浅谈心形线

浅谈心形线
History of cardioids
The 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 means 'heart-shaped'.
Its length had been found by La Hire in 1708, 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 La Hire.
There 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 6πa2.
1.摘要：In geometry, a cardioid is 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 one cusp or as a member of the family of limaçons. It is also a type of sinusoidal spiral, and is the inverse 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，而且是关于著名数学家，笛卡尔的。

笛卡尔于1596年出生在法国，欧洲大陆爆发黑死病时他流浪到瑞典，认识了瑞典一个小公国18岁的公主克里斯汀，后成为她的数学老师，日日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(1-sinθ）。

国王看不懂，觉得他们俩之间并不是总是说情话的，大发慈悲就把这封信交给一直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来，看到图形，她开心极了，她知道恋人仍然爱着她，原来方程的图形是一颗心的形状。

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，无奈斯人已故，先她走一步了，徒留她孤零零在人间...
据说这封享誉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且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：
4. 心形线性质
1.对于任意斜率，心形线都拥有三条互相平行的切线
2.心形线包围的面积是6*PI*a^2。

3.心形线的周长为16*a
5.教材相关：
例1：计算心形线r=a(1+cos θ) （a>0）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. ⎰⎰
+==ππ
θθθ0220
2
)cos 1(2
12
d a d r A
例2求心形线 )0)(cos 1(>+=a a r θ的全长(图7.4-4). 解 θθsin )(a r -='，根据对称性，有 ⎰
'+=π
θθθ0
22)()(2
d r r s
θθθθπ
d a ⎰+++=0
22sin cos 2cos 12 ⎰
+=π
θθ0
cos 222d a
⎰=π
θθ
)2(cos 4d a ⎰
=π
θθ
2
cos 4d a
π
θ02sin 8⎥⎦⎤⎢⎣
⎡
=a =a 8.
例3 求心形线)cos 1(4ϕρ+=与射线0=ϕ、2/πϕ=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 )cos (cos 42ϕϕ+=，)cos 1(sin 4ϕϕ+=y ，旋转体体积
dx y V 28
0⎰=π=ϕϕϕϕϕππd )cos 21(sin )cos 1(sin 64220
2
/+⋅+-⎰
=π160
6.应用：
1.心形线麦克风
y
图7.4-4
US664A大学声音动态心形线,最常见的单向麦克风是一个心形线麦克风，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，通常被用来作为声音或语音麦克风，因为它们是在拒绝来自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号麦克风叠加，因为声波从后面来，正波会和反波相抵消。

The Application of Cardioid Shape
1)Cardioid microphones have a heart-shaped pickup pattern. Probably the most common microphones use today. They reject sound coming from the back of a microphone and are progressively more sensitive to sounds as the direction 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. 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 value of frequency changes. In other words, the shape of the graph depends on the amount of frequency that the microphone detects. The distance between the source of frequency and the microphone also affect the cardioid pattern: The further the distance the lower the frequency. As you can see from the pictures below, the cardioid microphone gives p = 1 + sin θ. Unidirectional Microphone pick up cardioids pattern at 0 degree because it is at the highest sensitivity and no sensitivity at 180degree.
Even though unidirectional microphone only detects frequency coming from one direction, but the sound coming from behind the microphone is also detected. This is why there are two loops at the bottom.
You have probably seen a cardioid microphone in action when a reporter moves a handheld microphone back and forth in a live TV interview. This is because cardioid microphones can exhibit a "proximity effect" - a boost in low-to-mid frequencies - as the distance between the sound source and the microphone decreases. This can sometimes cause unwanted distortion of the audio signal if the distance between the sound source and microphone is not controlled.
2)I found one experiment that uses the application of cardioid pattern in Loud Speaker.
It is remarkable that cardioid microphones are used very often to reject incoming sounds from the rear of the microphone. In sound reproduction however, cardioid loudspeakers are less common. One example is the Philips-Bosch cardioid column loudspeaker. These speakers make use of ‘acoustic filters’ in the form of slits in the outer part of the enclosure. The proposed optimisation technique can be applied to other array types as well (e.g. bass arrays). Due to their ‘unidirectional’ behaviou r these cardioid loudspeaker arrays are expected to have many acoustic benefits: improved indoor bass response, higher direct-to-reverberant ratio, higher gain before- feedback, improved echo-reduction in delayed set-ups, etc.
In theory, a cardioid loudspeaker can simply be made with two opposite polarity monopole sources separated by a distance Dl. The signal to one of the sources should be delayed by a time Dl /c, where c is the speed of sound, as shown in Fig 1. The sound field of this cardioid is given by
Where r1 and r2 are the distances from the monopole loudspeakers to a far field receiver position and k the wave number (2f/c). The complex factors Afront and Aback are given by
with =1 in this situation.
Fig 1: Cardioid source made by two closely spaced monopole loudspeakers with intermediate distance l, processed with complex weighting factors Afront and Aback. For small loudspeaker distances compared to the wavelength (k l<<1) the sound field of the cardioid source can be approximated in the far field (r0>>l) by
Where r0 is the distance from the centre between the two loudspeakers to a far field position at angle
. In this expression the directivity D is given by
Note that the cardioid frequency response is proportional with the intermediate distance l and with the wave number. Consequently, for small values of k l the cardioid source becomes very inefficient.
In the case of axial symmetric sources the directivity factor Q is given by3
Hence,
Using the above equation, we find that Q=3 for the cardioid source (=1).
Inserting =0 or =1/3 in the directivity D equation, the well-known dipole (figure-8) and hypercardioid characteristics are obtained, respectively. It can be shown by differentiating the equation above that for =1/3 (i.e., hypercardioid source) the directivity factor is maximum.
3) Radio Direction finding(Yagi-Uda Antenna)
Direction finding refers to the establishment of the direction from which a received signal was transmitted. This can refer to radio or other forms of wireless communication. Direction finding often requires an antenna that is directional - that is, more sensitive in certain directions than in others. Many antenna designs exhibit this property. For example, a Yagi antenna has quite pronounced directionality, so the source of a transmission can be determined simply by pointing it in the direction where the maximum signal level is obtained. However, to establish direction to great accuracy requires much more sophisticated techniques.
Yagi-Uda Antenna(TV)
How does Yagi-Uda Antenna works to produce Cardioid pattern?here I use Antenna array to explain the answer. An array of antenna elements is a spatially extended collection of N similar radiators or elements, where N is a countable number bigger than 1, and the term "similar radiators" means that all the elements have the same polar radiation patterns, orientated in the same direction in 3-d space. The elements don't have to be spaced on a regular grid, neither do they have to have the same terminal voltages, but it is assumed that they are all fed with the same frequency and that one can define a fixed amplitude and phase angle for the drive voltage of each element. The polar radiation pattern of a single element is called the "element pattern". It is possible for the array to be built recursively; for example the element may itself be an array, as would be the case if we had an array of Yagi-Uda antennas. A Yagi-Uda antenna may be thought of as an array of dipoles with different amplitudes and phases of the dipole currents. The array pattern is the polar radiation pattern which would result if the elements were replaced by isotropic radiators, having the same amplitude and phase of excitation as the actual elements, and spaced at points on a grid corresponding to the far field phase centres of the elements.
The radiated field strength at a certain point in space, assumed to be in the far field, is calculated by adding the contributions of each element to the total radiated fields. The field strengths fall off as 1/r where r is the distance from the isotrope to the field point. We must take into account any phase angle of the isotrope excitation, and also the phase delay which is due to the time it takes the signal to get from the source to the field point. This phase delay is expressed as 2 Pi radians times (r/lambda) where lambda is the free space wavelength of the radiation. Contours of equal field strength may be interpreted as an amplitude polar radiation pattern. Contours of the squared modulus of the field strength may be interpreted as a power polar radiation pattern.
By changing the wavelenght, amplitude and the phase, the pattern will change and a heart shaper pattern is obtained at some point.
参考文献
《数学故事》
/wiki
<<高等数学>>
<<数学与美>>
/Personal/D.Jefferies/antarray.html
/Personal/D.Jefferies/yagiuda.html
/blog/polar-coordinates-and-cardioid-microphones/246
/cardioidmicrophone
design and application of DDS-controlled,cardioid loudspeaker arrays <<by Evert Start and Gerald van Beuningen>>]
The Result 结论
心形线是一种广泛应用于生活的曲线,它不仅用在话筒,手机,发动机等处均有应用,还在示爱方面有奇效。

我们小组从心形线的故事来历，实际应用，教材相关，图形画法,性质介绍等方面来肤浅的介绍了一下心形线，使大家对心形线有一个感性的认识。

不足之处，还请海涵。