Cone

A ( finite ) cone or cone is a geometric body that arises when one connects all points of an in-plane, bounded round flat piece straight with a point (peak or apex) outside the plane. The patch is called base whose boundary line, the directrix and the point of the tip or the apex of the cone. Thus, a cone having a tip ( the apex ), an edge ( the trajectory ), and two surfaces ( the cladding and the base ).

The distance between the tip and base is the height of the cone and is always perpendicular to the base surface, especially at very oblique cones. The links of the tip with the trajectory hot surface lines, their union forms the conical shell or the shell surface.

Straight and helical bevel

When the delivery is in the geometry of a cone, the special case of the right circular cone is often meant. Under a circular cone is defined as a body that is defined by a circle ( base circle or base circle ) and a point outside the plane of the circle ( tip of the cone ).

The plane in which the base circle is located, is called a basis (county ) level. With the radius r of the cone is normally understood the radius of the base circle. The line through the center of the base circle and the tip is called the axis of the cone. The height h of the cone, the spacing of the tip from the base plane; This distance shall be measured perpendicular to the base plane.

If the axis is perpendicular to the base plane, then there is a right circular cone or reamer. Otherwise, one speaks of an oblique circular cone or elliptical cone. Each elliptical cone has two directions in which a plane is to be cut with a circle; this fact makes the stereographic projection of a circular loyalty advantage.

The term " rotary cone" suggests that there is a body of revolution. It is formed by rotation of a right-angled triangle about one of its other two sides. In this case, the metal lines (ie, the links of the (boundary ) points of the base circle with the tip ) also called generating (m ) because they " produce " the mantle. The opening angle is twice the angle between the generatrices and the axis of a rotary cone. The angle between the generatrices and the axis is half the aperture angle.

A rotary cone with opening angle of 60 ° is called equilateral cone. This designation is explained as follows: If you cut such a cone with a plane containing the axis, we obtain an equilateral triangle.

Especially in the art, the word cone (from Latin conus ) is used for the rotating cone. The corresponding adjective referred conical objects with the shape of a cone or a rotary (rotary) truncated cone.

Particularly in connection with conic sections, the word " cone" is also used in the sense of the undermentioned double cone.

Formulas

• Volume of any circular cone:

• Surface area of ​​the lateral surface of a right circular cone:

• Total surface of a right circular cone:

• Generating line of a right circular cone:

• Height of a right circular cone:

• A right circular cone radius:

• Moment of inertia of a filled circular cone ( the axis of rotation through the tip and base side center):

• Moment of inertia of a hollow circular cone ( the axis of rotation through the tip and base side center):

• Opening angle

The opening angle is twice the angle between the generatrices and the axis of a rotary cone.

Using the trigonometric functions, the following formulas result.

Evidence

Volume

In elementary geometry of the volume formula is often justified by the principle of Cavalieri. One compares the given circular cone with a pyramid of the same base and height. For planes parallel to the base surface at any distance follows from the laws of similarity or the central extension that the corresponding cut surfaces equal in area possess. Therefore, the two bodies must match in volume. The pyramids of the base G and height h valid volume formula

Can therefore be transmitted to the cone. Along with the formula for the circular surface obtained

It is also possible to approximate the cone by a pyramid with a regular n-gon as a base ( as n approaches infinity).

Another proof ( specifically illustrated for the right circular cone here ) sets the integral calculus as a tool. It is a Cartesian coordinate system is used, with the apex at the origin ( 0 | 0) and the center of the base circle at the point ( h | 0) lie. One can now put together the cones think of an infinite number of infinitesimal cylindrical discs ( infinitesimal ) Height ( thickness ) dx. Since the distance of such a cylindrical disk x is given by the cone tip by the coordinate, is considered by the theorem:

The total volume of the rotary cone corresponds to the entirety of all these infinitesimal cylinder. To calculate the definite integral of one forms with the integration limits 0 and h:

Thus one comes to the well-known formula:

Lateral surface

The lateral surface of a right circular cone is curved, but can be unwound to a circular sector. The radius of this sector is in agreement with the length of a generatrix of the cone ( s ). The central angle of the circular sector can be determined by a ratio equation. It is related to the 360 ° angle as the circular arc length ( the circumference of the base circle ) for the entire circumference of a circle of radius p

The desired area of ​​the lateral surface is now obtained from the formula for the area of ​​a circular sector.

Central angle

The center angle of the outgoing formula can be calculated.

Double Cone

A double cone is formed as a surface of revolution of a straight line to a not at right angles intersecting axis. It creates two rotating cone with the same opening angle and a common axis, which touch at the top. If you cut such an infinite double cone with a plane caused the so-called conic sections: circle, ellipse, parabola, hyperbola.

Generalizations

They generalize the properties of the ( infinite) cone to consist of beams with a common starting point for the conical volume, which then, for example, also an infinite pyramid belongs. Of particular interest is the convex cones, which play a role in the linear optimization.

The term of the order cone is important: If we define a partial order means, wherein a convex and closed cone, so is this reflexive, antisymmetric, transitive, and multiplicative and additive tolerated. This is such a partial order is a generalization of the ( componentwise ) arithmetic partial order, which is the positive orthant based. One possible definition of such a cone is:

Be a real Banach space and a non-empty subset of. is called cone if the following conditions are met:

Is omitted, the fourth condition is obtained, a possible definition of a wedge.

• As a further generalization of the cone can permit arbitrary bases. The cone is obtained then the convex hull of the base and another point outside the surface ( the apex ). In this sense, a pyramid, a cone over a polygon.
• In the synthetic geometry of the cone concept is defined for certain quadratic quantities in projective geometry of arbitrary dimension. See Quadratic lot # cone.

In topology is meant by the cone over a topological space the space obtained from the product by identifying all points in (the " apex ").

The corresponding " double cone " ( by additional identifying ) is also referred to as device to attach or suspension.