Parabolic arbelos

definition

If you replace all semicircular arcs in the Arbelos figure with parabolic arcs , the width (= length of the base line) is always twice as large as the height (= length of the vertex ordinate), the modified figure is called Parabolic Arbelos .

( Note : In a modified definition, Sondow uses the term Parbelos , which differs from the above definition.)

property

The area measure of the green colored parabolic arbelos (first picture) is equal to the area measure of the green colored parabolic segment over AD with the height DB (second picture).

proof

The Archimedes' formula for parabolic segments is used, which Archimedes proves in one of his main writings with the title Quadrature of the Parabola , Latin De quadratura parabolae . This formula says that the area of a parabolic segment with the base g and the height h has the dimension number . ${\ displaystyle \ textstyle {\ frac {2} {3}}}$ ${\ displaystyle \ mathrm {gh}}$

In the following denote the length of the route AD. Since the basic figures of the Parabolic Arbelos are geometrically similar to one another for all lengths of the segment AB, the length of the segment AB is selected to be 2 in the following without restricting the general validity. ${\ displaystyle p}$

Area measure of the parabolic segment over AD:

{\ displaystyle {\ frac {2} {3}} \ cdot p \ cdot {\ frac {p} {2}} = {\ frac {2} {3}} \ cdot {\ frac {p ^ {2} } {2}} = {\ frac {p ^ {2}} {3}}}

Area dimension of the parabolic segment over DB:

{\ displaystyle {\ frac {2} {3}} \ cdot (2-p) \ cdot {\ frac {2-p} {2}} = {\ frac {2} {3}} \ cdot {\ frac {(2-p) ^ {2}} {2}} = {\ frac {(2-p) ^ {2}} {3}}}

The sum of both areas is:

{\ displaystyle {\ frac {1} {3}} \ cdot (p ^ {2} + (2-p) ^ {2}) = {\ frac {2} {3}} \ cdot (p ^ {2 } -2p + 2)}

The area measure of the parabolic segment over AB is:

{\ displaystyle {\ frac {2} {3}} \ cdot 2 \ cdot 1 = {\ frac {4} {3}}}

As a measure of the area of the Parbelos this results in:

{\ displaystyle {\ frac {4} {3}} - {\ frac {2} {3}} \ cdot (p ^ {2} -2p + 2) = {\ frac {4} {3}} - { \ frac {2} {3}} p ^ {2} + {\ frac {4} {3}} p - {\ frac {4} {3}} = - {\ frac {2} {3}} p ^ {2} + {\ frac {4} {3}} p = {\ frac {2} {3}} p \ cdot (2-p)}

According to the Archimedean formula for parabolic segments, the area of the parabolic segment with the base side length and the height also has the dimension . ${\ displaystyle p}$ ${\ displaystyle (2-p)}$ ${\ displaystyle \ textstyle {\ frac {2} {3}} p \ cdot (2-p)}$