In three-dimensional space, a regulus ''R'' is a set of
skew lines
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 sa ...
, 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 regulus ''S''. In ℝ
3 the union ''R'' ∪ ''S'' is the
ruled surface
In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the ...
of a
hyperboloid of one sheet
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 def ...
.
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''.
According to
Charlotte Scott
Charlotte Angas Scott (8 June 1858 – 10 November 1931) was a British mathematician who made her career in the United States and was influential in the development of American mathematics, including the mathematical education of women. Scott ...
, "The regulus supplies extremely simple proofs of the properties of a conic...the theorems of Chasles,
Brianchon, and
Pascal ..."
In a
finite geometry
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
PG(3, ''q''), a regulus has ''q'' + 1 lines. For example, in 1954
William Edge described a pair of reguli of four lines each in PG(3,3).
Robert J. T. Bell
Robert J. T. Bell Royal Society of Edinburgh, RSE FRSE (15 January 1876 – 8 September 1963) was a Scottish people, Scottish mathematician. He held the positions of Professor of Pure and Applied Mathematics and Dean of the Faculty of Arts and Sci ...
described how the regulus is generated by a moving straight line. First, the hyperboloid
is factored as
:
Then two systems of lines, parametrized by λ and μ satisfy this equation:
:
and
:
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
In classical 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.
Analytic geometry is used in physics and engineerin ...
, 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).
Robert J. T. Bell
Robert J. T. Bell Royal Society of Edinburgh, RSE FRSE (15 January 1876 – 8 September 1963) was a Scottish people, Scottish mathematician. He held the positions of Professor of Pure and Applied Mathematics and Dean of the Faculty of Arts and Sci ...
(1910
An Elementary Treatise on Co-ordinate Geometry of Three Dimensions
page 148, via Internet Archive
The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
See also
*
References
{{Reflist
*
H. G. Forder (1950) ''Geometry'', page 118, Hutchinson's University Library.
Geometry
Quadrics