Cavalieri's principle

The Cavalieri Principle (also known as the Cavalieri Theorem or Cavalierian Principle ) is a statement from geometry that goes back to the Italian mathematician Bonaventura Cavalieri .

General

The Cavalieri principle states:

Two bodies have the same volume if all their intersecting surfaces in planes parallel to a base plane at the same heights have the same area.

Another formulation is:

If two bodies between mutually parallel planes , as well , and they are from any of these plane parallel cut so that equally large sectional areas are formed, the bodies have the same volume. ${\displaystyle E_{1}}$ ${\displaystyle E_{2}}$ ${\displaystyle E'}$

A simple illustration of the idea is provided by a block of square notes that are twisted to form a screw: it has the same volume as the cuboid that results from normal stacking. For the application of the Cavalieri principle, the notes of the twisted pile can vary in shape and size.

Classification and history

In the modern approach to analytical geometry and measure theory , Cavalieri's principle is a special case of Fubini's theorem . Cavalieri himself had no rigorous proof of the principle, but used it to justify his method of indivisibilien , which he presented in 1635 in Geometria indivisibilibus and in 1647 in Exercitationes Geometricae . With this he was able to calculate the volumes for some bodies and go beyond the results of Archimedes and Kepler . The idea of ​​reducing the calculation of volumes to areas represented an important step in the development of integral calculus .

From the Cavalieri principle it can be deduced that the volume of a 'vertically stretched' body (with a constant base area) is proportional to its height. As an example: A body, the height of which is doubled in this way, can be constructed using 2 identical starting bodies by first merging all equivalent cut surfaces and stacking them in the corresponding order of the starting body (both starting bodies are virtually pushed into one another).

Application examples

cylinder

cylinder

The sections of a cylinder with planes perpendicular to the axis of rotation are circular disks with area , if the radius of the base is designated. According to Cavalieri's principle, the volume of the cylinder is equal to that of a cuboid of the same height , the base area of ​​which has the same area, for example the edge lengths and . The volume of the cylinder is accordingly . ${\displaystyle \pi r^{2}}$ ${\displaystyle r}$ ${\displaystyle h}$ ${\displaystyle r}$ ${\displaystyle \pi r}$ ${\displaystyle \pi r^{2}\cdot h}$

Hemisphere

Vertical (above) and horizontal (below) sections through hemisphere and reference body

The intersection of a hemisphere of radius with a plane that runs parallel to the base in height is, according to the Pythagorean theorem, a circle with a radius ${\displaystyle r}$ ${\displaystyle h}$

${\displaystyle r'={\sqrt {r^{2}-h^{2}}}.}$

The area of ​​the cut surface is accordingly

${\displaystyle \pi \cdot (r')^{2}=\pi \cdot (r^{2}-h^{2}).}$

In this example, the comparison body is a cylinder with the same base area and height as the hemisphere, from which a circular cone standing on its tip was cut out. The cutting area in height is a circular ring with an outer radius and an inner radius , so the area is also ${\displaystyle h}$ ${\displaystyle r}$ ${\displaystyle h}$

${\displaystyle \pi \cdot r^{2}-\pi \cdot h^{2}=\pi \cdot (r^{2}-h^{2}).}$

So the two bodies fulfill the Cavalieri principle and therefore have the same volume. The volume of the reference body is the difference between the volumes of the cylinder and the cone, i.e.

${\displaystyle \pi \cdot r^{2}\cdot r-{\frac {1}{3}}\cdot \pi \cdot r^{2}\cdot r={\frac {2}{3}}\pi \cdot r^{3}.}$

Doubling provides the well-known formula for the spherical volume.

Relation to integral calculus

Difference between the integrals and the integral of the difference

The idea behind Cavalieri's principle is often found in integral calculus . The equation provides an example for dimensions that are one smaller, i.e. lengths of the intersections of straight lines with two surfaces

${\displaystyle \int _{a}^{b}(f(x)-g(x))\,\mathrm {d} x=\int _{a}^{b}f(x)\,\mathrm {d} x-\int _{a}^{b}g(x)\,\mathrm {d} x}$

which essentially states that the area between the function graphs of and is the same as the area under the function graph of the difference ; but this latter surface is characterized precisely by the fact that its vertical sections are of the same length as the sections of . ${\displaystyle A_{1}}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle A_{2}}$ ${\displaystyle x\mapsto f(x)-g(x)}$ ${\displaystyle A_{1}}$

In the modern theoretical approach, however, the relationship between integral and area or volume is typically established differently; the Cavalieri principle is less important.

Relation to measure theory

Cavalieri's theorem in the elementary form described above is a special case of the following more general theorem, which in turn is a special case of Fubini's theorem :

Be measurable . Then are also and for almost everyone or measurable (over or ) and it applies ${\displaystyle A\subset \mathbb {R} ^{p}\times \mathbb {R} ^{q}}$ ${\displaystyle A_{x}=\{y\in \mathbb {R} ^{q}\mid (x,y)\in A\}}$ ${\displaystyle A_{y}=\{x\in \mathbb {R} ^{p}\mid (x,y)\in A\}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \mathbb {R} ^{q}}$ ${\displaystyle \mathbb {R} ^{p}}$

${\displaystyle \lambda ^{p+q}(A)=\int _{\mathbb {R} ^{p}}\lambda ^{q}(A_{x})d^{p}x}$or , ${\displaystyle \lambda ^{p+q}(A)=\int _{\mathbb {R} ^{q}}\lambda ^{p}(A_{y})d^{q}y}$

where the -dimensional Lebesgue measure (volume) denotes. In particular, the following applies: is also measurable and applies to almost everyone , so is . The same applies to and . ${\displaystyle \lambda ^{k}}$ ${\displaystyle k}$ ${\displaystyle B\subset \mathbb {R} ^{p+q}}$ ${\displaystyle \lambda ^{q}(A_{x})=\lambda ^{q}(B_{x})}$ ${\displaystyle x}$ ${\displaystyle \lambda ^{p+q}(A)=\lambda ^{p+q}(B)}$ ${\displaystyle A_{y}}$ ${\displaystyle B_{y}}$

An analogous statement applies to any product dimensions .

Remarks

1. ↑ This condition also implies that the two bodies have the same height.

Web links

• HTML5 app on the Cavalieri principle by Walter Fendt