Cubic Curves in The Plane of A Triangle
Suppose that ABC is a triangle with sidelengths a = |BC|, b = |CA|, c = |AB|. Relative to ABC, many named cubics pass through well known points. Examples shown below use two kinds of homogeneous coordinates: trilinear and barycentric.
To convert from trilinear to barycentric in a cubic equation, substitute as follows:
x → bcx, y → cay, z → abz;
to convert from barycentric to trilinear, use
x → ax, y → by, z → cz.
Many equations for cubics have the form
f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,z,x,y) = 0.
In the examples below, such equations are written more succinctly in "cyclic sum notation", like this:
= 0.
The cubics listed below can be defined in terms of the isogonal conjugate, denoted by X*, of a point X not on a sideline of ABC. A construction of X* follows. Let LA be the reflection of line XA about the internal angle bisector of angle A, and define LB and LC analogously. Then the three reflected lines concur in X*. In trilinear coordinates, if X = x:y:z, then X* = 1/x:1/y:1/z.
Read more about this topic: Cubic Plane Curve
Famous quotes containing the words cubic, curves and/or plane:
“Mining today is an affair of mathematics, of finance, of the latest in engineering skill. Cautious men behind polished desks in San Francisco figure out in advance the amount of metal to a cubic yard, the number of yards washed a day, the cost of each operation. They have no need of grubstakes.”
—Merle Colby, U.S. public relief program (1935-1943)
“For a hundred and fifty years, in the pasture of dead horses,
roots of pine trees pushed through the pale curves of your ribs,
yellow blossoms flourished above you in autumn, and in winter
frost heaved your bones in the groundold toilers, soil makers:
O Roger, Mackerel, Riley, Ned, Nellie, Chester, Lady Ghost.”
—Donald Hall (b. 1928)
“Have you ever been up in your plane at night, alone, somewhere, 20,000 feet above the ocean?... Did you ever hear music up there?... Its the music a mans spirit sings to his heart, when the earths far away and there isnt any more fear. Its the high, fine, beautiful sound of an earth-bound creature who grew wings and flew up high and looked straight into the face of the future. And caught, just for an instant, the unbelievable vision of a free man in a free world.”
—Dalton Trumbo (19051976)