pencil of lines
be equations of some lines. Use the short notations .
If the lines and have an intersection point , then, by the parent entry (http://planetmath.org/LineThroughAnIntersectionPoint), the equation
with various real values of and can any line passing through the point ; this set of lines is called a pencil of lines.
Proof. If the line belongs to the fan of lines determined by the lines and , i.e. all the three lines have a common point, there must be the identity
i.e. there exist three real numbers , , , which are not all zeroes, such that the equation
has nontrivial solutions . By linear algebra, it follows that the determinant of this group of equations has to vanish.
Suppose conversely that the determinant vanishes. This implies that the above group of equations has a nontrivial solution . Thus we can write the identic equation (3). Let e.g. . Solving (3) for yields
which shows that the line belongs to the fan determined by the lines and ; so the lines pass through a common point.
- 1 Lauri Pimiä: Analyyttinen geometria. Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).
|Title||pencil of lines|
|Date of creation||2013-03-22 18:09:03|
|Last modified on||2013-03-22 18:09:03|
|Last modified by||pahio (2872)|