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What is the Heart Curve?
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The heart curve is a closed line, which has the shape of a heart.
The heart is well known as a figure on playings cards besides diamonds,
cross and spades. |
If you speak about a heart, you rather mean the heart figure than the
heart shaped curve.
... ... |
In the simplest case, a heart is formed by a square standing on its
point and two semi-circles sitting on the sides. Characteristics of the
heart figure obviously are a groove above and a point below. |
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A heart figure develops also if you set two semi-circles upon a triangle.
But here you get two unpleasant corners.
Obviously you expect that the sides are rounded. |
... ...
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If the point below is missing then don't speak of a heart but rather
of a heart shaped figure. This form, however, is more similar to
the human heart.
The figure on the left is formed by three semi-circles. |
Drawn Heart Curves top
Method 1
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1 Draw an isosceles triangle.
2 Draw the perpendiculars to the legs.
They produce a second isosceles triangle.
3 Draw two semi-circles upon the legs of the (now yellow) triangle. |
Method 2
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1 Draw two touching equal circles.
2 Draw the common tangent.
3 Draw two further (outer) tangents from one point of the tangent. |
Method 3
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1 Draw a square.
2 Draw four equal circles. The centres are the corners of the square
and the common radius is "half square side". |
Method 4
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1 Draw an ellipse.
2 Turn it about 45°.
3 Reflect it.
4 Form two hearts. Mark one. |
Method 5
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1 Draw the graph of f(x)=sin(x), 0<x<pi/2.
2 Turn the curve 90°. Reflect this curve.
3 Form a triangle of these two curves and a straight line.
4 Set two semi-circles upon the triangle. |
Method 6
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1 Draw the graph of f(x)=sin(x), -pi/2<x<pi/2.
2 Turn the curve 90°. Reflect this curve.
3 Form a triangle of these two curves and a straight line.
4 Set two semi-circles upon the triangle. |
Calculated
Heart Curves top
It is a challenge to find formulas which produce
hearts.
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You can describe method 4 by formulas.
The black ellipse has the formula 2x²-2xy+y²-1=0.The domain
is {x| x>=0}.
The red ellipse has the formula 2x²+2xy+y²-1=0. The domain
is {x| x<=0}. |
You can leave out the domain, if you isolate y and use the function f(x)=|x|.
Then y=|x|+sqrt(1-x²) and y=|x|-sqrt(1-x²) describe a heart
curve [(7)].
More curves
Sources:
(Picture 1) Book 8, Eugen Beutel (1901) (2) Buch 4, Aufgabe
8.5.5.,
(3) MathWorld (simplified formulas), (4) H.-J. Caspar's
web site (URL below) (5) My heart ;-)
From the desk of Torsten Sillke
Threedimensional
If you choose y=0 respectively x=0, you get the equation of the 2D
heart above on the left.
Source: Gabriel Taubin [for example MathWorld (URL below)]
The graphs were made by the freeware program "winplot" (Version 23.05.2000,
URL below).
Winword Hearts top
And how do artists design a heart?
The heart appears as a well known figure in character sets of programs
under MS Windows. There it is a figure of playings cards besides
diamonds, cross and spades.
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Here is a choice of well known character sets. |
The sets are Normaler Text, Arial, Courier New, Estrangelo Edessa,
Lucida Console, Symbol, Times New Roman, Webdings.
If you increase the letters from 12 to 72 you recognize the shapes.
The originally black colour is replaced by the heart colour red.
The upper part of the heart figure is formed by curves similar to arcs.
The lower lines do not approach linearly to the point but usually are at
first inward and then outward curved. That gives the heart a special
sweep.
Found at Unicode Standard, Version 4.0
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1 BLACK HEART SUIT 2665
2 WHITE HEART SUIT 2664
3 HEAVY BLACK HEART 2764 |
Pupils' Hearts top
And how do students draw a heart?
23 students of HS Lohfeld at Bad Salzuflen (Germany) were to draw a
simple heart.
The results are:
Klasse 7a, Jg.2003/2004 - thank you.
Heart Curve or Cardioid top
How to produce it
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Draw a circle (on the left, yellow) and roll an equal circle on it.
Fix one point on the moving circle line and follow this point. It describes
the heart curve or cardioid (on the right). |
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Another way of producing a heart
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A cardioid can also be seen as the envelope of circles.
Draw a (yellow) circle and a fixed point P on the circle line.
All circles that pass through the fixed point P and have their centres
on the (yellow) circle have a cardioid as an envelope. |
Area and perimeter of the heart curve
Use the polar form r=2a[1+cos (t)] as the simplest equation for
calculating the area A and the perimeter U. The origin of a coordinate
system lies in the point of the cardioid.
The perimeter is a rational number. A square with the side 4a has the
same one.
Mandelbrot Set and Cardioid
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The "main figure" of the Mandelbrot set has the form of the cardioid.
The German name Apfelmännchen (apple man) uses this shape.
Actually the main figure is a cardioid. The points of the Mandelbrot
set, which have convergent sequences, lie inside a cardioid.
Source: (5), page 208ff. There you find a proof and more references. |
The picture was taken from my page Mandelbrot
Set.
Catacaustic and Cardioid
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If light is falling on a spheric mirror (wedding ring in the sun light),
the reflecting rays form a special surface, the catacaustic. It isn't a
cardioid but a nephroid.
A cardioid develops as an envelope, if the rays start at a point on
the circle and are reflected at the circle (drawing on the right). |
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You find more on my German page Ringe.
Characteristic Curve of a Microphone
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Microphones have a certain characteristic curve. In the plane it is
a circle for the "sound-pressure-receiver" and similar to a lying eight
figure for the "sound-velocity-receiver".
Special receivers like condenser microphones have both capacities. Their
characteristic curve develops by overlaying to a cardioid. |
Source: (6), page 550
The Broken Heart top
The broken heart (Das gebrochene Herz) is a tangram game.
... ... |
Lay inside a square two circles and draw some lines. A heart develops,
which is divided in nine pieces.
The fun is to lay a heart with the pieces or to discover new figures
like those on the right. |
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Source: (1) page 22, (2) page 140-145
The Woven Heart top
1 Draw a square and two half circles sitting on it.
2 Cut it in along the red line.
3 Copy it. Paint the paper in two different colours or start
with coloured paper.
4 Insert the blue piece in the green piece.
5,6 The heart works fine with a larger number of slits as well.
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If you like have a look at the heart basket
page made by Christopher Hamkins. |
Tesselation of Hearts top
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1 Give a spiral.
2 Reflect the spiral on the end point.
3 Connect both spirals to get a double spiral.
4 Reflect the double spiral. It forms a heart together with the first
double spiral.
Many hearts lead to a tesselation (on the right). |
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The idea for this drawing came from a window grill in Venice (June 2004):
Venice is rich with heart grills.
Rosettes of Heart Figures
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four-leaf clover
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Once again a Venice photo- brightened up with some red.
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Six waffles
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A pair of swans during the courtship display |
Photos on the web site fotocommunity (URL below)
References top
(1) Pieter van Delft, Jack Botermans: Denkspiele der Welt, München
1998 ISBN 3-88034-87-0]
(2) Karl-Heinz Koch: ...lege Spiele, Köln 1987 (dumont taschenbuch1480)
[ISBN 3-7701-2097-3]
(3) Heinz Nickel u.a: Algebra und Geometrie für Ingenieur- und
Fachschulen, Frankfurt / Zürich 1966
(4) Hans Schupp, Heinz Dabrock: Höhere Kurven, BI Wissenschaftsverlag
1995 [ISBN 3-411-17221-5]
x^2 + 2( 3/5 (x^2)^(1/3) - y )^2 = 1
(5) Herbert Zeitler: Über die Hauptkörper spezieller Funktionen,
MNU, Jg.52, 1999, Heft4
(6) Bergmann-Schaefer: Lehrbuch der Expermentalphysik, Berlin, NewYork
1975 [ISBN 3 11 004861 2]
(7) Norbert Herrmann: Mathematik ist überall, Oldenbourg Verlag
2004 [ISBN 3-486-57583-X]
y = |x| +- sqrt(1 - x^2)
y = 2/3 ( (x^2 + |x| - 6)/(x^2 + |x| + 2) +-
sqrt(36 - x^2) ) (siehe auch Webseite von Thomas Jahre)
(8) Eugen Beutel: Algebraische Kurven, G.J. Göschen, Leipzig 1909-11
(x^2 + y^2 - 1)^3 = 4x^2y^3
(9) Ulrich Graf: Kabarett der Mathematik, Dresden: L. Ehlermann,
1942 Hardcover, 1.Auflage. (1943 Hardcover. 2. Auflage.)
(10) Michael Zettler: Und noch ein Herz. PM 6/99 Seite 274
y = sqrt(1 - (|x|-1)^2), y = arccos(1 -
|x|) - pi
(11) Thomas Hechinger: ... und noch ein weiteres Herz. PM
2/00 Seite 67
y = sqrt(1 - (|x|-1)^2), y = -3 sqrt(1
- sqrt(|x|/2))
(12) Mitteilung von Torsten Sillke:
x^2 + 2 (y - p*|x|^q)^2 = 1 (siehe
Schupp / Eisemann)
r = 2 sin^2(phi/4) = 1 - cos(phi/2) mit |phi|
<= pi (nach Eisemann)
r = |phi|/pi
mit |phi| <= pi. (Archimedische Spirale)
r = (1 - |phi|)(1 + 3|phi|)
mit |phi| <= 1. (nach Caspar)
(13) El-Milick, Elements d'Algebre Ornementale,
Paris, 1936:
y=(x)^(2/3)+(a²-x²)^(1/2) und y=x^(2/3)-(a²-x²)^(1/2)
and a=2
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Heart Curves on the Internet
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German:
Armin Dietz
Das Herzsymbol (Herkunft, Geschichte
und Bedeutung)
Christian Ucke und Christian Engelhardt
Kaustik
in der Kaffeetasse (.pdf-Datei)
[auch erschienen in: Physik in unserer Zeit, 29 (1998), Seite 120 bis
122]
Die Lutherrose
Ernst-Moritz-Arndt-Universität Greifswald
Herzkartenentwurf
von Stabius-Werner (Mathematik und Kunst) |
Dies ist die Formel des Randes des
"Herzkartenentwurf von Stabius-Werner"
X = t sin( 3.14 sin(t)/t
)
Y = - abs(t) cos( 3.14 sin(t)/t )
-pi <= t <= pi
(Mitteilung von Torsten Sillke) |
Fotocommunity
Frühlingsgefühle
(Thomas Th.) Zeichen
der Liebe (Heidi Zimmerli)
H.-J. Caspar
Kurven
x = a (-phi² + 40 phi +1200) sin(pi*phi/180)
y = a (-phi² + 40 phi +1200) cos(pi*phi/180)
Jan Schormann
u.a. Bézierkurven
Friedrich Krause
Problem
5: Kurvendiskussion des Herzens
Lösung
des Herzens
y = sqrt(|x|) +- sqrt(1 - x^2)
Michael Holzapfel
Herzkurve
aus zwei Funktionsteilen
y = sqrt(1 - (|x|-1)^2), y = arccos(1 -
|x|) - pi (Halbkreise und Sinuskurve)
(x^2 + y^2)(x^2 + y^2 - 2ax) - a^2y^2 =
0 (Kardioide)
NN (Matheplanet)
Geometrie
in der Teetasse
Torsten Sillke
Herzkurven
Wikipedia
Herz (Symbol)
English:
Alex Bogomolny (Cut The Knot!)
Hearty Munching
on Cardioids
Eric W. Weisstein (MathWorld)
Cardioid
Heart Curve
(x²+y²-1)³-x²y³ (siehe
auch Beutel)
x=sin(t)cos(t)ln|t| ; y=|t|0.3[cos(t)]0.5
mit 0<=t<=1
Bonne Projection
Heart Surface
[x²+(9/4)y²+z²]³-x²z³-(8/90)y²z³=0
(2x²+2y²+z²-1)³-(1/10)x²z³-y²z³=0
Circle
Catacaustic
graphicssoft.about.com
Four
Ways to 3D Hearts: Part 1
Ivars Peterson (MathTrek)
Algebraic
Hearts
Jan Wassenaar
cardioid
JOC/EFR (School of Mathematics and Statistics, University of St Andrews,
Scotland)
Cardioid
Kurt Eisemann:
x^2 + (y - 3/4 (x^2)^(1/3))^2 = 1 (Footnote)
r = sin^2( pi/8 - phi/4 ) (Footnote)
Les Reid (SMSU)
Problem#27
Problem#38
Richard Parris (peanut Software)
Program WINPLOT
Wikipedia
Heart (symbol)
Xah Lee
Cardioid
(History, Description, Formulas, Properties, Related Web Sites)
French:
Robert FERRÉOL (mathcurve)
CARDIOIDE,
DOUBLE-COEUR
Thanks to Torsten Sillke for several hints.
Gail from Oregon Coast - thank you for supporting me in my translation.
Found on the Internet ;-):
the nerdy way of drawing a heart.
http://www.mathematische-basteleien.de/heart.htm
Must they do EVERYTHING in math? ><;; lol.
Feedback: Email address on my main page
This page is also available in German.
URL of my Homepage:
http://www.mathematische-basteleien.de/
©
Jürgen Köller
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