Regulus (geometry)

Last updated July 16, 2024

In three-dimensional space, a regulusR is a set of skew lines, every point of which is on a transversal which intersects an element of R only once, and such that every point on a transversal lies on a line of R.

The set of transversals of R forms an opposite regulusS. In $\mathbb {R} ^{3}$ the union R ∪ S is the ruled surface of a hyperboloid of one sheet.

Three skew lines determine a regulus:

The locus of lines meeting three given skew lines is called a regulus. Gallucci's theorem shows that the lines meeting the generators of the regulus (including the original three lines) form another "associated" regulus, such that every generator of either regulus meets every generator of the other. The two reguli are the two systems of generators of a ruled quadric.^[1]

According to Charlotte Scott, "The regulus supplies extremely simple proofs of the properties of a conic...the theorems of Chasles, Brianchon, and Pascal ..."^[2]

In a finite geometry PG(3, q), a regulus has q + 1 lines.^[3] For example, in 1954 William Edge described a pair of reguli of four lines each in PG(3,3).^[4]

Robert J. T. Bell described how the regulus is generated by a moving straight line. First, the hyperboloid ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}\ =\ 1$ is factored as

\left({\frac {x}{a}}+{\frac {z}{c}}\right)\left({\frac {x}{a}}-{\frac {z}{c}}\right)\ =\ \left(1+{\frac {y}{b}}\right)\left(1-{\frac {y}{b}}\right).

Then two systems of lines, parametrized by λ and μ satisfy this equation:

{\frac {x}{a}}+{\frac {z}{c}}\ =\ \lambda \left(1+{\frac {y}{b}}\right),\quad {\frac {x}{a}}-{\frac {z}{c}}\ =\ {\frac {1}{\lambda }}\left(1-{\frac {y}{b}}\right)

and

{\frac {x}{a}}-{\frac {z}{c}}\ =\ \mu \left(1+{\frac {y}{b}}\right),\quad {\frac {x}{a}}+{\frac {z}{c}}\ =\ {\frac {1}{\mu }}\left(1-{\frac {y}{b}}\right).

No member of the first set of lines is a member of the second. As λ or μ varies, the hyperboloid is generated. The two sets represent a regulus and its opposite. Using analytic geometry, Bell proves that no two generators in a set intersect, and that any two generators in opposite reguli do intersect and form the plane tangent to the hyperboloid at that point. (page 155).^[5]

Related Research Articles

In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola.

<span class="mw-page-title-main">Ellipsoid</span> Quadric surface that looks like a deformed sphere

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

<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 geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.

In mathematics, the matrix representation of conic sections permits the tools of linear algebra to be used in the study of conic sections. It provides easy ways to calculate a conic section's axis, vertices, tangents and the pole and polar relationship between points and lines of the plane determined by the conic. The technique does not require putting the equation of a conic section into a standard form, thus making it easier to investigate those conic sections whose axes are not parallel to the coordinate system.

In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.

In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points $A$ , $B$ , $C$ , $D$ on a line, their cross ratio is defined as

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.

<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 three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more dimensions. Two lines are skew if and only if they are not coplanar.

<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.

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.

In projective geometry, a dual curve of a given plane curve $C$ is a curve in the dual projective plane consisting of the set of lines tangent to $C$ . There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If $C$ is algebraic then so is its dual and the degree of the dual is known as the class of the original curve. The equation of the dual of $C$ , given in line coordinates, is known as the tangential equation of $C$ . Duality is an involution: the dual of the dual of $C$ is the original curve $C$ .

Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system $that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces, the ellipsoidal coordinate system is based on confocal quadrics.$

<span class="mw-page-title-main">Polar curve</span>

In algebraic geometry, the first polar, or simply polar of an algebraic plane curve C of degree n with respect to a point Q is an algebraic curve of degree n−1 which contains every point of C whose tangent line passes through Q. It is used to investigate the relationship between the curve and its dual, for example in the derivation of the Plücker formulas.

<span class="mw-page-title-main">Projective harmonic conjugate</span> Point found separated from another, given a point pair

In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following construction:

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.

<span class="mw-page-title-main">Confocal conic sections</span> Conic sections with the same foci

In geometry, two conic sections are called confocal if they have the same foci.

A frequently studied problem in finite geometry is to identify ways in which an object can be covered by other simpler objects such as points, lines, and planes. In projective geometry, a specific instance of this problem that has numerous applications is determining whether, and how, a projective space can be covered by pairwise disjoint subspaces which have the same dimension; such a partition is called a spread. Specifically, a spread of a projective space , where $is an integer and a division ring, is a set of -dimensional subspaces, for some such that every point of the space lies in exactly one of the elements of the spread.$

In Euclidean geometry, a triangle conic is a conic in the plane of the reference triangle and associated with it in some way. For example, the circumcircle and the incircle of the reference triangle are triangle conics. Other examples are the Steiner ellipse, which is an ellipse passing through the vertices and having its centre at the centroid of the reference triangle; the Kiepert hyperbola which is a conic passing through the vertices, the centroid and the orthocentre of the reference triangle; and the Artzt parabolas, which are parabolas touching two sidelines of the reference triangle at vertices of the triangle.

References

↑ H. S. M. Coxeter (1969) Introduction to Geometry, page 259, John Wiley & Sons
↑ Charlotte Angas Scott (1905) The elementary treatment of the conics by means of the regulus, Bulletin of the American Mathematical Society 12(1): 1–7
↑ Albrecht Beutelspacher & Ute Rosenbaum (1998) Projective Geometry, page 72, Cambridge University Press ISBN 0-521-48277-1
↑ W. L. Edge (1954) "Geometry of three dimensions over GF(3)", Proceedings of the Royal Society A 222: 262–86 doi : 10.1098/rspa.1954.0068
↑ Robert J. T. Bell (1910) An Elementary Treatise on Co-ordinate Geometry of Three Dimensions, page 148, via Internet Archive

H. G. Forder (1950) Geometry, page 118, Hutchinson's University Library.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] H. S. M. Coxeter (1969) Introduction to Geometry, page 259, John Wiley & Sons

[2] Charlotte Angas Scott (1905) The elementary treatment of the conics by means of the regulus, Bulletin of the American Mathematical Society 12(1): 1–7

[3] Albrecht Beutelspacher & Ute Rosenbaum (1998) Projective Geometry, page 72, Cambridge University Press ISBN 0-521-48277-1

[4] W. L. Edge (1954) "Geometry of three dimensions over GF(3)", Proceedings of the Royal Society A 222: 262–86 doi : 10.1098/rspa.1954.0068

[5] Robert J. T. Bell (1910) An Elementary Treatise on Co-ordinate Geometry of Three Dimensions, page 148, via Internet Archive

[1]

[2]

[3]

[4]

[5]

Regulus (geometry)

See also

Related Research Articles

References