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, t ...
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''.
According to
Charlotte Scott, "The regulus supplies extremely simple proofs of the properties of a conic...the theorems of Chasles,
Brianchon, and
Pascal
Pascal, Pascal's or PASCAL may refer to:
People and fictional characters
* Pascal (given name), including a list of people with the name
* Pascal (surname), including a list of people and fictional characters with the name
** Blaise Pascal, Frenc ...
..."
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 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 enginee ...
, 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 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