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.
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
- Cartesian coordinates:
- Polar coordinates:
- Parametric equation:
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.
- The arc length of the Pascal screws is
- A loop is created for values , and at least one more indentation.
- For values , the area of the screw approximates that of a corresponding circle (with radius and center ) to less than 1%.
Pascal snail as Trisektrix
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 .
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
- 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 )