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1、浅谈心形线History of cardioidsThe cardioid, a name first used by de Castillon in a paper in the Philosophical Transactions of the Royal Societyin 1741, is a curve that is the locus of a point on the circumference of circle rolling round the circumference of a circle of equal radius Of course the name mea
2、ns eart-shaped1.Its length had been found by La Hire in 170& and he therefore has some claim to be the discoverer of the curve In the notation given above the length is 16a It is a special case of the Limacon of Pascal (Etienne Pascal) and so, in a sense, its study goes back long before Castillon or
3、 La HireThere are exactly three parallel tangents to the cardioid with any given gradient. Also the tangents at the ends of any chord through the cusp point are at right angles. The length of any chord through the cusp point is 4a and the area of the cardioid is 6na2.1 .摘要:In geometry, a cardioid is
4、 the curve traced by a point on the edge of a circular wheel that is rolling around a fixed wheel of the same size. The resulting curve is roughly heart-shaped, with a cusp at the place where the point touches the fixed wheel.The cardioid is a roulette, and can be viewed as either an epicycloid with
5、 one cusp or as a member of the family of lima? ons. It is also a type of sinu soidal spiral, and is the in verse curve of a parabola with the focus as the center of inversion.3.引言：(1)来历：心形线的外形就像一颗红心，让人不免产生浪漫的联系。事实上，心形线的背 后确实有一段浪漫感人的故事，而且是关于箸名数学家，笛卡尔的。笛卡尔于1596年出生 在法国，欧洲大陆爆发黑死病吋他流浪到瑞典，认识了瑞典一个小公国18岁的公
6、主克里斯 汀，后成为她的数学老师，ULI相处使他们彼此产生爱慕之心，公主的父亲国王知道了后勃 然大怒，下令将笛卡尔处死，后因女儿求情将其流放回法国，克里斯汀公主也被父亲软禁起 来。笛卡尔回法国后不久便染上重病，他日日给公主写信，因被国王拦截，克里斯汀一直没 收到笛卡尔的信。笛卡尔在给克里斯汀寄出第十三封信后就气绝身亡了，这第十三封信内容 只有短短的一个公式：r=a(1sin8)。国王看不懂，觉得他们俩Z间并不是总是说情话的， 人发慈悲就把这封信交给一肓闷闷不乐的克里斯汀，公主看到后，立即明了恋人的意图，她 马上着手把方程的图形画出來，看到图形，她开心极了，她知道恋人仍然爱着她，原來方程 的图形
7、是一颗心的形状。这也就是著名的“心形线”。国王死后，克里斯汀登基，立即派人在欧洲四处寻找心上人，无奈斯人己故，先她走一步 了，徒留她孤零零在人I可据说这封享巻世界的另类情书还保存在欧洲笛卡尔的纪念馆里。数学故事(2) 建立：心形线，是一个圆上的固定一点在它绕着与其相切H半径相同的另外一个圆周 滚动时所形成的轨迹。(3) 图像示意：(4)图像:* 1 1 7二 1+ Cos 3丿/ 1 1 00.5-0.5X-0.571.50.5X0000.511.5-2V* 、/P 二=1- CC7、 -0.57P=L-sin 0/*/X0.5p = 4-sin 610.54. 心形线性质1 对于任意斜率，心
8、形线都拥冇三条互相平行的切线2. 心形线包围的面积是6*PI*aA2o3. 心形线的周长为16*a5. 教材相关：例1：计算心形线r=a(1+cos0) (a0)所围图形的面积解图形对称于极轴，因此所求面积是极轴以上部分面积的两倍.A = 2 +厂d &( Q 2(1 + cos &)2 d&5(1 + 2cos + cos2 0)d0=672 |+2cos+ 1 cos2 dO 2 2 3iTTQ=a2+ 2sin&+ sin2 Tier|_24Jo2例2求心形线r = a(l + cos0)(a 0)的全长(图7. 4-4).解 Fsin0,根据对称性,有S = 2帥&) + 厂，2(%2
9、a P V1 + cos2 0 + 2cos0 + sin2 OdO=2a ( a/2 + 2cos 0d 0 prQ=4町(cos pT Q=4a cosI冬I 7 4-48。sin 2 Jo=Sa 例3求心形线 = 4(l + cos0)与射线0 = 0、(p2围成的绕极轴旋转形成的旋转体体积 解 心形线的参数方程为X =4(cos + cos2) , y = 4sin(l + cos),旋转体体积V = 7T y2dx = -64 k sinll + cos siny(l + 2cos0)d0 = 16O/r6.应用：1 心形线麦克风US664A人学声音动态心形线，最常见的单向麦克风是一
10、个心形线麦克风，如此命名是因为灵 敏度模式是心形的。心形线麦克风是相似，但有严格的前敏感区和后敏感区。心形线麦克风 是一个类似超心型，除了有更多的前低后皮卡和皮卡。这三种模式，通常被用來作为声音或 语音麦克风，因为它们是在拒绝来口其他方向的声音有很好的效果。心形线麦克风实际上是一个全方位的数字，8号麦克风亞加，因为声波从示面来，正波会和 反波相抵消。The Application of Cardioid Shapel) Cardioid microphones have a heart-shaped pickup patter n. Probably the most comm on micr
11、ophones use today They reject sound coming from the back of a microph one and are progressively more sen sitive to sounds as the directio n approaches the front of the microphone. This is a unidirectional microphone, which mean it only can pick up sound form one direction the front of the microphone
12、. The cardioid pickup pattern resembles the shape of a human heart, hence the name, Cardioid There two types of cardioid pattern: Supercardioid and hypercardiod These two types of cardioid pattern will be shown below in the Cardioid Loudspeaker Application.The heart-shape pattern changes as the valu
13、e of frequency changes In other words, the shape of the graph depends on the amou nt of freque ncy that the micropho ne detects The distance betwee n the source of freque ncy and the microph one also affect the cardioid pattern: The further the distanee the lower the frequency. As you can see from t
14、he pictures below, the cardioid micropho ne gives p = 1 + si n 8. Un idirectio nal Microph one pick up cardioids patter n at 0 degree because it is at the highest sensitivity and no sensitivity at 180degree.Cardioid Microphone .Sound Waves DmnGclEve n though unidirecti onal microph one only detects freque ncy comi ng from one directi on, but the sound comi ng from behind
