Many types of heart curves are seen on the sites of Heart Curve - Mathematische Basteleien, First heart Curve - Wolfram|Alpha and Heart Curve -- from Wolfram MathWorld as representations. Also, an interesting heart curve and its equation can be seen on the site of Equation of Heart Curve.
Heart curves are expressed in various types. In this page, some types are tried to be drawn by simple calculations. The most simple way is thought to be that the each part of a circle is pulled up by some rate into a heart curve.
On the other hand, the methods that many kinds of the better shaped heart curves are made from Cardioid are described in , and in .
As the first method, the each part of the circle is pulled up by the value of y=bx (b>0) in the right half plane of the x-y coordinates and is pulled up by the value of y=-bx (b>0) in the left half plane.
Such a method is expressed into the following equation.
If we calculate Eq.(1) as varying the several values of b with the use of computer, the each heart-like curve is drawn as shown in Fig.1.
If we use (b>0) in the right half plane of the x-y coordinates and use (b>0) in the left half plane in spite of the usage of y=bx (b>0) and y=-bx respectively in the previous section, the equation expressing another type of heart curve is obtained in the following;
If we calculate Eq.(2) as varying the several values of b with the use of computer, the each heart-like curve is drawn as shown in Fig.2.
Heart curve seen in the site of "Mathematical Heart Shape" ,which we found in June 9, 2009, is calculated with the same skill as in ours fundamentally as described below.
The equation used in this site is substantially expressed in the followings;
If we replace the variable y in Eq.(3) into and the variable x into , and if we take the constant b as , Eq.(3) is led to Eq.(2). If we show the figure of hearts around the constant value of as the additional and the inserted figures to Fig.2, Fig.3 is obtained.
4. The other methods
The other closed curve, as an example of an ellipse, is used in spite of the circle as a base figure. When asymmetry of right and left is allowed in heart curve, an inclined ellipse may also be useful.
Moreover, if the pulling-up functions are variously selected, many sort of heart curves may be obtained.
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|[Addition] By chance, a cow with heart-marked forehead as displayed above has been found in some sites I (Here in.) and II (Here in.). The design of the heart-shaped mark may tend to stick to the forehead of a cow.|