Limaçon

The Pascalian worm, also Pascalian Limaçon, is a special plane curve, or more precisely an algebraic curve of 4th order. The cardioid represents a special case of Pascal's snail

It is named after the French jurist Étienne Pascal, father of the mathematician, physicist and philosopher Blaise Pascal, although Albrecht Dürer drawn already half a century ago in his book Underweysung of measurement ( p. 40) for the first time and because of the guides its construction "Spider line" has been called.

Equations of Pascal's snail

• Cartesian coordinates:

• Polar Coordinates:

• Parameter equation:

Properties of Pascal's snail

• The following geometric properties can be used to define the curve: Given a circle of diameter A, a point A on the circle, and a positive real number b. Then lie for an arbitrary point B of the circle, the two points P and P ' lie on the line AB and B have the distance b, on the Pascal's snail. So this is a special case of the general conchoid.

• The area enclosed by the Pascal's snail surface has the surface area. It should be noted that will be counted twice for the area of ​​the inner loop, since the points are circulated by the bend twice within this loop.

• Is the arc length of Pascal's snail

• For values ​​of a loop, for at least one indentation formed yet.

• For values ​​of the surface area of ​​the screw of a circle corresponding approaches ( with radius and center ) to less than 1%.