Ungula

In solid geometry, an ungula is a section or part of a solid of revolution, cut off by a plane oblique to its base. A common instance is the spherical wedge. The term ungula refers to the hoof of a horse, an anatomical feature that defines a class of mammals called ungulates.

The volume of an ungula of a cylinder was calculated by Grégoire de Saint Vincent. Two cylinders with equal radii and perpendicular axes intersect in four double ungulae. The bicylinder formed by the intersection had been measured by Archimedes in The Method of Mechanical Theorems, but the manuscript was lost until 1906.

A historian of calculus described the role of the ungula in integral calculus:

Grégoire himself was primarily concerned to illustrate by reference to the ungula that volumetric integration could be reduced, through the ductus in planum, to a consideration of geometric relations between the lies of plane figures. The ungula, however, proved a valuable source of inspiration for those who followed him, and who saw in it a means of representing and transforming integrals in many ingenious ways.

Given a solid determined by Q(x,y,z) = constant (often a quadratic form), a method of integrating the volume of the ungula determined by the plane z = 0 and the inclined plane z = ky, uses polar coordinates $x=r\sin \theta ,\ \ y=r\sin \theta .$ For a fixed θ, r can be found from Q and intersection with z = ky, and an area A(θ) computed. Then the volume of the ungula is

$V=\int _{0}^{\pi }A(\theta )\ d\theta .$

For a cylindrical ungula (radius 1), the intersection is at z = k sin θ, and $A={\frac {k}{2}}\sin \theta .$ Then

$V={\frac {k}{2}}\int _{0}^{\pi }\sin \theta \ d\theta \ =\ -{\frac {k}{2}}\cos \theta \ {\Bigg \vert }_{0}^{\pi }=k.$.

For a cone of height h on the unit disk, the intersection is found from $h(1-r)=kr\sin \theta \ \ {\text{giving}}\ \ r=(1+{\frac {k}{h}}\sin \theta )^{-1}.$ Using the area of a triangle with base 1 and height z produces a volume

$V={\frac {h}{2}}\int _{0}^{\pi }{\frac {\sin \theta \ d\theta }{1+{\frac {k}{h}}\sin \theta }}.$

References

• William Vogdes (1861) An Elementary Treatise on Measuration and Practical Geometry via Google Books