In three-dimensional space, a regulus ''R'' is a set of
skew lines
In three-dimensional geometry, skew lines are two Line (geometry), lines that do not Line-line intersection, intersect and are not Parallel (geometry), parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges ...
, 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
the union ''R'' ∪ ''S'' is the
ruled surface of a
hyperboloid of one sheet
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface (mathematics), surface generated by rotating a hyperbola around one of its Hyperbola#Equation, principal axes. A hyperboloid is the surface obtained ...
.
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, "The regulus supplies extremely simple proofs of the properties of a conic...the theorems of Chasles,
Brianchon, and
Pascal ..."
In a
finite geometry 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 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 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 engineering, and als ...
, 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 (1910]
An Elementary Treatise on Co-ordinate Geometry of Three Dimensions
page 148, via Internet Archive
The Internet Archive is an American 501(c)(3) organization, non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including web ...
See also
*
Spread (projective geometry)
*
References
{{Reflist
*
H. G. Forder (1950) ''Geometry'', page 118, Hutchinson's University Library.
Geometry
Quadrics