Pascal snail

The Pascal snail , also called Pascal limaçon , is a special flat curve , more precisely an algebraic curve of the 4th order. The cardioid is a special case of the Pascal snail.

Pascal snails

It is named after the French lawyer Étienne Pascal , the father of the mathematician, physicist and philosopher Blaise Pascal , although Albrecht Dürer drew it for the first time half a century earlier in his book Underweysung of Measurement and called it "spider line" because of the auxiliary lines in his construction .

Pascal snail equations

Pascal screw with parameters and

{\ displaystyle a}

{\ displaystyle b}

Cartesian coordinates: ${\ displaystyle (x ^ {2} + y ^ {2} -ax) ^ {2} -b ^ {2} (x ^ {2} + y ^ {2}) \, = \, 0}$
Polar coordinates: ${\ displaystyle r = a \ cos \ varphi + b}$
Parametric equation: ${\ displaystyle x = a (\ cos t) ^ {2} + b \ cos t; \ qquad y = a \ cos t \ sin t + b \ sin t}$

Properties of the Pascal snail

The following geometrical property can be used to define the curve: Given a circle with diameter a, a point A on this circle and a positive real number b. Then for an arbitrary point B of the circle the two points P and P ', which lie on the straight line AB and have the distance b from B, lie on the Pascal screw. It is therefore a special case of the general conchoid .
The area enclosed by the Pascal snail has the area . It should be noted that the area of the inner loop is counted twice, as the points inside this loop are covered twice by the curve. ${\ displaystyle {\ tfrac {1} {2}} a ^ {2} \ pi + b ^ {2} \ pi}$ ${\ displaystyle b <a}$
The arc length of the Pascal screws is ${\ displaystyle \ int _ {0} ^ {2 \ pi} \ mathrm {\ sqrt {a ^ {2} + 2abcos {\ varphi} + b ^ {2}}} \, \ mathrm {d} \ varphi}$
A loop is created for values , and at least one more indentation. ${\ displaystyle b <a}$ ${\ displaystyle b <2a}$
For values , the area of the screw approximates that of a corresponding circle (with radius and center ) to less than 1%. ${\ displaystyle b> 7a}$ ${\ displaystyle r = b}$ ${\ displaystyle M (1 | 0)}$

Pascal snail as Trisektrix

Pascal snail as Trisektrix with

{\ displaystyle a = | AB |, \, b = | MB |}

The Pascal snail with the parameter relationship is also known as a trisectrix , as it can be used to divide an angle into three. To do this, choose a point B on one of the legs of the given angle with point A and construct a Pascal snail with | AB | as the diameter of its associated circle with center M and the radius of this circle as the distance parameter b. The circle A with radius b intersects the second leg in C. now intersects the track CM the inner loop of the Pascal snail in P 'and the track AP' angle formed by the segment AB is one-third of the output angle, ie . ${\ displaystyle b = {\ tfrac {a} {2}}}$ ${\ displaystyle \ alpha}$ ${\ displaystyle \ beta}$ ${\ displaystyle \ beta = {\ tfrac {\ alpha} {3}}}$

Modern application in astronomy

The shadows of rotating black holes can be described with a very high degree of accuracy by limaçons, which is a very great simplification compared to the computationally intensive ray tracing method.

literature

Higher math at your fingertips . Vieweg, 1974, pp. 719–722 ( excerpt (Google) )
IN Bronshtein, KA Semendyayev, Gerhard Musiol, Heiner Mühlig: Handbook of Mathematics . Springer, 2015, pp. 98–99 ( excerpt (Google) )
EH Lockwood: Book of Curves . Cambridge University Press, 1961, pp. 44-51
J. Dennis Lawrence: A Catalog of Special Plane Curves . Dover Publications, 1972, pp. 113–117 ( excerpt (Google) )
Robert C. Yates: Curves and their Properties . National Council of Teachers of Mathematics, 1974, pp. 148-151

Web links

Commons : Pascalsche Schnecke - collection of images, videos and audio files

Dürer's construction of Pascal's snail in his underweysung the measurement (p. 40) (PDF; 197 kB)
John J. O'Connor, Edmund F. Robertson : Pascal's Limacon. In: MacTutor History of Mathematics archive .
Eric W. Weisstein : Limacon . In: MathWorld (English).
Pascal limaçon in the Encyclopaedia of Mathematics

Individual evidence

↑ Albrecht Dürer : Underweysung the measurement . P. 40
↑ Andreas de Vries: Shadows of rotating black holes approximated by Dürer-Pascal limaçons (PDF) In: Ralf Wilhelm Muno (Ed.): Annual journal of the Bochumer Interdisciplinary Society eV 2003 . Ibidem-Verlag, Stuttgart 2005, ISBN 3-89821-456-7 , pp. 1–20 ( excerpt from Google Books )

[1] Albrecht Dürer : Underweysung the measurement . P. 40

[2] Andreas de Vries: Shadows of rotating black holes approximated by Dürer-Pascal limaçons (PDF) In: Ralf Wilhelm Muno (Ed.): Annual journal of the Bochumer Interdisciplinary Society eV 2003 . Ibidem-Verlag, Stuttgart 2005, ISBN 3-89821-456-7 , pp. 1–20 ( excerpt from Google Books )