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In
solid geometry In mathematics, solid geometry or stereometry is the traditional name for the geometry of three-dimensional, Euclidean spaces (i.e., 3D geometry). Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), inc ...
, an ungula is a region 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 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 rum ...
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 yea ...
, an anatomical feature that defines a class of mammals 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, ...
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). Th ...
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 Pascalbr>Lettre de Dettonville a Carcavi
describes the onglet and double onglet, link from HathiTrust
The bicylinder formed by the intersection had been measured by Archimedes 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 Greek polymath Ar ...
, 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 ...
, 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'', ...

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Cylindrical ungula

A cylindrical ungula of base radius ''r'' and height ''h'' has volume : V = r^2 h,.
at The Engineering Toolbox
Its total surface area is : A = \pi r^2 + \pi r \sqrt + 2 r h, the surface area of its curved sidewall is : A_s = 2 r h , and the surface area of its top (slanted roof) is : A_t = \pi r \sqrt .


Proof

Consider a cylinder x^2 + y^2 = r^2 bounded below by plane z = 0 and above by plane z = k y where ''k'' is the slope of the slanted roof: : k = . Cutting up the volume into slices parallel to the ''y''-axis, then a differential slice, shaped like a triangular prism, has volume : A(x) \, dx where : A(x) = \sqrt \cdot k \sqrt = k (r^2 - x^2) is the area of a right triangle whose vertices are, (x, 0, 0), (x, \sqrt, 0) , and (x, \sqrt, k \sqrt), and whose base and height are thereby \sqrt and k \sqrt, respectively. Then the volume of the whole cylindrical ungula is : V = \int_^r A(x) \, dx = \int_^r k (r^2 - x^2) \, dx : \qquad = k \Big( ^2 x^r - \Big x^3\Big^r \Big) = k (2 r^3 - r^3) = k r^3 which equals : V = r^2 h after substituting r k = h. A differential surface area of the curved side wall is : dA_s = k r (\sin \theta) \cdot r \, d\theta = k r^2 (\sin \theta) \, d\theta , which area belongs to a nearly flat rectangle bounded by vertices (r \cos \theta, r \sin \theta, 0), (r \cos \theta, r \sin \theta, k r \sin \theta), (r \cos (\theta + d\theta), r \sin (\theta + d\theta), 0), and (r \cos (\theta + d\theta), r \sin (\theta + d\theta), k r \sin (\theta + d\theta)), and whose width and height are thereby r \, d\theta and (close enough to) k r \sin \theta, respectively. Then the surface area of the wall is : A_s = \int_0^\pi dA_s = \int_0^\pi k r^2 (\sin \theta) \, d\theta = k r^2 \int_0^\pi \sin \theta \, d\theta where the integral yields - cos \theta0^\pi = - 1 - 1= 2 , so that the area of the wall is : A_s = 2 k r^2 , and substituting r k = h yields : A_s = 2 r h. The base of the cylindrical ungula has the surface area of half a circle of radius ''r'': \pi r^2 , 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 r \sqrt, so that its area is : A_t = \pi r \cdot r \sqrt = \pi r \sqrt and substituting k r = h yields : A_t = \pi r \sqrt. ∎ Note how the surface area of the side wall is related to the volume: such surface area being 2kr^2, multiplying it by dr 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 o ...
, whose integral is k r^3, the volume. When the slope ''k'' equals 1 then such ungula is precisely one eighth of a bicylinder, whose volume is r^3. One eighth of this is r^3.


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 : V = where : H = is the height of the cone from which the ungula has been cut out, and : I = \int_0^\pi \sin \theta \, d\theta . The surface area of the curved sidewall is : A_s = I . 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: :\lim_ \Big(I - \Big) = \lim_ \Big( \int_0^\pi \sin \theta \, d\theta - \Big) = 0 so that :\lim_ V = \cdot = k r^3 , :\lim_ A_s = \cdot = 2 k r^2 , and : \lim_ A_t = \pi r^2 = \pi r^2 \sqrt = \pi r \sqrt, which results agree with the cylindrical case.


Proof

Let a cone be described by : 1 - = where ''r'' and ''H'' are constants and ''z'' and ''ρ'' are variables, with : \rho = \sqrt, \qquad 0 \le \rho \le r and : x = \rho \cos \theta, \qquad y = \rho \sin \theta . Let the cone be cut by a plane : z = k y = k \rho \sin \theta. Substituting this ''z'' into the cone's equation, and solving for ''ρ'' yields : \rho_0 = 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 : z_0 = H \Big(1 - \Big) . So along the direction of angle ''θ'', a cross-section of the conical ungula looks like the triangle : (0,0,0) - (\rho_0 \cos \theta, \rho_0 \sin \theta, z_0) - (r \cos \theta, r \sin \theta, 0) . Rotating this triangle by an angle d\theta about the ''z''-axis yields another triangle with \theta + d\theta, \rho_1, z_1 substituted for \theta, \rho_0, and z_0 respectively, where \rho_1 and z_1 are functions of \theta + d\theta instead of \theta. Since d\theta is infinitesimal then \rho_1 and z_1 also vary infinitesimally from \rho_0 and z_0, 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 r d\theta, a length at the top of \Big(\Big) r d\theta, and altitude \sqrt, so the trapezoid has area :A_T = \sqrt = r\,d\theta \sqrt. An altitude from the trapezoidal base to the point (0,0,0) 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: : V = \int_0^\pi \sqrt r\,d\theta = \int_0^\pi r^2 d\theta = \int_0^\pi (2 H - k y_0) y_0 \,d\theta where : y_0 = \rho_0 \sin \theta = = Substituting the right hand side into the integral and doing some algebraic manipulation yields the formula for volume to be proven. For the sidewall: :A_s = \int_0^\pi A_T = \int_0^\pi r \sqrt\,d\theta = \int_0^\pi (2 H - z_0) y_0 \,d\theta and the integral on the rightmost-hand-side simplifies to H^2 r I. ∎ As a consistency check, consider what happens when ''k'' goes to infinity; then the conical ungula should become a semi-cone. : \lim_ \Big(I - \Big) = 0 : \lim_ V = \cdot = \Big( \pi r^2 H\Big) which is half of the volume of a cone. : \lim_ A_s = \cdot = \pi r \sqrt which is half of the surface area of the curved wall of a cone.


Surface area of top part

When k = H / r , the "top part" (i.e., the flat face that is not semicircular like the base) has a parabolic shape and its surface area is : A_t = r \sqrt . When k < H / r then the top part has an elliptic shape (i.e., it is less than one-half of an ellipse) and its surface area is : A_t = \pi x_ (y_1 - y_m) \sqrt \Lambda where : x_ = \sqrt , : y_1 = , : y_m = , : \Lambda = - \arcsin (1 - \lambda) - \sin (2 \arcsin (1 - \lambda)) , and : \lambda = . When k > H / r then the top part is a section of a hyperbola and its surface area is : A_t = \sqrt (2 C r - a J) where : C = = y_m , : y_1 is as above, : y_2 = , : a = , : \Delta = , : J = B + \log \Biggr, \Biggr, , where the logarithm is natural, and : B = \sqrt .


See also

* Spherical wedge * Steinmetz solid


References

{{Reflist


External link

* William Vogdes (1861
An Elementary Treatise on Measuration and Practical Geometry
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