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Conical surface

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In geometry, a conical surface is an unbounded surface in three-dimensional space formed from the union of infinite lines that pass through a fixed point and a space curve.

Definitions

[edit]

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. The directrix is often taken as a plane curve, in a plane not containing the apex, but this is not a requirement.^[1]

In general, a conical surface consists of two congruent unbounded halves joined by the apex. Each half is called a nappe, and is the union of all the rays that start at the apex and pass through a point of some fixed space curve.^[2] Sometimes the term "conical surface" is used to mean just one nappe.^[3]

Special cases

[edit]

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 or double cone.^[2] 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 an elliptic cone.^[4]

Equations

[edit]

A conical surface $S$ can be described parametrically as

S(t,u)=v+uq(t)

where $v$ is the apex and $q$ is the directrix.^[5]

Related surface

[edit]

Conical surfaces are ruled surfaces, surfaces that have a straight line through each of their points.^[6] Patches of conical surfaces that avoid the apex are special cases of developable surfaces, surfaces that can be unfolded to a flat plane without stretching. When the directrix has the property that the angle it subtends from the apex is exactly $2\pi$ , then each nappe of the conical surface, including the apex, is a developable surface.^[7]

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 a cylindrical surface is just a special case of a conical surface.^[8]

References

[edit]

^ Adler, Alphonse A. (1912), "1003. Conical surface", The Theory of Engineering Drawing, D. Van Nostrand, p. 166
^ ^a ^b Wells, Webster; Hart, Walter Wilson (1927), Modern Solid Geometry, Graded Course, Books 6-9, D. C. Heath, pp. 400–401
^ Shutts, George C. (1913), "640. Conical surface", Solid Geometry, Atkinson, Mentzer, p. 410
^ Young, J. R. (1838), Analytical Geometry, J. Souter, p. 227
^ Gray, Alfred (1997), "19.2 Flat ruled surfaces", Modern Differential Geometry of Curves and Surfaces with Mathematica (2nd ed.), CRC Press, pp. 439–441, ISBN 9780849371646
^ Mathematical Society of Japan (1993), Ito, Kiyosi (ed.), Encyclopedic Dictionary of Mathematics, Vol. I: A–N (2nd ed.), MIT Press, p. 419
^ Audoly, Basile; Pomeau, Yves (2010), Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells, Oxford University Press, pp. 326–327, ISBN 9780198506256
^ Giesecke, F. E.; Mitchell, A. (1916), Descriptive Geometry, Von Boeckmann–Jones Company, p. 66

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from Grokipedia

A conical surface is a ruled surface in three-dimensional Euclidean space formed by the union of all straight lines, known as generators, that pass through a fixed point called the apex or vertex and intersect a fixed curve referred to as the directrix. This construction generates an unbounded surface that extends infinitely in both directions from the apex unless bounded by the directrix.^[1] In its most common form, the directrix is a circle, producing a circular conical surface, or simply a cone, where the generators sweep around the circumference of the base circle while fixed at the apex.^[1] Conical surfaces are classified as right if the apex lies along the axis perpendicular to the plane of the directrix at its center, or oblique otherwise; they may also feature elliptical or other conic directrices, leading to elliptic cones.^[1] These surfaces are quadratic in nature, with the infinite double cone—comprising two nappes sharing the apex—serving as a fundamental quadric surface in analytic geometry.^[1] Mathematically, a right circular conical surface can be represented parametrically as

x = \frac{h-u}{h} r \cos\theta

y = \frac{h-u}{h} r \sin\theta

z = u

for height

h

, base radius

r

u \in [0,h]

, and

\theta \in [0,2\pi)

, or implicitly as

\frac{x^2 + y^2}{c^2} = (z - z_0)^2

where

c = r/h

and

z_0 = h

.^[1] Conical surfaces exhibit zero Gaussian curvature, confirming their developable ruled nature, and their lateral surface area (excluding the base) is given by

\pi r \sqrt{r^2 + h^2}

πrr2+h2

.^[1] They are central to conic sections, where plane intersections yield circles, ellipses, parabolas, or hyperbolas depending on the angle and position relative to the nappes.^[1]

Definitions and Fundamentals

Formal Definition

A conical surface is a ruled surface in three-dimensional Euclidean space, defined as the locus of all points lying on straight lines—known as rulings—that pass through a fixed point, called the vertex or apex, and intersect a fixed curve, termed the directrix, which does not contain the vertex.^[2]^[3] This construction ensures that every ruling connects the vertex to a point on the directrix, forming an unbounded surface unless intersected by a plane.^[1] While the term "cone" is sometimes used interchangeably, it often refers to the solid region bounded by the conical surface and a base plane intersecting the directrix, distinguishing the surface itself as the boundary rather than the volume it encloses.^[1]^[3] As a foundational concept, a ruled surface is any surface that can be parameterized as the union of a family of straight lines, providing the geometric framework for the conical surface's linear generators.^[4] The directrix serves as the guiding curve that shapes the surface's profile, without passing through the vertex.^[2] For instance, a right circular conical surface arises when the directrix is a circle and the rulings are generated by rotating a straight line around an axis passing through the vertex and the circle's center.^[1] Conical surfaces with conic directrices represent special cases of quadric surfaces.^[5]

Key Components

A conical surface is fundamentally composed of three primary elements: the vertex, the generating lines, and the directrix. The vertex serves as the fixed point in three-dimensional space from which all other components emanate, acting as the apex or singular point of convergence for the surface's structure.^[6]^[7] This point is essential, as it ensures that the surface is ruled, meaning it is formed exclusively by straight lines passing through it. Without the vertex, the defining characteristic of the conical form would be lost, distinguishing it from other ruled surfaces like cylinders.^[8] The generating lines, also known as rulings or generatrices, constitute an infinite family of straight lines that all pass through the vertex and sweep out the surface. Each generating line connects the vertex to a point on the directrix, and their collective union forms the unbounded conical surface, which typically extends infinitely in both directions from the vertex, creating two nappes or sheets.^[7]^[8] These lines are the dynamic elements that "generate" the surface as they move while maintaining contact with the fixed directrix, providing the linear texture inherent to cones.^[6] The directrix is a fixed curve in space—often, but not necessarily, planar—that is intersected by every generating line at exactly one point, with the critical condition that the curve does not pass through the vertex itself.^[7]^[8] This curve dictates the shape and extent of the surface, serving as the guiding path for the rulings. In contexts where the directrix is closed and lies in a plane, it is sometimes referred to as the base curve, particularly for finite cones like those with circular or elliptical bases; however, for general conical surfaces, the directrix may be a non-planar space curve, broadening the applicability beyond simple right cones.^[6] Together, these components uniquely determine the conical surface: for any given vertex and directrix, there exists precisely one such surface formed by the rulings connecting them, ensuring no ambiguity in the geometric construction.^[9] This uniqueness underscores the conical surface's role as a fundamental ruled surface in geometry, where the interplay of the fixed point and curve yields a structure invariant under homotheties centered at the vertex.^[9]

Geometric Properties

Generating Elements

A conical surface is generated by the motion of a straight line, known as the generatrix, that passes through a fixed point called the vertex and intersects a fixed curve termed the directrix.^[10] This process traces out the surface as the generatrix sweeps along the directrix, with the vertex remaining stationary. For instance, rotating a line around the vertex while ensuring it touches the directrix at varying points constructs the surface in a rotational manner, or skewing the line can produce more general forms.^[10] The generating elements form a one-parameter family of rulings, which are the straight lines comprising the surface.^[11] Each ruling in this family passes through the vertex and is parameterized by a single variable corresponding to its position along the directrix. When the directrix is a smooth curve not passing through the vertex, each ruling intersects the directrix at exactly one point, ensuring the surface is fully covered without overlaps or gaps.^[10] The vertex serves as the only singular point on the conical surface, where all rulings converge, leading to a breakdown in the usual differentiability properties away from this apex.^[12] This convergence creates a ray-like structure emanating from the vertex, visually resembling a bundle of lines radiating outward to meet the directrix, which underscores the surface's ruled nature.^[12]

Developability and Metrics

A conical surface is a developable surface, meaning it is a ruled surface with zero Gaussian curvature everywhere except possibly at the vertex, allowing it to be isometrically mapped onto a plane without distortion. This property arises because the surface can be generated by straight lines (rulings) passing through a fixed vertex, ensuring that the product of the principal curvatures is zero, as one principal curvature vanishes along the rulings. Such surfaces, including cones, cylinders, and tangent developables, can be constructed from flat sheets like paper or metal, preserving lengths and angles during the mapping.^[13] For a general conical surface, the unrolling process maps it to a portion of the plane in which the image of the directrix forms the outer boundary and the generators correspond to straight lines radiating from the image of the vertex. Geodesics on the original surface correspond to straight lines in this planar development. Intrinsic metrics, such as lengths along the generators, are preserved under unrolling, facilitating the computation of distances and areas on the surface.^[13]

Algebraic Representation

General Equation

A conical surface is represented algebraically as a special case of a quadric surface, given by the general second-degree equation in three variables:

Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0,

where the coefficients are chosen such that the equation describes a conical type, specifically a degenerate quadric with a singular point known as the vertex.^[14]^[15] Conical surfaces are classified among quadric surfaces by the presence of a singular point at the vertex, where the partial derivatives vanish simultaneously, and by discriminant conditions that identify degeneracy. In matrix terms, the quadric is expressed in homogeneous coordinates as

\mathbf{X}^T Q \mathbf{X} = 0

, where

\mathbf{X} = [x, y, z, 1]^T

and

Q

is the symmetric 4×4 coefficient matrix; a conical surface corresponds to

\operatorname{rank}(Q) = 3

, distinguishing it from full-rank (rank 4) non-degenerate quadrics like ellipsoids or hyperboloids.^[14]^[15] When the vertex is positioned at the origin, the linear and constant terms vanish (

G = H = I = J = 0

), simplifying the equation to a homogeneous quadratic form. For an elliptic cone aligned with the coordinate axes, this takes the canonical form

\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0,

where

a, b, c > 0

determine the semi-axes, with two positive and one negative coefficients (up to overall sign and permutation of variables), corresponding to the signature (2,1) of the quadratic form matrix.^[16]^[15] To obtain this canonical form from the general equation, the coordinate system is transformed via translation to shift the vertex to the origin, followed by rotation to diagonalize the quadratic form matrix through orthogonal transformation, aligning the principal axes with the coordinate axes. This process reduces the equation to its standard type, facilitating analysis in computational geometry contexts where matrix operations enable efficient classification and intersection computations.^[14]^[15]

Specific Forms

A conical surface can be represented parametrically as

\vec{r}(u,v) = \vec{V} + u \cdot \vec{d}(v)

(u,v)=V

+u⋅d

(v), where $\vec{V}$

is the position vector of the vertex, $u \geq 0$ is a scaling parameter along each generating line (ruling), and $\vec{d}(v)$

(v) is a vector function parameterizing the directions from the vertex to the directrix curve, with $v$ varying over the parameter domain of the directrix.^[1] This form captures the ruled nature of the surface, with each fixed $v$ tracing a straight line from the vertex. For a right circular cone with vertex at the origin and semi-vertical angle $\alpha$ , an example uses parameters $t \geq 0$ (replacing $u$ ) and $\theta \in [0, 2\pi)$ , yielding $\vec{r}(\theta, t) = (t \cos \theta, t \sin \theta, t \cot \alpha)$

(θ,t)=(tcosθ,tsinθ,tcotα).^[1] In cylindrical coordinates $(r, \theta, z)$ , the equation for a right circular cone with vertex at the origin and axis along the positive $z$ -direction simplifies to $z = r \cot \alpha$ , where $\alpha$ is the semi-vertical angle and $\theta \in [0, 2\pi)$ . This relation holds because the radial distance $r$ increases linearly with height $z$ along the generators. For oblique cones, where the generators do not form right angles with the base plane, the implicit form incorporates linear terms into the general quadratic equation of a quadric surface, such as $ax^2 + by^2 + cz^2 + dxy + eyz + fzx + gx + hy + iz + j = 0$ , with the surface degenerating into a pair of nappes meeting at the vertex. The parametric form $\vec{r}(u,v) = u \cdot \vec{d}(v)$

(u,v)=u⋅d

(v) (assuming vertex at origin for simplicity) derives from the definition: select points on the directrix curve $\vec{c}(v)$

(v), then form rays from the vertex through these points, parameterizing each ray as scalar multiples of the direction vectors $\vec{d}(v) = \vec{c}(v) - \vec{V}$

(v)=c

(v)−V

.^[17] In computer graphics, the vector form position = vertex + $s$ \cdot direction_vector is commonly used for ray-cone intersections and rendering, where $s \geq 0$ scales along the generator and the direction_vector varies to trace the surface.^[18]

Special Cases and Variations

Circular Cones

A circular cone is a conical surface whose directrix is a circle, exhibiting rotational symmetry about its axis. This symmetry distinguishes it from more general elliptic or hyperbolic cones, allowing for uniform generating lines at a constant angle to the axis. In applications ranging from geometry to optics, circular cones serve as foundational models due to their isotropic properties.^[1] The right circular cone represents the standard form, where the axis is perpendicular to the plane of the circular base, positioning the apex directly above the base center. The semi-vertical angle α is the fixed angle between the axis and any generating line from the apex to the base circumference. For a right circular cone with apex at the origin and axis along the positive z-direction, the surface equation is given by

z = \frac{\sqrt{x^2 + y^2}}{\tan \alpha},

z=tanαx2+y2

,
describing the set of points where the radial distance in the xy-plane scales linearly with height z according to the angle α. This equation captures the linear taper from the apex, with the base at z = h forming a circle of radius r = h tan α.^[1]^[19]^[20] In contrast, an oblique circular cone maintains a circular directrix but tilts the axis away from perpendicularity to the base plane, offsetting the apex from the base center. This obliqueness results in generating lines of varying lengths, while preserving the circular nature of the directrix in its own plane; however, projections of the base onto planes perpendicular to the axis or along the line of sight often appear elliptical due to the angular distortion. The surface remains ruled, with properties like slant height varying azimuthally around the axis.^[21]^[1] Specific to circular cones, the aperture is defined as twice the semi-vertical angle, or 2α, representing the maximum angle subtended between two opposite generating lines at the apex. This measure quantifies the cone's angular width and is invariant under translation along the axis. For eccentricity in circular cases, the base directrix has zero eccentricity, reflecting perfect circularity, while cross-sections parallel to the base remain circles (e = 0); oblique or tilted sections yield ellipses with eccentricity e < 1, determined by the cutting plane's angle relative to α.^[22]^[23] Although the focus here is on the surface, the enclosed solid right circular cone has volume V = (1/3) π r² h, where r is the base radius and h the perpendicular height; for oblique variants, the formula uses the same perpendicular height despite varying slant heights along the surface, which affect lateral area computations. The slant height l = √(r² + h²) = h / cos α emphasizes the surface's generative aspect, influencing metrics like lateral surface area π r l.^[1] In ray tracing applications, circular conical surfaces exhibit advantageous optical properties due to their axial symmetry, enabling efficient ray propagation along generators without deviation in azimuthal directions. Cone tracing algorithms extend standard ray tracing by modeling light as conical bundles rather than infinitesimal rays, improving sampling of indirect illumination and shadows in scenes with circular cone primitives; this approach reduces aliasing by accounting for the cone's aperture 2α to approximate penumbra and soft shadows. Such methods are particularly effective for rendering reflections and refractions on conical optics, as rays intersecting the surface follow predictable bilinear paths.^[24]^[25]

Elliptic and Hyperbolic Cones

An elliptic cone is a type of conical surface formed by the union of straight lines connecting a fixed vertex to points on an elliptic directrix curve, resulting in elliptical cross-sections that are bounded.^[26] Unlike circular cones, these exhibit asymmetry due to the ellipse's varying major and minor axes, leading to non-uniform expansion along the rulings from the vertex. The standard Cartesian equation for an elliptic cone aligned with the z-axis is

\frac{x^2}{a^2} + \frac{y^2}{b^2} = \left( \frac{z}{c} \right)^2,

where

a

b

, and

c > 0

determine the scaling along each axis.^[16] A hyperbolic cone arises similarly, but with a hyperbolic directrix, producing cross-sections that are hyperbolas unbounded in both positive and negative directions along the transverse axis. This configuration exhibits asymmetry due to the differing curvatures along the principal axes, but maintains symmetry in positive and negative directions. Its standard equation is

\frac{x^2}{a^2} - \frac{y^2}{b^2} = \left( \frac{z}{c} \right)^2.

Both elliptic and hyperbolic cones exhibit asymptotic behavior, extending infinitely along their generating rulings (straight lines from the vertex) without bound, forming an unbounded ruled surface.^[1] Intersections with planes parallel to the directrix plane yield conic sections similar to the directrix itself, scaled proportionally by the distance from the vertex; for instance, at height

z = kz_0

, the cross-section is a scaled version of the ellipse or hyperbola at

z = z_0

.^[16]

Quadric Surfaces

A conical surface is classified as a degenerate quadric surface in three-dimensional space, characterized by the singularity arising from the rank of the associated symmetric matrix being less than full.^[27] Specifically, when the rank is 3, the quadric becomes singular yet irreducible, resulting in a cone (or cylinder, depending on the signature), as opposed to full rank 4 for nondegenerate cases like ellipsoids or hyperboloids.^[27] Degeneracy to a pair of planes occurs at rank 1 or 2.^[27] The singularity of a conical surface is detected via the discriminant of the quadric equation, where the determinant Δ of the 4×4 symmetric matrix in homogeneous coordinates equals zero, indicating the presence of singular points.^[28] This condition ensures the quadratic form defines a cone with a vertex, distinguishing it from smooth quadrics. Conical surfaces emerge as limiting cases of other quadrics; for instance, a cone can be obtained by setting the constant term to zero in the equation of a hyperboloid of one or two sheets, effectively "pinching" the surface at the vertex.^[29] Similarly, it serves as a boundary between elliptic and hyperbolic paraboloids in the parameter space of quadric forms. In projective geometry, conical surfaces are naturally represented using homogeneous coordinates

[x : y : z : w]

, where the equation

\mathbf{X}^T A \mathbf{X} = 0

(with

A

symmetric and

\det(A) = 0

) defines a quadric cone apexed at a singular point, such as the origin in affine space.^[30] This framework unifies affine cones with their projective closures, treating lines through the vertex as generators. Affine transformations preserve the conical type within the classification of quadrics, mapping singular irreducible surfaces of rank 3 to equivalent forms while maintaining the degeneracy structure.^[31]

Ruled Surfaces

A ruled surface is a surface in three-dimensional space that can be generated by the motion of a straight line, known as a ruling or generator, along a curve, such that every point on the surface lies on one of these lines.^[4] Conical surfaces form a specific subclass of ruled surfaces, characterized by all rulings passing through a fixed point called the apex, creating a union of lines emanating from that point to trace a directrix curve.^[3] In contrast to conical surfaces, more general ruled surfaces may feature multiple families of rulings; for instance, a hyperboloid of one sheet is a doubly ruled surface with two distinct reguli, where lines from each family intersect those of the other without all converging to a single point./01%3A_Vectors_in_Euclidean_Space/1.06%3A_Surfaces) This multiplicity allows for greater flexibility in form compared to the singly ruled structure of cones.^[32] Conical surfaces share the property of developability with other ruled surfaces like cylinders, meaning they possess zero Gaussian curvature and can be isometrically mapped onto a plane, facilitating applications in manufacturing and design without distortion.^[13] As explored in the geometric properties section, this developability arises from the rulings lying within a single tangent plane at each point.^[33] In modern architecture, ruled surfaces—such as hyperbolic paraboloids—extend the principles of ruled geometry beyond traditional cones, enabling efficient, lightweight structures like tensile roofs and shells.^[34]

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