Trilinear coordinates

Trilinear coordinates (more precisely homogeneous trilinear coordinates ) are triangular section a of Julius Plucker , imported tool to the position of a point with respect to a triangle to describe.

Definition and spelling

A triangle ABC is given. For any point P of the plane of hot three real numbers , and (homogeneous) trilinear coordinates of P, if there is a different from 0 real number is, so that ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle r}$

${\displaystyle x\,=\,rd_{BC};\quad y\,=\,rd_{CA};\quad z\,=\,rd_{AB}}$

applies. Here denote , and the signed distances of point P from the straight line BC, CA or AB. The size is given a positive sign if P is on the same side of BC as corner A, and a negative sign if P and A are on different sides of BC. The other two signs are determined accordingly. ${\displaystyle d_{BC}}$ ${\displaystyle d_{CA}}$ ${\displaystyle d_{AB}}$ ${\displaystyle d_{BC}}$

The totality of the trilinear coordinates of a point is written either as an ordered triplet or in the form . ${\displaystyle (x,y,z)}$ ${\displaystyle x:y:z}$

Trilinear coordinates are not clearly defined: Multiplication with any real number other than 0 yields trilinear coordinates of the given point.

Examples

• The corners A, B and C of the given triangle have the trilinear coordinates , respectively . ${\displaystyle (1,0,0)}$ ${\displaystyle (0,1,0)}$ ${\displaystyle (0,0,1)}$
• The inscribed center of a triangle has the trilinear coordinates because it has the same distance from all three sides of the triangle. ${\displaystyle (1,1,1)}$
• For the center of gravity of a triangle, the trilinear coordinates are equivalent or or . In this case, a, b, c for the side lengths , , for the sizes of the internal angles and for the cosecant . ${\displaystyle \left({\frac {1}{a}},\,{\frac {1}{b}},\,{\frac {1}{c}}\right)}$ ${\displaystyle \left(bc,\,ca,\,ab\right)}$ ${\displaystyle \left(\csc \alpha ,\,\csc \beta ,\,\csc \gamma \right)}$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \gamma }$ ${\displaystyle \csc }$

Relation to the barycentric coordinates

There is a simple relationship between the trilinear coordinates and the barycentric coordinates , which are also frequently used in triangular geometry : If the trilinear coordinates are given by, then barycentric coordinates are obtained , where a, b and c stand for the side lengths. ${\displaystyle \left(x,y,z\right)}$ ${\displaystyle \left(ax,by,cz\right)}$

Formulas

In many cases, trilinear coordinates enable the use of algebraic methods in triangular geometry. For example, there are three points , and with the trilinear coordinates ${\displaystyle P_{1}}$ ${\displaystyle P_{2}}$ ${\displaystyle P_{3}}$

${\displaystyle x_{1}:y_{1}:z_{1}}$

${\displaystyle x_{2}:y_{2}:z_{2}}$

${\displaystyle x_{3}:y_{3}:z_{3}}$

if and collinear when the determinant

${\displaystyle D=\left|{\begin{matrix}x_{1}&y_{1}&z_{1}\\x_{2}&y_{2}&z_{2}\\x_{3}&y_{3}&z_{3}\end{matrix}}\right|}$

equals 0. The dual statement for this sentence is also correct: three straight lines passing through the equations

${\displaystyle x_{1}\alpha +y_{1}\beta +z_{1}\gamma =0}$,

${\displaystyle x_{2}\alpha +y_{2}\beta +z_{2}\gamma =0}$,

${\displaystyle x_{3}\alpha +y_{3}\beta +z_{3}\gamma =0}$

are given have a common point if and only if applies. ${\displaystyle D=0}$

literature

• William Allen Whitworth: Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions . Cambridge, 1866 ( online copy in the internet archive )
• Oene Bottema : Topics in Elementary Geometry . Springer, 2008, ISBN 9780387781310 , pp. 25-28

Web links

• Eric W. Weisstein : TrilinearCoordinates . In: MathWorld (English).