In solid geometry, an ungula is a region of a solid of revolution, cut off by a plane oblique to its base.[1] 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.[2] Two cylinders with equal radii and perpendicular axes intersect in four double ungulae.[3] 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.[4]: 146 

Cylindrical ungula

edit
 
Ungula of a right circular cylinder.

A cylindrical ungula of base radius r and height h has volume

 ,.[5]

Its total surface area is

 ,

the surface area of its curved sidewall is

 ,

and the surface area of its top (slanted roof) is

 .

Proof

edit

Consider a cylinder   bounded below by plane   and above by plane   where k is the slope of the slanted roof:

 .

Cutting up the volume into slices parallel to the y-axis, then a differential slice, shaped like a triangular prism, has volume

 

where

 

is the area of a right triangle whose vertices are,  ,  , and  , and whose base and height are thereby   and  , respectively. Then the volume of the whole cylindrical ungula is

 
 

which equals

 

after substituting  .

A differential surface area of the curved side wall is

 ,

which area belongs to a nearly flat rectangle bounded by vertices  ,  ,  , and  , and whose width and height are thereby   and (close enough to)  , respectively. Then the surface area of the wall is

 

where the integral yields  , so that the area of the wall is

 ,

and substituting   yields

 .

The base of the cylindrical ungula has the surface area of half a circle of radius r:  , and the slanted top of the said ungula is a half-ellipse with semi-minor axis of length r and semi-major axis of length  , so that its area is

 

and substituting   yields

 . ∎

Note how the surface area of the side wall is related to the volume: such surface area being  , multiplying it by   gives the volume of a differential half-shell, whose integral is  , the volume.

When the slope k equals 1 then such ungula is precisely one eighth of a bicylinder, whose volume is  . One eighth of this is  .

Conical ungula

edit
 
Ungula of a right circular cone.

A conical ungula of height h, base radius r, and upper flat surface slope k (if the semicircular base is at the bottom, on the plane z = 0) has volume

 

where

 

is the height of the cone from which the ungula has been cut out, and

 .

The surface area of the curved sidewall is

 .

As a consistency check, consider what happens when the height of the cone goes to infinity, so that the cone becomes a cylinder in the limit:

 

so that

 ,
 , and
 ,

which results agree with the cylindrical case.

Proof

edit

Let a cone be described by

 

where r and H are constants and z and ρ are variables, with

 

and

 .

Let the cone be cut by a plane

 .

Substituting this z into the cone's equation, and solving for ρ yields

 

which for a given value of θ is the radial coordinate of the point common to both the plane and the cone that is farthest from the cone's axis along an angle θ from the x-axis. The cylindrical height coordinate of this point is

 .

So along the direction of angle θ, a cross-section of the conical ungula looks like the triangle

 .

Rotating this triangle by an angle   about the z-axis yields another triangle with  ,  ,   substituted for  ,  , and   respectively, where   and   are functions of   instead of  . Since   is infinitesimal then   and   also vary infinitesimally from   and  , so for purposes of considering the volume of the differential trapezoidal pyramid, they may be considered equal.

The differential trapezoidal pyramid has a trapezoidal base with a length at the base (of the cone) of  , a length at the top of  , and altitude  , so the trapezoid has area

 .

An altitude from the trapezoidal base to the point   has length differentially close to

 .

(This is an altitude of one of the side triangles of the trapezoidal pyramid.) The volume of the pyramid is one-third its base area times its altitudinal length, so the volume of the conical ungula is the integral of that:

 

where

 

Substituting the right hand side into the integral and doing some algebraic manipulation yields the formula for volume to be proven.

For the sidewall:

 

and the integral on the rightmost-hand-side simplifies to  . ∎

As a consistency check, consider what happens when k goes to infinity; then the conical ungula should become a semi-cone.

 
 

which is half of the volume of a cone.

 

which is half of the surface area of the curved wall of a cone.

Surface area of top part

edit

When  , the "top part" (i.e., the flat face that is not semicircular like the base) has a parabolic shape and its surface area is

 .

When   then the top part has an elliptic shape (i.e., it is less than one-half of an ellipse) and its surface area is

 

where

 ,
 ,
 ,
 , and
 .

When   then the top part is a section of a hyperbola and its surface area is

 

where

 ,
  is as above,
 ,
 ,
 ,
 ,

where the logarithm is natural, and

 .

See also

edit

References

edit
  1. ^ Ungula at Webster Dictionary.org
  2. ^ Gregory of St. Vincent (1647) Opus Geometricum quadraturae circuli et sectionum coni
  3. ^ Blaise Pascal Lettre de Dettonville a Carcavi describes the onglet and double onglet, link from HathiTrust
  4. ^ Margaret E. Baron (1969) The Origins of the Infinitesimal Calculus, Pergamon Press, republished 2014 by Elsevier, Google Books preview
  5. ^ Solids - Volumes and Surfaces at The Engineering Toolbox
edit
  NODES
HOME 1
inspiration 1
languages 1
Note 2
os 28
web 1