64 0.004994 0.001267 0.001258 0.000749 128 0.021239 0.005021 0.005830 0.003441 256 0.091427 0.025601 0.026613 0.018417 512 0.255989 0.131983 0.123238 0.115229 1024 1.146842 0.538035 0.557338 0.759640 2048 4.808548 2.592261 2.306600 5.595165 4096 20.190101 10.937208 9.588313 45.934461","breadcrumbs":"iOverlay » Performance Comparison » Lines Net Test","id":"11","title":"Lines Net Test"},"12":{"body":"Spiral Test Links iOverlay(Swift) iOverlay(Rust) MT off iOverlay(Rust) MT on Clipper2(C++) 2 0.000006 0.000003 0.000003 0.000002 4 0.000010 0.000005 0.000005 0.000004 8 0.000019 0.000010 0.000010 0.000007 16 0.000040 0.000020 0.000020 0.000014 32 0.000095 0.000050 0.000049 0.000031 64 0.000218 0.000133 0.000133 0.000083 128 0.000490 0.000305 0.000305 0.000202 256 0.001105 0.000699 0.000702 0.000476 512 0.003385 0.001723 0.001673 0.001195 1024 0.006174 0.003801 0.003693 0.002941 2048 0.013200 0.008436 0.008434 0.007578 4096 0.026065 0.016422 0.016581 0.020287 8192 0.055944 0.034739 0.034528 0.054647 16384 0.112375 0.068730 0.068337 0.181050 32768 0.237412 0.146693 0.144771 0.606854 65536 0.473847 0.297990 0.295959 2.013809 131072 0.981729 0.632649 0.631458 6.547658 262144 1.979541 1.292599 1.299418 21.171540 524288 4.101912 2.703499 2.704447 72.147615 1048576 8.254108 5.407953 5.457118 259.866180","breadcrumbs":"iOverlay » Performance Comparison » Spiral Test","id":"12","title":"Spiral Test"},"13":{"body":"Windows Test Squares iOverlay(Swift) iOverlay(Rust) MT off iOverlay(Rust) MT on Clipper2(C++) 8 0.000016 0.000006 0.000006 0.000008 32 0.000062 0.000023 0.000023 0.000028 128 0.000276 0.000104 0.000105 0.000112 512 0.001306 0.000544 0.000545 0.000507 2048 0.005543 0.002436 0.002413 0.002454 8192 0.023805 0.011186 0.011143 0.012364 32768 0.104822 0.054641 0.053747 0.076850 131072 0.445712 0.243877 0.232455 0.568316 524288 1.862371 0.995490 0.978971 4.142673 2097152 7.657815 4.160511 4.089460 33.165570 8388608 30.833973 17.234603 16.098006 265.387333","breadcrumbs":"iOverlay » Performance Comparison » Windows Test","id":"13","title":"Windows Test"},"14":{"body":"Nested_Squares Test Squares iOverlay(Swift) iOverlay(Rust) MT off iOverlay(Rust) MT on Clipper2(C++) 4 0.000022 0.000010 0.000010 0.000012 8 0.000045 0.000019 0.000019 0.000023 16 0.000098 0.000039 0.000038 0.000050 32 0.000238 0.000084 0.000083 0.000118 64 0.000624 0.000216 0.000210 0.000291 128 0.001853 0.000589 0.000593 0.000806 256 0.002398 0.001964 0.001883 0.003415 512 0.005013 0.002736 0.002724 0.015989 1024 0.011067 0.005108 0.005558 0.081267 2048 0.023194 0.011339 0.010744 0.461883 4096 0.050601 0.026789 0.026263 2.347209 8192 0.111691 0.052864 0.053851 10.612424 16384 0.252357 0.134364 0.137991 46.205474 32768 0.581991 0.292781 0.287206 251.260857 65536 1.368794 0.781251 0.767102 3502.233611 131072 3.250746 1.688798 1.640510 ---- 262144 8.529555 4.809529 4.726510 ---- 524288 21.177892 10.273028 10.513923 ----","breadcrumbs":"iOverlay » Performance Comparison » Nested Squares Test","id":"14","title":"Nested Squares Test"},"15":{"body":"Stars Rotation Subject Star First Radius: Second Radius: Rotation Speed: Corners Count: Clip Star First Radius: Second Radius: Rotation Speed: Corners Count: Operation: Xor Union Intersect Difference InverseDifference Subject Clip","breadcrumbs":"iOverlay » Documentation","id":"15","title":"iOverlay"},"16":{"body":"","breadcrumbs":"iOverlay » Documentation » Filling Rules » Filling Rules","id":"16","title":"Filling Rules"},"17":{"body":"Even-Odd","breadcrumbs":"iOverlay » Documentation » Filling Rules » Even-Odd","id":"17","title":"Even-Odd"},"18":{"body":"Non-Zero","breadcrumbs":"iOverlay » Documentation » Filling Rules » Non-Zero","id":"18","title":"Non-Zero"},"19":{"body":"Even-Odd : Only odd numbered sub-regions are filled Non-Zero : Only non-zero sub-regions are filled","breadcrumbs":"iOverlay » Documentation » Filling Rules » Filling Rules:","id":"19","title":"Filling Rules:"},"2":{"body":"Operations : union, intersection, difference, and exclusion. Polygons : with holes, self-intersections, and multiple paths. Simplification : removes degenerate vertices and merges collinear edges. Fill Rules : even-odd and non-zero.","breadcrumbs":"iOverlay » Features","id":"2","title":"Features"},"20":{"body":"","breadcrumbs":"iOverlay » Documentation » Overlay Rules » Overlay Rules","id":"20","title":"Overlay Rules"},"21":{"body":"","breadcrumbs":"iOverlay » Documentation » Overlay Rules » Union, A or B","id":"21","title":"Union, A or B"},"22":{"body":"","breadcrumbs":"iOverlay » Documentation » Overlay Rules » Intersection, A and B","id":"22","title":"Intersection, A and B"},"23":{"body":"","breadcrumbs":"iOverlay » Documentation » Overlay Rules » Difference, A - B","id":"23","title":"Difference, A - B"},"24":{"body":"","breadcrumbs":"iOverlay » Documentation » Overlay Rules » Inverse Difference, B - A","id":"24","title":"Inverse Difference, B - A"},"25":{"body":"","breadcrumbs":"iOverlay » Documentation » Overlay Rules » Exclusion, A xor B","id":"25","title":"Exclusion, A xor B"},"26":{"body":"","breadcrumbs":"iOverlay » Documentation » Contours » Contours","id":"26","title":"Contours"},"27":{"body":"Contour In the context of the Overlay Graph , contours are used to represent the boundaries of geometric objects. These contours are classified into two types: outer contours and inner contours.","breadcrumbs":"iOverlay » Documentation » Contours » Outer and Inner","id":"27","title":"Outer and Inner"},"28":{"body":"An outer contour is a sequence of points ordered in a clockwise direction. The outer contour defines the external boundary of a shape, enclosing the exterior space.","breadcrumbs":"iOverlay » Documentation » Contours » Outer Contour:","id":"28","title":"Outer Contour:"},"29":{"body":"An inner contour is a sequence of points ordered in a counterclockwise direction. Inner contours represent enclosed areas within an outer contour, often referred to as \"holes\" or \"caves.\" Both outer and inner contours must be simple, meaning they must not self-intersect and must not share edges with other contours.","breadcrumbs":"iOverlay » Documentation » Contours » Inner Contour (Hole or Cave):","id":"29","title":"Inner Contour (Hole or Cave):"},"3":{"body":"Swift Version: iShape-Swift/iOverlay Rust Version: iShape-Rust/iOverlay JS Version: iShape-Rust/iShape-js","breadcrumbs":"iOverlay » Source Code","id":"3","title":"Source Code"},"30":{"body":"Example An Overlay Graph is a data structure representing the intersections and overlays of two geometric objects ( A and B ) defined by closed contours in 2D space. The graph is constructed by dividing all the segments of the object contours into non-intersecting parts, where segments can only touch at their endpoints. Each segment in the graph contains the following properties: For each side of the segment, it stores information about its membership to object A and object B Segments do not intersect each other, but they may touch at their endpoints. for more Overlay Graph examples see Shape Editor","breadcrumbs":"iOverlay » Documentation » Overlay Graph » Overlay Graph","id":"30","title":"Overlay Graph"},"31":{"body":"","breadcrumbs":"iOverlay » Documentation » Overlay Graph » Filter Segments","id":"31","title":"Filter Segments"},"32":{"body":"The resulting segments of C must not be inside body B and must belong to body A on one side. The side associated solely with body A will represent the inner part of the resulting shape. Example","breadcrumbs":"iOverlay » Documentation » Overlay Graph » Difference, C = A - B","id":"32","title":"Difference, C = A - B"},"33":{"body":"The resulting segments of C must not be inside body A and must belong to body B on one side. The side associated solely with body B will represent the inner part of the resulting shape. Example","breadcrumbs":"iOverlay » Documentation » Overlay Graph » Difference, C = B - A","id":"33","title":"Difference, C = B - A"},"34":{"body":"Example The resulting segments of C must belong to either body A or body B , or to both. The opposite side of each segment must not belong to anybody. The side associated with one of the bodies will represent the inner part of the resulting shape.","breadcrumbs":"iOverlay » Documentation » Overlay Graph » Union, C = A or B","id":"34","title":"Union, C = A or B"},"35":{"body":"Example The resulting segments of C must belong to both bodies A and B . The opposite side of each segment must not belong to both bodies simultaneously. The side associated with both bodies A and B will represent the inner part of the resulting shape.","breadcrumbs":"iOverlay » Documentation » Overlay Graph » Intersection, C = A and B","id":"35","title":"Intersection, C = A and B"},"36":{"body":"Example The resulting segments of C must belong to either body A or body B , but not to both simultaneously. The opposite side of each segment must either belong to both bodies or to neither. The side associated with one of the bodies ( A or B ) will represent the inner part of the resulting shape.","breadcrumbs":"iOverlay » Documentation » Overlay Graph » Exclusion, C = A xor B","id":"36","title":"Exclusion, C = A xor B"},"37":{"body":"Once we apply boolean filter to Overlay Graph , we can begin extract contours.","breadcrumbs":"iOverlay » Documentation » Extract Shapes » Extract Shapes","id":"37","title":"Extract Shapes"},"38":{"body":"","breadcrumbs":"iOverlay » Documentation » Extract Shapes » Build Contour","id":"38","title":"Build Contour"},"39":{"body":"Extract Contour","breadcrumbs":"iOverlay » Documentation » Extract Shapes » Outer Contour","id":"39","title":"Outer Contour"},"4":{"body":"Stars Rotation Subject Star First Radius: Second Radius: Rotation Speed: Corners Count: Clip Star First Radius: Second Radius: Rotation Speed: Corners Count: Operation: Xor Union Intersect Difference InverseDifference Subject Clip","breadcrumbs":"iOverlay » Demo","id":"4","title":"iOverlay"},"40":{"body":"Extract Contour The algorithm starts by selecting the leftmost node and proceeds by choosing the topmost segment connected to that node. The process continues by traversing to the next node along the selected segment. At each node, the algorithm selects the next segment by rotating around the current node in a counterclockwise direction and taking the first nearest segment. To prevent segments from being visited twice, each segment is marked as visited upon traversal. This process continues until the contour is complete, forming either an outer or inner contour. By following this approach, outer contours are extracted in a clockwise direction, while inner contours are extracted in a counterclockwise direction.","breadcrumbs":"iOverlay » Documentation » Extract Shapes » Inner Contour","id":"40","title":"Inner Contour"},"41":{"body":"Define Contour To define a contour, the algorithm begins by identifying the leftmost and topmost segment in the contour. The classification of the contour is determined as follows: If the left-top side of the segment is classified as the outer side, then the contour is an outer contour. If the left-top side of the segment is classified as the inner side, then the contour is an inner contour. This method ensures each contour is correctly classified based on its orientation in 2D space.","breadcrumbs":"iOverlay » Documentation » Extract Shapes » Define Contour","id":"41","title":"Define Contour"},"42":{"body":"Define Shape A shape is defined as a group of contours, where the first contour is always an outer contour, and the subsequent contours (if any) are inner contours.","breadcrumbs":"iOverlay » Documentation » Extract Shapes » Define Shape","id":"42","title":"Define Shape"},"43":{"body":"Matching Contours To match inner contours to their corresponding outer contour, a line is drawn from any point on the inner contour. The first segment encountered from an outer contour beneath this point belongs to the outer contour that contains the inner contour. This outer contour serves as the container for the inner contour.","breadcrumbs":"iOverlay » Documentation » Extract Shapes » Matching Contours","id":"43","title":"Matching Contours"},"44":{"body":"","breadcrumbs":"iOverlay » Documentation » Extract Shapes » Define Segment under Point","id":"44","title":"Define Segment under Point"},"45":{"body":"Segment Under Point To determine whether a segment AB is below a point P , one may be tempted to compute the value of ym at the point of intersection M , where a vertical line is dropped from P onto AB (i.e., xp = xm): $$ y_{m} = \\frac{y_{a} - y_{b}}{x_{a} - x_{b}}\\cdot(x_{m} - x_{a}) + y_{a} $$ However, this approach can introduce precision issues due to the division involved. A more reliable method involves using the order of traversal around the vertices of the triangle APB . If segment AB is below point P , the vertices A , P , and B will appear in a clockwise order. This method uses the cross product of vectors PA and PB : $$ a \\times b = a_x b_y - a_y b_x $$ Since this method avoids division, it eliminates precision issues, making it stable for determining whether a segment is below a point.","breadcrumbs":"iOverlay » Documentation » Extract Shapes » Segment under Point","id":"45","title":"Segment under Point"},"46":{"body":"When multiple segments are positioned below point P , we need to determine which segment is the closest to P . This scenario can be divided into three distinct cases based on the configuration of the segments relative to P .","breadcrumbs":"iOverlay » Documentation » Extract Shapes » Selecting the Closest Segment under Point","id":"46","title":"Selecting the Closest Segment under Point"},"47":{"body":"Left Case When both segments share a common left vertex A , we check the positions of their right endpoints. If the vertices B0 , B1 , and A form a clockwise pattern, then AB0 is closer to P than AB1 .","breadcrumbs":"iOverlay » Documentation » Extract Shapes » Left Case","id":"47","title":"Left Case"},"48":{"body":"Right Case When both segments share a common right vertex B , we check the positions of their left endpoints. If the vertices A0 , A1 , and B form a clockwise pattern, then A0B is closer to P than A1B .","breadcrumbs":"iOverlay » Documentation » Extract Shapes » Right Case","id":"48","title":"Right Case"},"49":{"body":"Middle Case In this case, one of the vertices (e.g., A1 or B0 ) lies inside the opposite segment. We use the point-segment comparison method to determine which of the segments is closer to P .","breadcrumbs":"iOverlay » Documentation » Extract Shapes » Middle Case","id":"49","title":"Middle Case"},"5":{"body":"Stars Rotation Subject Star First Radius: Second Radius: Rotation Speed: Corners Count: Clip Star First Radius: Second Radius: Rotation Speed: Corners Count: Operation: Xor Union Intersect Difference InverseDifference Subject Clip","breadcrumbs":"iOverlay » Demo » Stars Rotation","id":"5","title":"iOverlay"},"50":{"body":"Eagle triangulation","breadcrumbs":"iTriangle » iTriangle","id":"50","title":"iTriangle"},"51":{"body":"Easy way to get your triangulation!","breadcrumbs":"iTriangle » Introduction","id":"51","title":"Introduction"},"52":{"body":"Delaunay triangulation Break into convex polygons Support any kind of polygons Self-Intersection Resolving","breadcrumbs":"iTriangle » Features","id":"52","title":"Features"},"53":{"body":"Swift Version: iShape-Swift/iTriangle Rust Version: iShape-Rust/iTriangle","breadcrumbs":"iTriangle » Source Code","id":"53","title":"Source Code"},"54":{"body":"","breadcrumbs":"iTriangle » Delaunay » Delaunay","id":"54","title":"Delaunay"},"55":{"body":"When creating a triangulation network, the Delaunay condition aims to form triangles such that their circumscribed circles do not contain any other points from the dataset. In simpler terms, it ensures that triangles are \"well-shaped\" rather than \"skinny,\" making the network more balanced and useful for various applications. Delaunay condition If the condition \\(\\alpha + \\beta < \\pi\\) holds, it implies that the point \\(P_{0}\\) will lie outside the circumscribed circle. This confirms that a pair of triangles satisfies the Delaunay condition. $$ \\alpha + \\beta < \\pi \\Rightarrow \\sin(\\alpha + \\beta) > 0 $$ $$ \\sin(\\alpha + \\beta) = \\sin(\\alpha)\\cos(\\beta) + \\cos(\\alpha)\\sin(\\beta) $$ Calculating \\(\\cos(\\alpha)\\) and \\(\\sin(\\alpha)\\): $$ \\cos(\\alpha) = \\frac{\\vec{a} \\cdot \\vec{b}}{|a||b|} = \\frac{a_{x}b_{x} + a_{y}b_{y}}{|a||b|} $$ $$ \\sin(\\alpha) = \\sqrt{1- \\cos^2(\\alpha)} = ... = \\frac{|a_{x}b_{y} - b_{x}a_{y}|}{|a||b|} = \\frac{|\\vec{a} \\times \\vec{b}|}{|a||b|} $$ Calculating \\(\\cos(\\beta)\\) and \\(\\sin(\\beta)\\): $$ \\cos(\\beta) = \\frac{\\vec{c} \\cdot \\vec{d}}{|c||d|} = \\frac{c_{x}d_{x} + c_{y}d_{y}}{|c||d|} $$ $$ \\sin(\\beta) = \\frac{|\\vec{c} \\times \\vec{d}|}{|c||d|} = \\frac{|c_{x}d_{y} - d_{x}c_{y}|}{|c||d|} $$ Final Equation: $$ \\sin(\\alpha + \\beta) = \\frac{|a_{x}b_{y} - b_{x}a_{y}|\\cdot(c_{x}d_{x} + c_{y}d_{y}) + (a_{x}b_{x} + a_{y}b_{y})\\cdot|c_{x}d_{y} - c_{x}d_{y}|}{|a||b||c||d|} > 0 $$ $$ |a_{x}b_{y} - b_{x}a_{y}|\\cdot(c_{x}d_{x} + c_{y}d_{y}) + (a_{x}b_{x} + a_{y}b_{y})\\cdot|c_{x}d_{y} - d_{x}c_{y}| > 0 $$ Or in vector form: $$ |\\vec{a} \\times \\vec{b}|(\\vec{c} \\cdot \\vec{d}) + (\\vec{a} \\cdot \\vec{b})|\\vec{c} \\times \\vec{d}| > 0 $$","breadcrumbs":"iTriangle » Delaunay » What is the Delaunay Condition?","id":"55","title":"What is the Delaunay Condition?"},"6":{"body":"Shapes Editor OverlayRule: Union Intersect Difference InverseDifference Xor Subject Clip FillRule: EvenOdd NonZero Snap to Grid: Fill Segments: Show Arrows: Prev Title Next","breadcrumbs":"iOverlay » Demo » Shapes Editor","id":"6","title":"iOverlay"},"7":{"body":"Benchmark project is here . 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This confirms that a pair of triangles satisfies the Delaunay condition. $$ \\alpha + \\beta < \\pi \\Rightarrow \\sin(\\alpha + \\beta) > 0 $$ $$ \\sin(\\alpha + \\beta) = \\sin(\\alpha)\\cos(\\beta) + \\cos(\\alpha)\\sin(\\beta) $$ Calculating \\(\\cos(\\alpha)\\) and \\(\\sin(\\alpha)\\): $$ \\cos(\\alpha) = \\frac{\\vec{a} \\cdot \\vec{b}}{|a||b|} = \\frac{a_{x}b_{x} + a_{y}b_{y}}{|a||b|} $$ $$ \\sin(\\alpha) = \\sqrt{1- \\cos^2(\\alpha)} = ... = \\frac{|a_{x}b_{y} - b_{x}a_{y}|}{|a||b|} = \\frac{|\\vec{a} \\times \\vec{b}|}{|a||b|} $$ Calculating \\(\\cos(\\beta)\\) and \\(\\sin(\\beta)\\): $$ \\cos(\\beta) = \\frac{\\vec{c} \\cdot \\vec{d}}{|c||d|} = \\frac{c_{x}d_{x} + c_{y}d_{y}}{|c||d|} $$ $$ \\sin(\\beta) = \\frac{|\\vec{c} \\times \\vec{d}|}{|c||d|} = \\frac{|c_{x}d_{y} - d_{x}c_{y}|}{|c||d|} $$ Final Equation: $$ \\sin(\\alpha + \\beta) = \\frac{|a_{x}b_{y} - b_{x}a_{y}|\\cdot(c_{x}d_{x} + c_{y}d_{y}) + (a_{x}b_{x} + a_{y}b_{y})\\cdot|c_{x}d_{y} - c_{x}d_{y}|}{|a||b||c||d|} > 0 $$ $$ |a_{x}b_{y} - b_{x}a_{y}|\\cdot(c_{x}d_{x} + c_{y}d_{y}) + (a_{x}b_{x} + a_{y}b_{y})\\cdot|c_{x}d_{y} - d_{x}c_{y}| > 0 $$ Or in vector form: $$ |\\vec{a} \\times \\vec{b}|(\\vec{c} \\cdot \\vec{d}) + (\\vec{a} \\cdot \\vec{b})|\\vec{c} \\times \\vec{d}| > 0 $$","breadcrumbs":"iTriangle » Delaunay » What is the Delaunay Condition?","id":"55","title":"What is the Delaunay Condition?"},"6":{"body":"Shapes Editor OverlayRule: Union Intersect Difference InverseDifference Xor Subject Clip FillRule: EvenOdd NonZero Snap to Grid: Fill Segments: Show Arrows: Prev Title Next","breadcrumbs":"iOverlay » Demo » Shapes Editor","id":"6","title":"iOverlay"},"7":{"body":"Benchmark project is here . 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This confirms that a pair of triangles satisfies the Delaunay condition. $$ \\alpha + \\beta < \\pi \\Rightarrow \\sin(\\alpha + \\beta) > 0 $$ $$ \\sin(\\alpha + \\beta) = \\sin(\\alpha)\\cos(\\beta) + \\cos(\\alpha)\\sin(\\beta) $$ Calculating \\(\\cos(\\alpha)\\) and \\(\\sin(\\alpha)\\): $$ \\cos(\\alpha) = \\frac{\\vec{a} \\cdot \\vec{b}}{|a||b|} = \\frac{a_{x}b_{x} + a_{y}b_{y}}{|a||b|} $$ $$ \\sin(\\alpha) = \\sqrt{1- \\cos^2(\\alpha)} = ... = \\frac{|a_{x}b_{y} - b_{x}a_{y}|}{|a||b|} = \\frac{|\\vec{a} \\times \\vec{b}|}{|a||b|} $$ Calculating \\(\\cos(\\beta)\\) and \\(\\sin(\\beta)\\): $$ \\cos(\\beta) = \\frac{\\vec{c} \\cdot \\vec{d}}{|c||d|} = \\frac{c_{x}d_{x} + c_{y}d_{y}}{|c||d|} $$ $$ \\sin(\\beta) = \\frac{|\\vec{c} \\times \\vec{d}|}{|c||d|} = \\frac{|c_{x}d_{y} - d_{x}c_{y}|}{|c||d|} $$ Final Equation: $$ \\sin(\\alpha + \\beta) = \\frac{|a_{x}b_{y} - b_{x}a_{y}|\\cdot(c_{x}d_{x} + c_{y}d_{y}) + (a_{x}b_{x} + a_{y}b_{y})\\cdot|c_{x}d_{y} - c_{x}d_{y}|}{|a||b||c||d|} > 0 $$ $$ |a_{x}b_{y} - b_{x}a_{y}|\\cdot(c_{x}d_{x} + c_{y}d_{y}) + (a_{x}b_{x} + a_{y}b_{y})\\cdot|c_{x}d_{y} - d_{x}c_{y}| > 0 $$ Or in vector form: $$ |\\vec{a} \\times \\vec{b}|(\\vec{c} \\cdot \\vec{d}) + (\\vec{a} \\cdot \\vec{b})|\\vec{c} \\times \\vec{d}| > 0 $$","breadcrumbs":"iTriangle » Delaunay » What is the Delaunay Condition?","id":"55","title":"What is the Delaunay Condition?"},"6":{"body":"Shapes Editor OverlayRule: Union Intersect Difference InverseDifference Xor Subject Clip FillRule: EvenOdd NonZero Snap to Grid: Fill Segments: Show Arrows: Prev Title Next","breadcrumbs":"iOverlay » Demo » Shapes Editor","id":"6","title":"iOverlay"},"7":{"body":"Benchmark project is here . 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This confirms that a pair of triangles satisfies the Delaunay condition. $$ \\alpha + \\beta < \\pi \\Rightarrow \\sin(\\alpha + \\beta) > 0 $$ $$ \\sin(\\alpha + \\beta) = \\sin(\\alpha)\\cos(\\beta) + \\cos(\\alpha)\\sin(\\beta) $$ Calculating \\(\\cos(\\alpha)\\) and \\(\\sin(\\alpha)\\): $$ \\cos(\\alpha) = \\frac{\\vec{a} \\cdot \\vec{b}}{|a||b|} = \\frac{a_{x}b_{x} + a_{y}b_{y}}{|a||b|} $$ $$ \\sin(\\alpha) = \\sqrt{1- \\cos^2(\\alpha)} = ... = \\frac{|a_{x}b_{y} - b_{x}a_{y}|}{|a||b|} = \\frac{|\\vec{a} \\times \\vec{b}|}{|a||b|} $$ Calculating \\(\\cos(\\beta)\\) and \\(\\sin(\\beta)\\): $$ \\cos(\\beta) = \\frac{\\vec{c} \\cdot \\vec{d}}{|c||d|} = \\frac{c_{x}d_{x} + c_{y}d_{y}}{|c||d|} $$ $$ \\sin(\\beta) = \\frac{|\\vec{c} \\times \\vec{d}|}{|c||d|} = \\frac{|c_{x}d_{y} - d_{x}c_{y}|}{|c||d|} $$ Final Equation: $$ \\sin(\\alpha + \\beta) = \\frac{|a_{x}b_{y} - b_{x}a_{y}|\\cdot(c_{x}d_{x} + c_{y}d_{y}) + (a_{x}b_{x} + a_{y}b_{y})\\cdot|c_{x}d_{y} - c_{x}d_{y}|}{|a||b||c||d|} > 0 $$ $$ |a_{x}b_{y} - b_{x}a_{y}|\\cdot(c_{x}d_{x} + c_{y}d_{y}) + (a_{x}b_{x} + a_{y}b_{y})\\cdot|c_{x}d_{y} - d_{x}c_{y}| > 0 $$ Or in vector form: $$ |\\vec{a} \\times \\vec{b}|(\\vec{c} \\cdot \\vec{d}) + (\\vec{a} \\cdot \\vec{b})|\\vec{c} \\times \\vec{d}| > 0 $$","breadcrumbs":"iTriangle » Delaunay » What is the Delaunay Condition?","id":"55","title":"What is the Delaunay Condition?"},"6":{"body":"Shapes Editor OverlayRule: Union Intersect Difference InverseDifference Xor Subject Clip FillRule: EvenOdd NonZero Snap to Grid: Fill Segments: Show Arrows: Prev Title Next","breadcrumbs":"iOverlay » Demo » Shapes Editor","id":"6","title":"iOverlay"},"7":{"body":"Benchmark project is here . 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