| 發布者 | 內容列 |
mathematical2005
Just can't stay away
註冊日: 2005-05-17
發表數: 117
China
|  Pick公式 |  | 設Y為平面上以格子點為頂點之單純多邊形,則面積為
A=(b/2)+i-1
其中b為邊界上的格子點數,i為內部的格子點數.
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The art of doing mathematics consists in finding that special case which contains all the germs of generality 
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| 2006-02-06 13:28 |  |
aa963854116
Home away from home
註冊日: 2006-01-06
發表數: 392
從未發現之89號星座
| | Re: Pick公式 | | 聽過,但是怎麼證 |
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| 2006-02-06 13:56 | |
訪客
| | Re: Pick公式 | | 等我一下,我馬上就打給你看 |
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| 2006-02-06 14:12 | |
mathematical2005
Just can't stay away
註冊日: 2005-05-17
發表數: 117
China
| | Re: Pick公式 | | 吾人將分三步證明
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The art of doing mathematics consists in finding that special case which contains all the germs of generality
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| 2006-02-06 14:32 | |
mathematical2005
Just can't stay away
註冊日: 2005-05-17
發表數: 117
China
| | Re: Pick公式 | | 1.最簡單的單純多邊形就是原子三角形,就是三個頂點之外,三邊及內部皆不含格子點之三角形
面積皆為0.5,也可用A=(b/2)+i-1來計算:3/2+0-1=0.5
so對原子三角形亦成立
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The art of doing mathematics consists in finding that special case which contains all the germs of generality
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| 2006-02-06 14:33 | |
mathematical2005
Just can't stay away
註冊日: 2005-05-17
發表數: 117
China
| | Re: Pick公式 | | 2.我們觀察到,對於任意單純多邊形都可以先分割成三角形(即三角形化),再進一步分割成原子三角形之組合(這叫做原子化)
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The art of doing mathematics consists in finding that special case which contains all the germs of generality
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| 2006-02-06 14:33 | |
mathematical2005
Just can't stay away
註冊日: 2005-05-17
發表數: 117
China
| | Re: Pick公式 | | 3.最後考慮任何單純多邊形.將它分割成c1,c2設C有b個邊界點,i個內點.且c1,c2 分別有b1,b2個邊界點,i1,i2個內點.再設c1,c2有b3個共同的邊界點,則
b= b1+ b2-2b3+2
i=i1+i2+b3-2
so (b/2)+i-1=(b1/2+i1-1)+(b2/2+i2-1)
因此Pick公式在分割下,具有加成性
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The art of doing mathematics consists in finding that special case which contains all the germs of generality
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| 2006-02-06 14:33 | |
mathematical2005
Just can't stay away
註冊日: 2005-05-17
發表數: 117
China
| | Re: Pick公式 | | 上述三個步驟綜合起來,我們就證明了Pick公式.
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The art of doing mathematics consists in finding that special case which contains all the germs of generality
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| 2006-02-06 14:33 | |
mathematical2005
Just can't stay away
註冊日: 2005-05-17
發表數: 117
China
| | Re: Pick公式 | | 不過聽說有一種圓形的證明方法
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The art of doing mathematics consists in finding that special case which contains all the germs of generality
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| 2006-02-06 14:34 | |
aa963854116
Home away from home
註冊日: 2006-01-06
發表數: 392
從未發現之89號星座
| | Re: Pick公式 | | OK |
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| 2006-02-06 14:41 | |
