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Diffstat (limited to 'upstream/clipper-6.4.2/Documentation/offset_triginometry3.svg')
| -rw-r--r-- | upstream/clipper-6.4.2/Documentation/offset_triginometry3.svg | 391 |
1 files changed, 391 insertions, 0 deletions
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" + 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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) > 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 > 120 deg. (or Ø < 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 + 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