The straight sides of the triangle can be measured accurately and the area of the triangle determined, but this method only gives an approximation of the circle's area because, again, it is dependent upon measurement of the circumference. .
Archimedes tried to use his spiral to improve upon this method. He started the drawing of the spiral at the center of a circle and rotated and opened the compass in such a way that the spiral reached the perimeter after one turn. This meant that the point where the spiral intersected the circle provided the point for one corner of the triangle. Since this triangle method can be carried out with equal accuracy with or without Archimedes' spiral, his method was really only of mathematical interest. However, Archimedes went on to determine a much more accurate value for pi, which advanced the determination of the area of circles in another way.
The reason parabolic spiral and hyperbolic spiral are so named is because their equation in polar system r*θ == 1 and r^2 == θ resembles the equation for hyperbola x*y == 1 and parabola x^2 == y in rectangular coordinates system.What's Fermat's involvement with parabolic spiral?.
Hyperbolic spiral is also called reciprocal spiral, because it is the inverse curve of Archemedes' spiral, with inversion center at the center. The inversion curve of all Archemedean spirals with respect to a circle on center is another Archemedean spiral. (see below).
Description.
Archimedean spiral is defined by the polar equation r == θn. Special names are given for some value of n.
n == 1, we have r == θ, Archimedes' spiral.
n== 1/2, we have r == Sqrt[θ], Fermat's spiral. (aka parabolic spiral).
n== 0, we have r = 1, circle.
n== -1/2, we have r == 1/Sqrt[θ], Lituus.
n== -1, we have r = 1/θ, reciprocal spiral. (aka hyperbolic spiral.).
Properties.
Inversion.
The inverse curve of a Archimedean spiral with respect to the center is another Archimedean spiral scaled.
