任意多边形面积求法的证明

Standard

对于一个边不自交的多边形,其面积等于沿着顺时针或逆时针方向遍历多边形端点,每遍历到相邻两点(用向量表示)叉积模长之和的一半的绝对值。

For any polygon, the area of it is half of the sum of the modulus of all adjacents points’ cross product in clockwise or counterclockwise.

证明:

proof:

我们先来看看由一条不自交的闭合曲线围成的面积如何来求

Firstly, let we see how to compute the area of an enclosed curve (not cross itself) in 2-D.

对于“简单的图形”我们可以直接用一元积分算出来,而稍稍复杂的可以通过二重积分算出,也可以运用格林公式转化为求曲线积分。

For a ‘simple region’, we can compute its area using single variable directly. As for a complicated one, double integral can figure it out, or use green formula to convert to line integral.

正式的证明如下:

formal proof:

多边形面积求法证明

多边形面积求法证明

附:面积的严格定义见这里

PS: The rigorous defination of area see here.

我们的故事,故事中的我们

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文学、电影、音乐、游戏…等等这些传递信息的媒介,都因为突破了现实的界限、使我们的视野更加开阔而充满魅力。我们的肉体被限制在世界上的某个地方 但心灵早已凭借着这些在四处漫游。当我们欣赏着音乐、文学、电影、玩着游戏的时候,就像置身于艺术描绘的世界,经历着和“主人公”一样的事情。读书时思考着作者提出的问题,仿佛他/她真的在和自己对话;听着动听的音乐,音乐家的热情真切就穿越时空来到我们的身边;看电影玩游戏到投入的时刻,我们像是钻入了那个世界,在危急关头拯救世界。

然而,在我们沉醉于人类文明辉煌的文化的同时,这世界还有这巨量我们无法解读的故事–当我们看到来自太空遥远星系的光芒时,恐怕很难意识到这些旅行了几千万年的电磁波究竟按照怎样的方式讲述了那个遥远年代的传奇故事;当我们手忙脚乱地研究着我们自己的身体构造时,又怎会知到一串串螺旋的DNA里隐藏着多少人类身世之谜………

我们的世界里有太多的故事,有的是人类按照自己现有的认知创造的故事,有的是非人类按照它们的方式创造的故事。无论是哪一种故事,发现并解读它们并撰写新的故事,是值得我们用尽一生一同努力的事情。

在Omegle上和一个黑人谈论音乐

Chat

Omegle是一个全球匿名聊天网站,选择你感兴趣的话题,匹配有共同兴趣的人。

和一个美国黑人谈论古典音乐:

You’re now chatting with a random stranger. Say hi!

You both like classical music.

You: hi

Stranger: Hi!

You: Do you like playing guitar

You: ?

You: classical guitar

Stranger: I don’t play guitar. My dad does.

You: I love guitar & piano

Stranger: I play violin, viola, cello, and flute

You: wow!

You: amazing

Stranger: I love piano! I don’t play I wish I did

You: I always want to learn piano, but I don’t have one…

Stranger: I only played flute for marching band, so I’m only fairly decent

You: but guitar is much cheaper, and I have one

Stranger: Sweet.

Stranger: Who’s your favorite composer?

You: Yanni

You: Jesse cook

You: Yanni plays new age music

Stranger: I’ve heard of Yanni. Not really familiar.

You: Jesse cook plays New Flamenco(using guitar)

You: Mozart

Stranger: Yas. Mozart is genius

You: is my favorite classical music composor

You: composer

Stranger: Very good choice.

Stranger: I play Mozart concertos all of the time.

Stranger: My favorite is Tchaikovsky

Stranger: and Dvorak

You: I have been listening his music since I was 8 years old

Stranger: but Mozart is great classical. Lol 8 years old was a pretty important time in his life.

Stranger: that’s freaking awesome.e

You: I only know one __ of Tchaikovsky, it is swan lake

You: well, English is not my native language…so I don’t know what should I use in ___ of the last scentence..

You: sentence

You: hello?

Stranger: sorry

Stranger: It’s a “work” or a “piece”

You: thx!

Stranger: probably work since it’s a full ballet

Stranger: no problem

Stranger: What is your first language? where are you from?

You: Chinese, China

You: and you ?

You: :)

Stranger: Oh that’s awesome! I’m American.

Stranger: Haitian american.

You: Haitian ?

You: what do you mean Haitian American?

You: let me google it..

Stranger: Yes. Sure. Just my parents are from Haiti. I’m black and it’s a pretty important part of my identity.

Stranger: I don’t just say American because I don’t identify with pure American culture so much

You: wow, you are black~

You: Djembe

You: do you like Djembe

You: ?

Stranger: I don’t know what that is…

You: a kind of drum

Stranger: I’m not African, sorry.

You: It’s just a cool combination of guitar and Djembe

Stranger: I looked it up

You: All the music I love(except some pop music) aren’t very popular in China, few people know them.

Stranger: you alright?

You: alright, I can enjoy them along~

Stranger: Really?

Stranger: Hm.

You: And I played clarinets long times ago

You: in primary school

Stranger: that’s cool, my dad plays clarinet

Stranger: It’s difficult

You: At that time, we have a orchestral band with many kids who can play saxophone, clarinet, trumpet, drum, trombone…

Stranger: I’m sorry; I actually have to go. It’s getting late where I am

Stranger: It was nice talking to you.

You: well, I’m so unfamiliar with the name of music instrument that I have to translate it

You: nice to talk to you too!

You: :)

Stranger: :)

You: Bye

Stranger: bye

Stranger has disconnected.

Jesse Cook:暴风骤雨般的吉他音乐

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New Flamenco是对古典Flamenco风格的现代诠释,而Jesse Cook 是New Flamenco的顶端人物。Jesse Cook的音乐常常由这几种乐器创作出来:打击乐器如非洲鼓等;弦乐:提琴、吉他等。其中以尼龙弦的古典吉他为主乐器。这种拉丁风格的flamenco rumba乐曲因此非常动感。

一直在听新弗拉门戈的吉他曲,再加上最近开始练吉他,现在真是越来越爱这种音乐了!

好几年前第一次听到Jesse Cook的Close to Madness和Mario Takes A Walk的时候感觉就是好激情!于是专门开了个Jesse的播放列表来,到后来又听到Bombay Diner和Bogota By Bus,到现在新发现的暴风骤雨(Tempest)和Viva以及Santa Marta,感觉他有太多质量超群的作品。后来又在Youtube上看到Live(演奏现场)的视频,越发的喜欢得不得了,有时竟然会不由自主的跟着音乐跳起来!

现场

现场

虽然新买的吉他是民谣吉他,但我已经下定决心练习古典吉他的弹奏方式,期待我的进展吧。

我的吉他

我的吉他

以下是Jesse Cook的介绍中的一段,引自豆瓣音乐的一个乐评:后起之秀,先声夺人

The Camargue位于法国南部小城Arles的南部,紧邻地中海。Jesse Cook就曾经在这里度过了许多个夏天,游逛于吉普赛人居住的区域的街道,在那里他被当地的那种氛围所感染。另外他的邻居,Nicolas Reyes,大名鼎鼎的Flamenco组合Gipsy Kings的主唱,也在一定程度上给予了Jesse Cook一定的影响。在父母离婚后,Jesse Cook跟随着母亲和姐姐回到了母亲的家乡加拿大,很快,他的母亲就发现了他的音乐才能。他被送进多伦多的一家音乐学院学习吉他。在多伦多,他得以继续着他的古典和爵士吉他的学业,随后,他把这些全部抛在脑后,让自己全心的投入到口述的传统吉普赛音乐中。这些经历使得他的音乐口味更加的广泛,而所带来的影响也体现在其音乐的融合和发展上。