Cone

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Cone
A right circular cone with the radius of its base r, its height h, its slant height c and its angle θ.
TypeSolid figure
Faces1 circular face and 1 conic surface
Euler char.2
Symmetry groupO(2)
Surface areaπr2 + πrℓ
Volume(πr2h)/3
A right circular cone and an oblique circular cone
A double cone (not shown infinitely extended)
3D model of a cone

A cone is a

geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex
.

A cone is formed by a set of

two-dimensional object in three-dimensional space. In the case of a solid object, the boundary formed by these lines or partial lines is called the lateral surface; if the lateral surface is unbounded, it is a conical surface
.

In the case of line segments, the cone does not extend beyond the base, while in the case of half-lines, it extends infinitely far. In the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a double cone on one side of the apex is called a nappe.

The axis of a cone is the straight line (if any), passing through the apex, about which the base (and the whole cone) has a circular symmetry.

In common usage in elementary

area, and that the apex lies outside the plane of the base). Contrasted with right cones are oblique cones, in which the axis passes through the centre of the base non-perpendicularly.[3]

Air traffic control tower in the shape of a cone, Sharjah Airport.

A cone with a polygonal base is called a pyramid.

Depending on the context, "cone" may also mean specifically a convex cone or a projective cone.

Cones can also be generalized to higher dimensions.

Further terminology

The perimeter of the base of a cone is called the "directrix", and each of the line segments between the directrix and apex is a "generatrix" or "generating line" of the lateral surface. (For the connection between this sense of the term "directrix" and the

directrix of a conic section, see Dandelin spheres
.)

The "base radius" of a circular cone is the radius of its base; often this is simply called the radius of the cone. The aperture of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angle θ to the axis, the aperture is 2θ. In optics, the angle θ is called the half-angle of the cone, to distinguish it from the aperture.

Illustration from Problemata mathematica... published in Acta Eruditorum, 1734
A cone truncated by an inclined plane

A cone with a region including its apex cut off by a plane is called a truncated cone; if the truncation plane is parallel to the cone's base, it is called a frustum.[1] An elliptical cone is a cone with an elliptical base.[1] A generalized cone is the surface created by the set of lines passing through a vertex and every point on a boundary (also see visual hull).

Measurements and equations

Volume

The volume of any conic solid is one third of the product of the area of the base and the height [4]

In modern mathematics, this formula can easily be computed using calculus — it is, up to scaling, the integral

Without using calculus, the formula can be proven by comparing the cone to a pyramid and applying Cavalieri's principle – specifically, comparing the cone to a (vertically scaled) right square pyramid, which forms one third of a cube. This formula cannot be proven without using such infinitesimal arguments – unlike the 2-dimensional formulae for polyhedral area, though similar to the area of the circle – and hence admitted less rigorous proofs before the advent of calculus, with the ancient Greeks using the method of exhaustion. This is essentially the content of Hilbert's third problem – more precisely, not all polyhedral pyramids are scissors congruent (can be cut apart into finite pieces and rearranged into the other), and thus volume cannot be computed purely by using a decomposition argument.[5]

Center of mass

The center of mass of a conic solid of uniform density lies one-quarter of the way from the center of the base to the vertex, on the straight line joining the two.

Right circular cone

Volume

For a circular cone with radius r and height h, the base is a circle of area and so the formula for volume becomes[6]

Slant height

The slant height of a right circular cone is the distance from any point on the circle of its base to the apex via a line segment along the surface of the cone. It is given by , where is the radius of the base and is the height. This can be proved by the Pythagorean theorem.

Surface area

The lateral surface area of a right circular cone is where is the radius of the circle at the bottom of the cone and is the slant height of the cone.[4] The surface area of the bottom circle of a cone is the same as for any circle, . Thus, the total surface area of a right circular cone can be expressed as each of the following:

  • Radius and height
(the area of the base plus the area of the lateral surface; the term is the slant height)
where is the radius and is the height.
Total surface area of a right circular cone, given radius 𝑟 and slant height ℓ
  • Radius and slant height
where is the radius and is the slant height.
  • Circumference and slant height
where is the circumference and is the slant height.
  • Apex angle and height
where is the apex angle and is the height.

Circular sector

The circular sector is obtained by unfolding the surface of one nappe of the cone:

  • radius R
  • arc length L
  • central angle φ in radians

Equation form

The surface of a cone can be parameterized as

where is the angle "around" the cone, and is the "height" along the cone.

A right solid circular cone with height and aperture , whose axis is the coordinate axis and whose apex is the origin, is described parametrically as

where range over , , and , respectively.

In implicit form, the same solid is defined by the inequalities

where

More generally, a right circular cone with vertex at the origin, axis parallel to the vector , and aperture , is given by the implicit vector equation where

where , and denotes the dot product.

Elliptic cone

elliptical cone quadric surface
An elliptical cone quadric surface

In the Cartesian coordinate system, an elliptic cone is the locus of an equation of the form[7]

It is an

affine image
of the right-circular unit cone with equation From the fact, that the affine image of a conic section is a conic section of the same type (ellipse, parabola,...), one gets:

  • Any plane section of an elliptic cone is a conic section.

Obviously, any right circular cone contains circles. This is also true, but less obvious, in the general case (see circular section).

The intersection of an elliptic cone with a concentric sphere is a spherical conic.

Projective geometry

cylinder
is simply a cone whose apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.

In

arctan, in the limit forming a right angle. This is useful in the definition of degenerate conics, which require considering the cylindrical conics
.

According to

axial pencils
(not in perspective) rather than the projective ranges used for the Steiner conic:

"If two copunctual non-costraight axial pencils are projective but not perspective, the meets of correlated planes form a 'conic surface of the second order', or 'cone'."[9]

Generalizations

The definition of a cone may be extended to higher dimensions; see convex cone. In this case, one says that a convex set C in the real vector space is a cone (with apex at the origin) if for every vector x in C and every nonnegative real number a, the vector ax is in C.[2] In this context, the analogues of circular cones are not usually special; in fact one is often interested in polyhedral cones.

An even more general concept is the

topological cone
, which is defined in arbitrary topological spaces.

See also

Notes

  1. ^ .
  2. ^ a b Grünbaum, Convex Polytopes, second edition, p. 23.
  3. ^ Weisstein, Eric W. "Cone". MathWorld.
  4. ^ .
  5. .
  6. .
  7. ^ Protter & Morrey (1970, p. 583)
  8. ^ Dowling, Linnaeus Wayland (1917-01-01). Projective Geometry. McGraw-Hill book Company, Incorporated.
  9. ^ G. B. Halsted (1906) Synthetic Projective Geometry, page 20

References

  • Protter, Murray H.; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading:
    LCCN 76087042

External links

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