What is "trilinear coordinates"

Coordinate \Co*["o]r"di*nate\, n.

1. A thing of the same rank with another thing; one two or more persons or things of equal rank, authority, or importance.

   It has neither co["o]rdinate nor analogon; it is absolutely one.
   --Coleridge.

2. pl. (Math.) Lines, or other elements of reference, by means of which the position of any point, as of a curve, is defined with respect to certain fixed lines, or planes, called co["o]rdinate axes and co["o]rdinate planes. See Abscissa. Note: Co["o]rdinates are of several kinds, consisting in some of the different cases, of the following elements, namely:

   1. (Geom. of Two Dimensions) The abscissa and ordinate of any point, taken together; as the abscissa PY and ordinate PX of the point P (Fig. 2, referred to the co["o]rdinate axes AY and AX.

   2. Any radius vector PA (Fig. 1), together with its angle of inclination to a fixed line, APX, by which any point A in the same plane is referred to that fixed line, and a fixed point in it, called the pole, P.

   3. (Geom. of Three Dimensions) Any three lines, or distances, PB, PC, PD (Fig. 3), taken parallel to three co["o]rdinate axes, AX, AY, AZ, and measured from the corresponding co["o]rdinate fixed planes, YAZ, XAZ, XAY, to any point in space, P, whose position is thereby determined with respect to these planes and axes.

   4. A radius vector, the angle which it makes with a fixed plane, and the angle which its projection on the plane makes with a fixed line line in the plane, by which means any point in space at the free extremity of the radius vector is referred to that fixed plane and fixed line, and a fixed point in that line, the pole of the radius vector. Cartesian co["o]rdinates. See under Cartesian. Geographical co["o]rdinates, the latitude and longitude of a place, by which its relative situation on the globe is known. The height of the above the sea level constitutes a third co["o]rdinate. Polar co["o]rdinates, co["o]rdinates made up of a radius vector and its angle of inclination to another line, or a line and plane; as those defined in (b) and (d) above. Rectangular co["o]rdinates, co["o]rdinates the axes of which intersect at right angles. Rectilinear co["o]rdinates, co["o]rdinates made up of right lines. Those defined in

      1. and (c) above are called also Cartesian co["o]rdinates.

         Trigonometrical co["o]rdinates or Spherical co["o]rdinates, elements of reference, by means of which the position of a point on the surface of a sphere may be determined with respect to two great circles of the sphere.

         Trilinear co["o]rdinates, co["o]rdinates of a point in a plane, consisting of the three ratios which the three distances of the point from three fixed lines have one to another.

In geometry, the trilinear coordinates x:y:z of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. They are often called simply "trilinears". The ratio x:y is the ratio of the perpendicular distances from the point to the sides ( extended if necessary) opposite vertices A and B respectively; the ratio y:z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C respectively; and likewise for z:x and vertices C and A.

In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (''a' '', ''b' '', ''c' ''), or equivalently in ratio form, ''ka' :kb' :kc' '' for any positive constant k. If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible for all three trilinears to be non-positive.