From f7b4cc602b9a646fbc66f3f17d6bb9c20efc3ead Mon Sep 17 00:00:00 2001
From: jaseg <git@jaseg.de>
Date: Sun, 24 Jan 2021 18:44:56 +0100
Subject: Initial commit

---
 .../Documentation/offset_triginometry3.svg         | 391 +++++++++++++++++++++
 1 file changed, 391 insertions(+)
 create mode 100644 upstream/clipper-6.4.2/Documentation/offset_triginometry3.svg

(limited to 'upstream/clipper-6.4.2/Documentation/offset_triginometry3.svg')

diff --git a/upstream/clipper-6.4.2/Documentation/offset_triginometry3.svg b/upstream/clipper-6.4.2/Documentation/offset_triginometry3.svg
new file mode 100644
index 0000000..52eff70
--- /dev/null
+++ b/upstream/clipper-6.4.2/Documentation/offset_triginometry3.svg
@@ -0,0 +1,391 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!-- Created with Inkscape (http://www.inkscape.org/) -->
+
+<svg
+   xmlns:dc="http://purl.org/dc/elements/1.1/"
+   xmlns:cc="http://creativecommons.org/ns#"
+   xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
+   xmlns:svg="http://www.w3.org/2000/svg"
+   xmlns="http://www.w3.org/2000/svg"
+   xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
+   xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
+   width="800"
+   height="600"
+   id="svg2"
+   version="1.1"
+   inkscape:version="0.48.2 r9819"
+   sodipodi:docname="offset_triginometry3.svg">
+  <defs
+     id="defs4" />
+  <sodipodi:namedview
+     id="base"
+     pagecolor="#ffffff"
+     bordercolor="#666666"
+     borderopacity="1.0"
+     inkscape:pageopacity="0.0"
+     inkscape:pageshadow="2"
+     inkscape:zoom="0.64"
+     inkscape:cx="-24.27"
+     inkscape:cy="218.1"
+     inkscape:document-units="px"
+     inkscape:current-layer="layer1"
+     showgrid="false"
+     inkscape:window-width="1280"
+     inkscape:window-height="753"
+     inkscape:window-x="-4"
+     inkscape:window-y="-4"
+     inkscape:window-maximized="1"
+     showborder="true"
+     inkscape:showpageshadow="true"
+     fit-margin-top="0"
+     fit-margin-left="0"
+     fit-margin-right="0"
+     fit-margin-bottom="0">
+    <inkscape:grid
+       type="xygrid"
+       id="grid3779"
+       empspacing="5"
+       visible="true"
+       enabled="true"
+       snapvisiblegridlinesonly="true" />
+  </sodipodi:namedview>
+  <metadata
+     id="metadata7">
+    <rdf:RDF>
+      <cc:Work
+         rdf:about="">
+        <dc:format>image/svg+xml</dc:format>
+        <dc:type
+           rdf:resource="http://purl.org/dc/dcmitype/StillImage" />
+        <dc:title></dc:title>
+      </cc:Work>
+    </rdf:RDF>
+  </metadata>
+  <g
+     inkscape:label="Layer 1"
+     inkscape:groupmode="layer"
+     id="layer1"
+     transform="translate(-407.6,90.95)">
+    <path
+       style="color:#000000;fill:none;stroke:#000000;stroke-width:1;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:4, 4;stroke-dashoffset:0;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate"
+       d="m 604.8,209.6 85.8,-55.7"
+       id="path3907"
+       inkscape:connector-curvature="0"
+       sodipodi:nodetypes="cc" />
+    <path
+       style="color:#000000;fill:none;stroke:#000000;stroke-width:1.8;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:none;stroke-dashoffset:0;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate"
+       d="M 420.4,326.1 603.7,10.08 757.6,263.6"
+       id="path2985"
+       inkscape:connector-curvature="0"
+       sodipodi:nodetypes="ccc" />
+    <text
+       xml:space="preserve"
+       style="font-size:20px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;text-indent:0;text-align:start;text-decoration:none;line-height:125%;letter-spacing:0px;word-spacing:0px;text-transform:none;direction:ltr;block-progression:tb;writing-mode:lr-tb;text-anchor:start;baseline-shift:baseline;color:#000000;fill:#000000;fill-opacity:1;fill-rule:nonzero;stroke:none;stroke-width:1px;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate;font-family:Tahoma;-inkscape-font-specification:Tahoma"
+       x="624.09735"
+       y="215.06464"
+       id="text3861"
+       sodipodi:linespacing="125%"><tspan
+         sodipodi:role="line"
+         id="tspan3863"
+         x="624.09735"
+         y="215.06464"
+         style="font-weight:bold;-inkscape-font-specification:Tahoma Bold">ß</tspan></text>
+    <path
+       sodipodi:nodetypes="cccccccc"
+       inkscape:connector-curvature="0"
+       id="path3857-2"
+       d="m 603,29.9 15,15.8 -10,0 0,155.1 -10,0 0,-155.1 -10,0 z"
+       style="color:#000000;fill:#ffff00;fill-opacity:1;stroke:#000000;stroke-width:1.776;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:none;stroke-dashoffset:0;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate"
+       inkscape:transform-center-x="-20.96"
+       inkscape:transform-center-y="-1.652" />
+    <text
+       xml:space="preserve"
+       style="font-size:10px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;text-indent:0;text-align:start;text-decoration:none;line-height:125%;letter-spacing:0px;word-spacing:0px;text-transform:none;direction:ltr;block-progression:tb;writing-mode:lr-tb;text-anchor:start;baseline-shift:baseline;color:#000000;fill:#000000;fill-opacity:1;fill-rule:nonzero;stroke:none;stroke-width:1px;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate;font-family:Tahoma;-inkscape-font-specification:Tahoma"
+       x="682.20001"
+       y="261.375"
+       id="text3952"
+       sodipodi:linespacing="125%"><tspan
+         sodipodi:role="line"
+         id="tspan3954"
+         x="682.20001"
+         y="261.375"
+         style="font-size:14px">delta</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:10px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;text-indent:0;text-align:start;text-decoration:none;line-height:125%;letter-spacing:0px;word-spacing:0px;text-transform:none;direction:ltr;block-progression:tb;writing-mode:lr-tb;text-anchor:start;baseline-shift:baseline;color:#000000;fill:#000000;fill-opacity:1;fill-rule:nonzero;stroke:none;stroke-width:1px;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate;font-family:Tahoma;-inkscape-font-specification:Tahoma"
+       x="698.20001"
+       y="151.875"
+       id="text3977"
+       sodipodi:linespacing="125%"><tspan
+         sodipodi:role="line"
+         id="tspan3979"
+         x="698.20001"
+         y="151.875"
+         style="font-size:14px">c</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:10px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;text-indent:0;text-align:start;text-decoration:none;line-height:125%;letter-spacing:0px;word-spacing:0px;text-transform:none;direction:ltr;block-progression:tb;writing-mode:lr-tb;text-anchor:start;baseline-shift:baseline;color:#000000;fill:#000000;fill-opacity:1;fill-rule:nonzero;stroke:none;stroke-width:1px;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate;font-family:Tahoma;-inkscape-font-specification:Tahoma"
+       x="599.76099"
+       y="3.3956299"
+       id="text3977-5-1-2"
+       sodipodi:linespacing="125%"
+       inkscape:transform-center-x="-52.52"
+       inkscape:transform-center-y="54.68"><tspan
+         sodipodi:role="line"
+         id="tspan3979-1-5-7"
+         x="599.76099"
+         y="3.3956299"
+         style="font-size:14px">b</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:14px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;text-indent:0;text-align:start;text-decoration:none;line-height:125%;letter-spacing:0px;word-spacing:0px;text-transform:none;direction:ltr;block-progression:tb;writing-mode:lr-tb;text-anchor:start;baseline-shift:baseline;color:#000000;fill:#000000;fill-opacity:1;fill-rule:nonzero;stroke:none;stroke-width:1px;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate;font-family:Tahoma;-inkscape-font-specification:Tahoma"
+       x="766"
+       y="-49.349998"
+       id="text4077"
+       sodipodi:linespacing="125%"><tspan
+         sodipodi:role="line"
+         x="766"
+         y="-49.349998"
+         id="tspan4087">When 'mitering' offset polygons, the maximum distance </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="-31.849998"
+         id="tspan3063">point 'b' can be from point 'a' is set by 'limit' where limit </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="-14.349998"
+         id="tspan3065">is a multiple of delta. Therefore, for any given angle</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="3.1500015"
+         id="tspan3067">we need to know if length(ab) &gt; limit * delta.</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="20.650002"
+         id="tspan4083" /><tspan
+         sodipodi:role="line"
+         x="766"
+         y="38.150002"
+         id="tspan3890">Find the largest angle ß (or smallest Ø since Ø = pi - ß) </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="55.650002"
+         id="tspan3904">for a given limit, expressing ß as sin(ß) or cos(ß) since</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="73.150002"
+         id="tspan3908">these can easily be derived from cross or dot products </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="90.650002"
+         id="tspan3916">respectively.</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="98.598343"
+         id="tspan3884"
+         style="font-size:4px"> </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="113.15"
+         id="tspan3888">angle(abc) = Ø/2 </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="130.64999"
+         id="tspan3083">length(ab) = limit * delta</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="148.14999"
+         id="tspan3087">length(ac) = delta</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="165.64999"
+         id="tspan3089">sin(Ø/2) = delta / (limit * delta) = 1 / limit</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="183.14999"
+         id="tspan3157">Given that sin(Ø/2) = sqrt((1-cos(Ø))/2) **</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="200.64999"
+         id="tspan3151">1 / limit = sqrt((1-cos(Ø))/2)</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="218.14999"
+         id="tspan3161">limit = sqrt(2 / (1-cos(Ø)))</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="235.64999"
+         id="tspan3815"
+         style="font-weight:normal">1-cos(Ø) = 2 / sqr(limit) </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="253.14999"
+         id="tspan3043">Since Ø = pi - ß ...</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="270.64999"
+         id="tspan3868">1-cos(pi - ß) = 2 / sqr(limit)</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="288.14999"
+         id="tspan3047">and given cos(pi-ß) = -cos(ß) ** ... </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="296.09836"
+         style="font-size:4px;font-weight:normal"
+         id="tspan3823"> </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="310.64999"
+         style="font-weight:bold"
+         id="tspan3122">1+cos(ß) = 2 / sqr(limit) </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="318.59836"
+         style="font-size:4px;font-weight:normal"
+         id="tspan3118"> </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="333.14999"
+         style="font-weight:normal"
+         id="tspan3114">cos(ß) = 2 / sqr(limit) - 1</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="350.64999"
+         style="font-weight:bold"
+         id="tspan3051" /><tspan
+         sodipodi:role="line"
+         x="766"
+         y="368.14999"
+         style="font-weight:normal"
+         id="tspan3066">Example: if miter limit = 2 (ie 2 times delta) then </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="385.64999"
+         style="font-weight:normal"
+         id="tspan3070">cos(ß) = (2 / 4) -1 = -0.5 and so ß = 120 degrees. </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="403.14999"
+         style="font-weight:normal"
+         id="tspan3072">Therefore, when ß &gt; 120 deg. (or Ø &lt; 60 deg.), the</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="420.64999"
+         style="font-weight:normal"
+         id="tspan3090">distance point 'b' would be from point 'a' would exceed </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="438.14999"
+         style="font-weight:normal"
+         id="tspan3094">the limit.</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="446.09836"
+         id="tspan3171"
+         style="font-size:4px;font-weight:bold">   </tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="451.09836"
+         style="font-size:4px;font-weight:bold"
+         id="tspan3049" /><tspan
+         sodipodi:role="line"
+         x="766"
+         y="462.78452"
+         id="tspan3163"
+         style="font-size:11px">** see http://en.wikipedia.org/wiki/List_of_trigonometric_identities</tspan><tspan
+         sodipodi:role="line"
+         x="766"
+         y="476.53452"
+         id="tspan3153" /><tspan
+         sodipodi:role="line"
+         x="766"
+         y="490.28452"
+         id="tspan3145" /><tspan
+         sodipodi:role="line"
+         x="766"
+         y="504.03452"
+         id="tspan4124"
+         style="font-weight:bold" /><tspan
+         sodipodi:role="line"
+         x="766"
+         y="517.78448"
+         id="tspan4116" /><tspan
+         sodipodi:role="line"
+         x="766"
+         y="531.53448"
+         id="tspan4095" /></text>
+    <path
+       style="color:#000000;fill:none;stroke:#000000;stroke-width:1;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:2, 2;stroke-dashoffset:0;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate"
+       d="m 684.7,144.1 -10.6,5.9 5.4,10.1"
+       id="path4091"
+       inkscape:connector-curvature="0"
+       sodipodi:nodetypes="ccc" />
+    <text
+       xml:space="preserve"
+       style="font-size:10px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;text-indent:0;text-align:start;text-decoration:none;line-height:125%;letter-spacing:0px;word-spacing:0px;text-transform:none;direction:ltr;block-progression:tb;writing-mode:lr-tb;text-anchor:start;baseline-shift:baseline;color:#000000;fill:#000000;fill-opacity:1;fill-rule:nonzero;stroke:none;stroke-width:1px;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate;font-family:Tahoma;-inkscape-font-specification:Tahoma"
+       x="588.19745"
+       y="208.67169"
+       id="text3977-1"
+       sodipodi:linespacing="125%"><tspan
+         sodipodi:role="line"
+         id="tspan3979-7"
+         x="588.19745"
+         y="208.67169"
+         style="font-size:14px">a</tspan></text>
+    <g
+       id="g3841"
+       transform="translate(400,-20.7)">
+      <path
+         sodipodi:nodetypes="cc"
+         inkscape:connector-curvature="0"
+         id="path3026"
+         d="M 175.6,276.8 233.7,175.7"
+         style="color:#000000;fill:none;stroke:#000000;stroke-width:1.8;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:7.2, 7.2;stroke-dashoffset:0;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate" />
+      <path
+         sodipodi:nodetypes="cc"
+         inkscape:connector-curvature="0"
+         id="path3809"
+         d="m 227.7,188.7 c 27.7,17.9 31.6,57.6 -0.5,76.7"
+         style="color:#000000;fill:none;stroke:#000000;stroke-width:1.81;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:7.24, 7.24;stroke-dashoffset:0;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate" />
+    </g>
+    <path
+       style="color:#000000;fill:#00ff00;fill-opacity:1;stroke:#000000;stroke-width:2.1383;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:none;stroke-dashoffset:0;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate"
+       d="m 724.8,212.4 -9.2,24.2 -5.6,-8.3 -46.8,31.5 5.4,8.3 -26,-0.8 9.4,-24.3 5.5,8.3 47,-31.3 -5.6,-8.3 z"
+       id="path3777-9"
+       inkscape:connector-curvature="0"
+       sodipodi:nodetypes="ccccccccccc"
+       inkscape:transform-center-x="-96.45"
+       inkscape:transform-center-y="-137.3" />
+    <text
+       xml:space="preserve"
+       style="font-size:14px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;text-indent:0;text-align:start;text-decoration:none;line-height:125%;letter-spacing:0px;word-spacing:0px;text-transform:none;direction:ltr;block-progression:tb;writing-mode:lr-tb;text-anchor:start;baseline-shift:baseline;color:#000000;fill:#000000;fill-opacity:1;fill-rule:nonzero;stroke:none;stroke-width:1px;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate;font-family:Tahoma;-inkscape-font-specification:Tahoma"
+       x="595.66949"
+       y="246.41019"
+       id="text3845"
+       sodipodi:linespacing="125%"><tspan
+         sodipodi:role="line"
+         id="tspan3847"
+         x="595.66949"
+         y="246.41019"
+         style="font-size:20px;font-weight:bold;-inkscape-font-specification:Tahoma Bold">Ø</tspan></text>
+    <text
+       sodipodi:linespacing="125%"
+       id="text3849"
+       y="246.41019"
+       x="595.66949"
+       style="font-size:14px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;text-indent:0;text-align:start;text-decoration:none;line-height:125%;letter-spacing:0px;word-spacing:0px;text-transform:none;direction:ltr;block-progression:tb;writing-mode:lr-tb;text-anchor:start;baseline-shift:baseline;color:#000000;fill:#000000;fill-opacity:1;fill-rule:nonzero;stroke:none;stroke-width:1px;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate;font-family:Tahoma;-inkscape-font-specification:Tahoma"
+       xml:space="preserve"><tspan
+         style="font-size:20px;font-weight:bold;-inkscape-font-specification:Tahoma Bold"
+         y="246.41019"
+         x="595.66949"
+         id="tspan3851"
+         sodipodi:role="line">Ø</tspan></text>
+    <path
+       style="color:#000000;fill:none;stroke:#000000;stroke-width:3.6;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:none;stroke-dashoffset:0;marker:none;visibility:visible;display:inline;overflow:visible;enable-background:accumulate"
+       d="M 530.8,333.3 603,210 677.5,330.2"
+       id="path2985-1-1"
+       inkscape:connector-curvature="0"
+       sodipodi:nodetypes="ccc" />
+  </g>
+</svg>
-- 
cgit