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Heart Curve

Nobuo YAMAMOTO

You can copy and use all the figures in this page freely.

1.   Preface

Many types of heart curves are seen on the sites of   Heart Curve - Mathematische Basteleien,   First heart Curve - Wolfram|Alpha,   Heart Curve -- from Wolfram MathWorld and How to draw a heart shaped figure 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 .

 Fig. 1
2.   Method 1

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.
,                       (1)
where b>0.
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.

 In purpose to calculate the numerical coordinates data of five species of heart curves as shown in Fig.1, a C++ program originated from Eq. (1) is given by .      Another C++ program treating a single heart curve is given by .      In the latter program, the constants are settled as b=0.8.      By executing the either C++ program, a common text file named "heart_curve.txt" including the calculated data is produced.      Each interval of these data is divided by 'comma'.      After moving these calculated data into an excel file, we obtain heart curves with the use of a graph wizard attached on the excel file.      A mask for the heart shaped trimming is prepared in this button , which is edited from the above obtained curve.      Please use this mask freely.

 Fig. 2 Fig. 3
3.   Method 2

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;
.            (2)
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;
.            (3)
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.

 In purpose to calculate the numerical coordinates data of five species of heart curves as shown in Fig.2, a C++ program originated from Eq. (2) is given by .      Another C++ program treating a single heart curve is given by .      By executing the either C++ program, a common text file named "heart_curve.txt" including the calculated data is produced.      Each interval of these data is divided by 'comma'.      After moving these calculated data into an excel file, we obtain heart curves with the use of a graph wizard attached on the excel file.      A mask for the heart shaped trimming is prepared in this button .
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.

[Application examples]
(If clicked, it will be expanded.)
 Taken in Dec. 11, '08., at Senbonmatsu farm, Tochigi, Japan.

 [Addition]  By chance, a cow with heart-marked forehead as displayed above has been found in this site..    The design of the heart-shaped mark may tend to stick to the forehead of a cow.

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Updated: 2008.01.14, edited by N. Yamamoto.
Revised on Mar. 02, 2015, Mar. 16, 2015, Nov. 09, 2015, Dec. 20, 2015, Mar. 28, 2020, May 05, 2020, Jan. 18, 2021, May 08, 2021 and Feb. 28, 2022.