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In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. Each of those lines is called a generatrix of the surface. Geometry (Greek γεωμετÏία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ...
An open surface with X-, Y-, and Z-contours shown. ...
A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ...
In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...
In general, a conical surface consists of two identical unbounded halves joined by the vertex. However, in some cases, these two halves may intersect, or even coincide. Every conic surface is ruled and developable. In geometry, a surface is ruled if through every point of there is a straight line that lies on . ...
A developable surface is a surface that can be flattened onto a plane without distortion (i. ...
In particular, if the directrix is a circle C, and the apex is located on the circle's axis (the line that contains the center of C and is perpendicular to its plane), one obtains the right circular conical surface. This special case is often called a cone, because it is one of the two distinct surfaces that bound the geometric solid of that name. This geometric object can also be described as the set of all points swept by a line that intercepts a the axis line and rotates around it; or the union of all lines that intersect the axis at a fixed point pand at a fixed angle θ. The aperture of the cone is the angle 2θ. A cone is a basic geometrical shape: see cone (solid). ...
In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ...
Rotation of a planar figure around a point Rotation of a planar body is the movement when points of the body travel in circular trajectories around a fixed point called the center of rotation. ...
More generally, when the directrix C is an ellipse, or any conic section, and the apex is an arbitrary point not on the plane of C, one obtains a conical quadric, which is a special case of a quadric. In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. ...
In mathematics, a conic section (or just conic) is a curved locus of points, fby intersecting a cone with a plane. ...
Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ...
A cylindrical surface can be viewed as a limiting case of a conical surface whose apex is moved off to infinity in a particular direction. Indeed, in projective geometry there is no difference between the cylindrical and conical surfaces, and the two halves of the latter become a single connected surface. Projective geometry can be thought of informally as the geometry which arises from placing ones eye at a point. ...
Equations
A conical surface S can be described parametrically as See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...
- S(t,u) = v + uq(t),
where v is the apex and q is the directrix.
A right circular cone whose axis is the Z coordinate axis, and whose apex is the origin, it is described parametrically as
- S(t,u) = (ucosθcost,ucosθsint,usinθ)
and in implicit form by S(x,y,z) = 0 where - S(x,y,z) = (x2 + y2)(cosθ)2 − z2(sinθ)2.
More generally, a right circular cone with vertex at the origin, axis parallel to the vector d, and aperture 2θ, is given by the implicit vector equation S(u) = 0 where Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ...
- S(u) = (u.d)2 − (d.d)(u.u)(cosθ)2
or - S(u) = u.d − | d | | u | cosθ
where u = (x,y,z), and u.d denotes the dot product. In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...
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