Isotropic line

Last updated

In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, and never with a definite quadratic form.

Using complex geometry, Edmond Laguerre first suggested the existence of two isotropic lines through the point (α, β) that depend on the imaginary unit i: [1]

First system:
Second system:

Laguerre then interpreted these lines as geodesics:

An essential property of isotropic lines, and which can be used to define them, is the following: the distance between any two points of an isotropic line situated at a finite distance in the plane is zero. In other terms, these lines satisfy the differential equation ds2 = 0. On an arbitrary surface one can study curves that satisfy this differential equation; these curves are the geodesic lines of the surface, and we also call them isotropic lines. [1] :90

In the complex projective plane, points are represented by homogeneous coordinates and lines by homogeneous coordinates . An isotropic line in the complex projective plane satisfies the equation: [2]

In terms of the affine subspace x3 = 1, an isotropic line through the origin is

In projective geometry, the isotropic lines are the ones passing through the circular points at infinity.

In the real orthogonal geometry of Emil Artin, isotropic lines occur in pairs:

A non-singular plane which contains an isotropic vector shall be called a hyperbolic plane. It can always be spanned by a pair n, m of vectors which satisfy
We shall call any such ordered pair n, m a hyperbolic pair. If V is a non-singular plane with orthogonal geometry and n ≠ 0 is an isotropic vector of V, then there exists precisely one m in V such that n, m is a hyperbolic pair. The vectors xn and ym are then the only isotropic vectors of V. [3]

Relativity

Isotropic lines have been used in cosmological writing to carry light. For example, in a mathematical encyclopedia, light consists of photons: "The worldline of a zero rest mass (such as a non-quantum model of a photon and other elementary particles of mass zero) is an isotropic line." [4] For isotropic lines through the origin, a particular point is a null vector, and the collection of all such isotropic lines forms the light cone at the origin.

Élie Cartan expanded the concept of isotropic lines to multivectors in his book on spinors in three dimensions. [5]

Related Research Articles

In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension D) in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D = 1 in the case of conic sections. When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a degenerate quadric or a reducible quadric.

<span class="mw-page-title-main">Hyperboloid</span> Unbounded quadric surface

In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.

In differential geometry, the second fundamental form is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by . Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold.

<span class="mw-page-title-main">Line (geometry)</span> Straight figure with zero width and depth

In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word line may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points.

In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective 3-space, . Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines in and points on a quadric in . A predecessor and special case of Grassmann coordinates, Plücker coordinates arise naturally in geometric algebra. They have proved useful for computer graphics, and also can be extended to coordinates for the screws and wrenches in the theory of kinematics used for robot control.

<span class="mw-page-title-main">Null vector</span> Vector on which a quadratic form is zero

In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0.

In mathematics, a versor is a quaternion of norm one. Each versor has the form

<span class="mw-page-title-main">Pencil (geometry)</span> Family of geometric objects with a common property

In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane.

<span class="mw-page-title-main">Three-dimensional space</span> Geometric model of the physical space

In geometry, a three-dimensional space is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region, a solid figure.

<span class="mw-page-title-main">Hyperboloid model</span> Model of n-dimensional hyperbolic geometry

In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S+ of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m-planes are represented by the intersections of (m+1)-planes passing through the origin in Minkowski space with S+ or by wedge products of m vectors. Hyperbolic space is embedded isometrically in Minkowski space; that is, the hyperbolic distance function is inherited from Minkowski space, analogous to the way spherical distance is inherited from Euclidean distance when the n-sphere is embedded in (n+1)-dimensional Euclidean space.

Affine geometry, broadly speaking, is the study of the geometrical properties of lines, planes, and their higher dimensional analogs, in which a notion of "parallel" is retained, but no metrical notions of distance or angle are. Affine spaces differ from linear spaces in that they do not have a distinguished choice of origin. So, in the words of Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps." Accordingly, a complex affine space, that is an affine space over the complex numbers, is like a complex vector space, but without a distinguished point to serve as the origin.

In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise it is a definite quadratic form. More explicitly, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if q(v) = 0. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form.

In mathematics and theoretical physics, a pseudo-Euclidean space of signature (k, n-k) is a finite-dimensional real n-space together with a non-degenerate quadratic form q. Such a quadratic form can, given a suitable choice of basis (e1, …, en), be applied to a vector x = x1e1 + ⋯ + xnen, giving which is called the scalar square of the vector x.

Non-linear least squares is the form of least squares analysis used to fit a set of m observations with a model that is non-linear in n unknown parameters (m ≥ n). It is used in some forms of nonlinear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations. There are many similarities to linear least squares, but also some significant differences. In economic theory, the non-linear least squares method is applied in (i) the probit regression, (ii) threshold regression, (iii) smooth regression, (iv) logistic link regression, (v) Box–Cox transformed regressors ().

<span class="mw-page-title-main">Conic section</span> Curve from a cone intersecting a plane

A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.

In geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space.

<span class="mw-page-title-main">Cayley–Klein metric</span>

In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance" where he calls the quadric the absolute. The construction was developed in further detail by Felix Klein in papers in 1871 and 1873, and subsequent books and papers. The Cayley–Klein metrics are a unifying idea in geometry since the method is used to provide metrics in hyperbolic geometry, elliptic geometry, and Euclidean geometry. The field of non-Euclidean geometry rests largely on the footing provided by Cayley–Klein metrics.

Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an n-dimensional base space Rp,q to null vectors in Rp+1,q+1. This allows operations on the base space, including reflections, rotations and translations to be represented using versors of the geometric algebra; and it is found that points, lines, planes, circles and spheres gain particularly natural and computationally amenable representations.

In mathematics and theoretical physics, a quasi-sphere is a generalization of the hypersphere and the hyperplane to the context of a pseudo-Euclidean space. It may be described as the set of points for which the quadratic form for the space applied to the displacement vector from a centre point is a constant value, with the inclusion of hyperplanes as a limiting case.

References

  1. 1 2 Edmond Laguerre (1870) "Sur l’emploi des imaginaires en la géométrie", Oeuvres de Laguerre 2: 89
  2. C. E. Springer (1964) Geometry and Analysis of Projective Spaces, page 141, W. H. Freeman and Company
  3. Emil Artin (1957) Geometric Algebra, page 119 via Internet Archive
  4. Encyclopedia of Mathematics World line
  5. Cartan, Élie (1981) [1938], The theory of spinors, New York: Dover Publications, p. 17, ISBN   978-0-486-64070-9, MR   0631850