add: hw1
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304 184 l
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108 264 l
|
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|
</path>
|
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|
<path stroke="black">
|
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|
32 216 m
|
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|
98.4 190 l
|
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|
</path>
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|
<path stroke="black">
|
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|
108 264 m
|
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|
32 216 l
|
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|
</path>
|
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|
<path stroke="black">
|
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|
108 264 m
|
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|
98.4 190 l
|
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|
</path>
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|
<path stroke="black">
|
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|
108 264 m
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|
228 244 l
|
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|
</path>
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|
<path stroke="black">
|
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|
228 244 m
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|
288 216 l
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|
</path>
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<path stroke="black">
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|
252 308 m
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|
228 244 l
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</path>
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<path stroke="black">
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252 308 m
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|
316 380 l
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|
</path>
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<path stroke="black">
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252 308 m
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|
108 264 l
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|
</path>
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|
<path stroke="black">
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|
388 300 m
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|
488 236 l
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|
</path>
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|
<path stroke="black">
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|
388 300 m
|
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|
316 380
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316 380 c
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</path>
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|
<path stroke="black">
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|
368 240 m
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|
488 236 l
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</path>
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<path stroke="black">
|
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|
368 240 m
|
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|
304 184 l
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|
</path>
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<path stroke="black">
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368 240 m
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552 204 l
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<use name="mark/circle(sx)" pos="388 300" size="normal" stroke="violet"/>
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<path stroke="black">
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|
488 236 m
|
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|
552 204 l
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</path>
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<use name="mark/circle(sx)" pos="304 184" size="normal" stroke="blue"/>
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<use name="mark/circle(sx)" pos="488 236" size="normal" stroke="blue"/>
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|
<use name="mark/circle(sx)" pos="288 216" size="normal" stroke="red"/>
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|
<path stroke="red">
|
||||||
|
144 164 m
|
||||||
|
64 208 l
|
||||||
|
34.338 223.243 l
|
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|
34.338 223.243 l
|
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|
</path>
|
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|
<path stroke="red">
|
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|
176 164 m
|
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|
259.069 289.936 l
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|
</path>
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|
<path stroke="red">
|
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|
320 304 m
|
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|
413.803 298.118 l
|
||||||
|
</path>
|
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|
<path stroke="red">
|
||||||
|
320 304 m
|
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|
193.362 311.606 l
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|
</path>
|
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|
<path stroke="red">
|
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|
328 260 m
|
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|
455.333 196.204 l
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|
</path>
|
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|
<text transformations="translations" pos="32 216" stroke="red" type="label" width="7.472" height="6.808" depth="0" valign="baseline">A</text>
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|
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||||||
|
<text matrix="1 0 0 1 -7.74742 31.4736" transformations="translations" pos="320 336" stroke="red" type="label" width="7.195" height="6.808" depth="0" valign="baseline">C</text>
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|
<text matrix="1 0 0 1 -73.6228 28.8556" transformations="translations" pos="438.081 204.848" stroke="red" type="label" width="7.61" height="6.808" depth="0" valign="baseline">D</text>
|
||||||
|
<text matrix="1 0 0 1 -58.3448 23.4305" transformations="translations" pos="543.702 207.121" stroke="red" type="label" width="6.78" height="6.808" depth="0" valign="baseline">E</text>
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|
<path stroke="red">
|
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|
529.948 258.011 m
|
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|
405.471 190.959 l
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||||||
|
</path>
|
||||||
|
</page>
|
||||||
|
</ipe>
|
521
HW1/hw1-polygon2-guard.ipe
Normal file
521
HW1/hw1-polygon2-guard.ipe
Normal file
|
@ -0,0 +1,521 @@
|
||||||
|
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h
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h
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h
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h
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|
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h
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0.4 -0.4 l
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0.4 0.4 l
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h
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<group>
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|
0.57 0.43 l
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0.43 0.57 l
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h
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0.57 -0.43 l
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0.43 -0.57 l
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|
-0.57 0.43 l
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|
h
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|
</path>
|
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</group>
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</symbol>
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|
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0 0 m
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|
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-1 -0.333 l
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h
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|
-1 -0.333 l
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|
h
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</symbol>
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<symbol name="arrow/fpointed(spx)">
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0 0 m
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h
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h
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|
-2 -0.333 l
|
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|
h
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|
</path>
|
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|
</symbol>
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|
<anglesize name="22.5 deg" value="22.5"/>
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||||||
|
<anglesize name="30 deg" value="30"/>
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|
<anglesize name="45 deg" value="45"/>
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|
<anglesize name="60 deg" value="60"/>
|
||||||
|
<anglesize name="90 deg" value="90"/>
|
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<use name="mark/disk(sx)" pos="387.322 644.678" size="normal" stroke="yellow"/>
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<use name="mark/disk(sx)" pos="539.322 636.678" size="normal" stroke="yellow"/>
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<use name="mark/disk(sx)" pos="443.322 548.678" size="normal" stroke="red"/>
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<use name="mark/disk(sx)" pos="491.322 480.678" size="normal" stroke="red"/>
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<use name="mark/disk(sx)" pos="259.322 652.678" size="normal" stroke="red"/>
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<use name="mark/disk(sx)" pos="367.322 724.678" size="normal" stroke="red"/>
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<use name="mark/disk(sx)" pos="387.322 624.678" size="normal" stroke="red"/>
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<use name="mark/disk(sx)" pos="427.322 704.678" size="normal" stroke="red"/>
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<use name="mark/disk(sx)" pos="195.322 544.678" size="normal" stroke="red"/>
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<use name="mark/disk(sx)" pos="119.322 612.678" size="normal" stroke="red"/>
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<use name="mark/disk(sx)" pos="112 320" size="normal" stroke="red"/>
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<path stroke="red">
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207.322 584.678 m
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|
112.65 564.301 l
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</path>
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<path stroke="red">
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|
207.322 584.678 m
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|
185.045 509.176 l
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</path>
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<path stroke="red" dash="dotted">
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|
351.322 480.678 m
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|
499.76 534.926 l
|
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|
</path>
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|
<path stroke="red">
|
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|
499.76 534.926 m
|
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|
459.322 520.678 l
|
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|
</path>
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|
<path stroke="red">
|
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|
427.322 704.678 m
|
||||||
|
464.177 630.745 l
|
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|
</path>
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|
<path stroke="red">
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|
503.322 748.678 m
|
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|
234.115 700.444 l
|
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|
</path>
|
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|
<path stroke="red">
|
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|
356.547 629.29 m
|
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|
404.44 652.678 l
|
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|
</path>
|
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|
<path stroke="red">
|
||||||
|
356.547 629.29 m
|
||||||
|
244.679 569.454
|
||||||
|
244.679 569.454
|
||||||
|
244.757 568.484 c
|
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|
</path>
|
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|
<path stroke="red" dash="dotted">
|
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|
356.547 629.29 m
|
||||||
|
237.585 657.425 l
|
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|
</path>
|
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|
<path stroke="red" dash="dotted">
|
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|
259.322 652.678 m
|
||||||
|
237.585 657.425 l
|
||||||
|
</path>
|
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|
<path stroke="red" dash="dotted">
|
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|
288 832 m
|
||||||
|
304 832 l
|
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|
</path>
|
||||||
|
<path stroke="red">
|
||||||
|
259.322 652.678 m
|
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|
237.585 657.425 l
|
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|
</path>
|
||||||
|
<path stroke="red">
|
||||||
|
288 832 m
|
||||||
|
304 832 l
|
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|
</path>
|
||||||
|
<text transformations="translations" pos="103.322 496.678" stroke="red" type="label" width="7.472" height="6.808" depth="0" valign="baseline">A</text>
|
||||||
|
<text transformations="translations" pos="571.322 496.678" stroke="red" type="label" width="7.057" height="6.808" depth="0" valign="baseline">B</text>
|
||||||
|
<text transformations="translations" pos="539.322 636.678" stroke="red" type="label" width="7.195" height="6.808" depth="0" valign="baseline">C</text>
|
||||||
|
<text transformations="translations" pos="227.322 784.678" stroke="red" type="label" width="7.61" height="6.808" depth="0" valign="baseline">D</text>
|
||||||
|
<text transformations="translations" pos="259.322 652.678" stroke="red" type="label" width="6.78" height="6.808" depth="0" valign="baseline">E</text>
|
||||||
|
<text transformations="translations" pos="295.609 832" stroke="red" type="label" width="7.472" height="6.808" depth="0" valign="baseline">A</text>
|
||||||
|
</page>
|
||||||
|
</ipe>
|
853
HW1/hw1.html
Normal file
853
HW1/hw1.html
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File diff suppressed because one or more lines are too long
38
HW1/hw1.md
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38
HW1/hw1.md
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|
@ -0,0 +1,38 @@
|
||||||
|
# HW1
|
||||||
|
## Problem 1
|
||||||
|
- 1. In this graph, there are 3 forced diagonals (7-12, 6-14 and 5-17) which divide this graph in to 4 parts. For the right most part it is a blunt nose fox so it has 4 ways of triangulations; for the second one there is only one way because it is a fox; the third one is again, a blunt nose fox with 4 ways; as for the last part we have to do case analysis:
|
||||||
|
- 1. use 4-17 but not 2-5: there are two convex polygon with 3 and 6 vertices, so there are 14*1=14 ways to triangulation
|
||||||
|
- 2. use 2-5 but not 4-17: for the graph under 2-5, it is a convex polygon with 4 vertices so there are 2 ways; for the upper part, 2-17 is a forced diagonal so it can be treated as a 4 vertices polygon so overall it has 2*2=4 ways
|
||||||
|
- 3. not use both 4-17 and 2-5: for the last situation, 3-17 is a forced diagonal so it has 5\*1\*1=5 ways
|
||||||
|
|
||||||
|
Overall, there are 4\*1\*4\*(14+4+5)=368 distinct ways to triangulation this graph
|
||||||
|
|
||||||
|
- 2. In this graph, there are a pair of diagonals that are mutual exclusive: 1-6 and 0-2, if we do case analysis based on that:
|
||||||
|
- 1. use 1-6: the left side is a convex polygon with 8 vertices so it has 6 ways; the right side is a blunt nose fox so it has 4 ways
|
||||||
|
- 2. use 0-2: then there are two forced diagonals (0-6 and 2-6). If we connect them, then there is a convex polygon with 7 vertices on the left and a fox on the right. In total, there should be 1\*1\*1\*42 ways
|
||||||
|
|
||||||
|
Over all, there are 132*4+42=570 distinct ways of triangulations.
|
||||||
|
|
||||||
|
## Problem 2
|
||||||
|
- 1.
|
||||||
|
- 1. ![2_1_1](2_1_1.png)
|
||||||
|
- 2. As the graph above, there are 6 reds, 6 blues and 7 yellows, then we put guards on either reds or blues and we can get 6 guards.
|
||||||
|
- 3. ![2_1_2](2_1_2.png)
|
||||||
|
If we put witness points near 18, 2, 7 and 10, and in between 4 and 5, we got 5 areas that does not overlap on any vertex, $w(P) \geq 5$. Also, we can find 5 guards A, B, C, D and E that cover P, thus $g(P) \leq 5$
|
||||||
|
|
||||||
|
$5 \leq w(p) \leq g(p) \leq 5$, and we can get $g(p) = 5$
|
||||||
|
- 4. ![2_1_3](2_1_3.png)
|
||||||
|
If we put witness points near 18, 2 and 7, and in between 4 and 5, we got 4 areas that does not overlap, so $w(P) \geq 4$. Also, we can find 4 guards A, B, C and D that cover P, thus $g(P) \leq 4$
|
||||||
|
|
||||||
|
$4 \leq w(p) \leq g(p) \leq 4$, and we can get $g(p) = 4$
|
||||||
|
- 2.
|
||||||
|
- 1. ![2_2_1](2_2_1.png)
|
||||||
|
- 2. As the graph above, there are 8 reds, 8 blues and 8 yellows, then we put guards on any of reds, yellows or blues and we can get 8 guards.
|
||||||
|
- 3. ![2_2_2](2_2_2.png)
|
||||||
|
If we put witness points near 3, 10, 17 and 23, and in between 13 and 14, we got 5 areas that does not overlap on any vertex, $w(P) \geq 5$. Also, we can find 5 guards A, B, C, D and E that cover P, thus $g(P) \leq 5$
|
||||||
|
|
||||||
|
$5 \leq w(p) \leq g(p) \leq 5$, and we can get $g(p) = 5$
|
||||||
|
- 4. ![2_2_2](2_2_2.png)
|
||||||
|
If we put witness points near 3, 10, 17 and 23, and in between 13 and 14, we got 5 areas that does not overlap, $w(P) \geq 5$. Also, we can find 5 guards A, B, C, D and E that cover P, thus $g(P) \leq 5$
|
||||||
|
|
||||||
|
$5 \leq w(p) \leq g(p) \leq 5$, and we can get $g(p) = 5$
|
Loading…
Reference in New Issue
Block a user