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+ xml:space="preserve"
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+ <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"
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+ 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"
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+ sodipodi:role="line"
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+ 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"
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+ 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"
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+ 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
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+ x="766"
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+ 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
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+ x="766"
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+ style="font-size:4px;font-weight:normal"
+ id="tspan3823"> </tspan><tspan
+ sodipodi:role="line"
+ x="766"
+ y="310.64999"
+ style="font-weight:bold"
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+ x="766"
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+ style="font-size:4px;font-weight:normal"
+ id="tspan3118"> </tspan><tspan
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+ x="766"
+ y="333.14999"
+ style="font-weight:normal"
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+ 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
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+ x="766"
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+ id="tspan3145" /><tspan
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+ x="766"
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+ id="tspan4124"
+ style="font-weight:bold" /><tspan
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+ x="766"
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+ 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"
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