In
solid geometry
In mathematics, solid geometry or stereometry is the traditional name for the geometry of Three-dimensional space, three-dimensional, Euclidean spaces (i.e., 3D geometry).
Stereometry deals with the measurements of volumes of various solid fig ...
, an ungula is a region of a
solid of revolution
In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the ''axis of revolution'') that lies on the same plane. The surface created by this revolution and which bounds the solid is the ...
, cut off by a plane oblique to its base. A common instance is the
spherical wedge
In geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune (termed the wedge's ''base''). The angle between the radii lying within the bounding semidisks is the dihedral . If is a semi ...
. The term ''ungula'' refers to the
hoof
The hoof (plural: hooves) is the tip of a toe of an ungulate mammal, which is covered and strengthened with a thick and horny keratin covering. Artiodactyls are even-toed ungulates, species whose feet have an even number of digits, yet the rumin ...
of a
horse
The horse (''Equus ferus caballus'') is a domesticated, one-toed, hoofed mammal. It belongs to the taxonomic family Equidae and is one of two extant subspecies of ''Equus ferus''. The horse has evolved over the past 45 to 55 million y ...
, an anatomical feature that defines a class of
mammal
Mammals () are a group of vertebrate animals constituting the class Mammalia (), characterized by the presence of mammary glands which in females produce milk for feeding (nursing) their young, a neocortex (a region of the brain), fur or ...
s called
ungulate
Ungulates ( ) are members of the diverse clade Ungulata which primarily consists of large mammals with hooves. These include odd-toed ungulates such as horses, rhinoceroses, and tapirs; and even-toed ungulates such as cattle, pigs, giraffes, cam ...
s.
The
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
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.
Blaise Pascal
Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer.
He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...
br>Lettre de Dettonville a Carcavi
describes the onglet and double onglet, link from HathiTrust
HathiTrust Digital Library is a large-scale collaborative repository of digital content from research libraries including content digitized via Google Books and the Internet Archive digitization initiatives, as well as content digitized locally ...
The
bicylinder
In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse.
The intersection of two cylinders is ...
formed by the intersection had been measured by
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
in
The Method of Mechanical Theorems
''The Method of Mechanical Theorems'' ( el, Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος), also referred to as ''The Method'', is one of the major surviving works of the ancient Greece, ancient G ...
, but the manuscript was lost until 1906.
A historian of
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
described the role of the ungula in
integral calculus
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
:
: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.
[ Margaret E. Baron (1969) ''The Origins of the Infinitesimal Calculus'', ]Pergamon Press
Pergamon Press was an Oxford-based publishing house, founded by Paul Rosbaud and Robert Maxwell, that published scientific and medical books and journals. Originally called Butterworth-Springer, it is now an imprint of Elsevier.
History
The cor ...
, republished 2014 by Elsevier
Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell'', the ScienceDirect collection of electronic journals, '' Trends'', th ...
Google Books preview
/ref>
Cylindrical ungula
A cylindrical ungula of base radius ''r'' and height ''h'' has volume
:,.
at The Engineering Toolbox
Its total surface area is
:,
the surface area of its curved sidewall is
:,
and the surface area of its top (slanted roof) is
:.
Proof
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
Shell may refer to:
Architecture and design
* Shell (structure), a thin structure
** Concrete shell, a thin shell of concrete, usually with no interior columns or exterior buttresses
** Thin-shell structure
Science Biology
* Seashell, a hard ou ...
, whose integral is , the volume.
When the slope ''k'' equals 1 then such ungula is precisely one eighth of a bicylinder
In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse.
The intersection of two cylinders is ...
, whose volume is . One eighth of this is .
Conical ungula
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
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
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
* Spherical wedge
In geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune (termed the wedge's ''base''). The angle between the radii lying within the bounding semidisks is the dihedral . If is a semi ...
* Steinmetz solid
In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse.
The intersection of two cylinders ...
References
{{Reflist
External link
* William Vogdes (1861
An Elementary Treatise on Measuration and Practical Geometry
via Google Books
Google Books (previously known as Google Book Search, Google Print, and by its code-name Project Ocean) is a service from Google Inc. that searches the full text of books and magazines that Google has scanned, converted to text using optical c ...
Geometric shapes
Euclidean solid geometry