108 lines
7.1 KiB
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108 lines
7.1 KiB
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<h2>Bézier Curves</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 363</h3><span style="width:300px;color:#666;">Published on Sunday, 18th December 2011, 10:00 am; Solved by 493</span></div>
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<div class="problem_content" role="problem">
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A cubic Bézier curve is defined by four points: P<sub>0</sub>, P<sub>1</sub>, P<sub>2</sub> and P<sub>3</sub>.</P>
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<P>
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The curve is constructed as follows:<BR>
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On the segments P<sub>0</sub>P<sub>1</sub>, P<sub>1</sub>P<sub>2</sub> and P<sub>2</sub>P<sub>3</sub> the points Q<sub>0</sub>,Q<sub>1</sub> and Q<sub>2</sub> are drawn such that P<sub>0</sub>Q<sub>0</sub>/P<sub>0</sub>P<sub>1</sub>=P<sub>1</sub>Q<sub>1</sub>/P<sub>1</sub>P<sub>2</sub>=P<sub>2</sub>Q<sub>2</sub>/P<sub>2</sub>P<sub>3</sub>=t (t in [0,1]).<BR>
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On the segments Q<sub>0</sub>Q<sub>1</sub> and Q<sub>1</sub>Q<sub>2</sub> the points R<sub>0</sub> and R<sub>1</sub> are drawn such that
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Q<sub>0</sub>R<sub>0</sub>/Q<sub>0</sub>Q<sub>1</sub>=Q<sub>1</sub>R<sub>1</sub>/Q<sub>1</sub>Q<sub>2</sub>=t for the same value of t.<BR>
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On the segment R<sub>0</sub>R<sub>1</sub> the point B is drawn such that R<sub>0</sub>B/R<sub>0</sub>R<sub>1</sub>=t for the same value of t.</BR>
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The Bézier curve defined by the points P<sub>0</sub>, P<sub>1</sub>, P<sub>2</sub>, P<sub>3</sub> is the locus of B as Q<sub>0</sub> takes all possible positions on the segment P<sub>0</sub>P<sub>1</sub>. (Please note that for all points the value of t is the same.)
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<BR>
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</p>
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<P><APPLET CODE=CabriJava.class WIDTH=390 HEIGHT=300 ALIGN=right hspace=20 archive=http://projecteuler.net/project/images/bezier/CabriJava.jar>
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<PARAM NAME=lang VALUE=en>
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<PARAM NAME=file VALUE=http://projecteuler.net/project/images/bezier/bezier.fig>
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<param name=loop value=true>
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</APPLET></center>
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</p>
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<P>
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In the applet to the right you can drag the points P<sub>0</sub>, P<sub>1</sub>, P<sub>2</sub> and P<sub>3</sub> to see what the Bézier curve (green curve) defined by those points looks like. You can also drag the point Q<sub>0</sub> along the segment P<sub>0</sub>P<sub>1</sub>.
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</P>
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<P>
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From the construction it is clear that the Bézier curve will be tangent to the segments P<sub>0</sub>P<sub>1</sub> in P<sub>0</sub> and P<sub>2</sub>P<sub>3</sub> in P<sub>3</sub>.
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<P>
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<BR />
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<BR />
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<BR />
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</P>
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<P>
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A cubic Bézier curve with P<sub>0</sub>=(1,0), P<sub>1</sub>=(1,<var>v</var>), P<sub>2</sub>=(<var>v</var>,1) and P<sub>3</sub>=(0,1) is used to approximate a quarter circle.<BR />
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The value <var>v</var></var><img src='images/symbol_gt.gif' width='10' height='10' alt='>' border='0' style='vertical-align:middle;' />0 is chosen such that the area enclosed by the lines OP<sub>0</sub>, OP<sub>3</sub> and the curve is equal to <sup>π</sup>/<sub>4</sub> (the area of the quarter circle).
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</P>
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<P>
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By how many percent does the length of the curve differ from the length of the quarter circle?<BR />
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That is, if L is the length of the curve, calculate 100*<sup>(L-π/2)</sup>/<sub>(π/2)</sub>.<BR />
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Give your answer rounded to 10 digits behind the decimal point.
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