Algebraic Curves in The Plane
Algebraic curves in the plane may be defined as the set of points (x, y) satisfying an equation of the form f(x, y)=0, where f is a polynomial function f:R2→R. If f is expanded as
If the origin (0, 0) is on the curve then a0=0. If b1≠0 then the implicit function theorem guarantees there is a smooth function h so that the curve has the form y=h(x) near the origin. Similarly, if b0≠0 then there is a smooth function k so that the curve has the form x=k(y) near the origin. In either case, there is a smooth map from R to the plane which defines the curve in the neighborhood of the origin. Note that at the origin
so the curve is non-singular or regular at the origin if at least one of the partial derivatives of f is non-zero. The singular points are those points on the curve where both partial derivatives vanish,
Read more about this topic: Singular Point Of A Curve
Famous quotes containing the words algebraic, curves and/or plane:
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“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 ground—old 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?... It’s the music a man’s spirit sings to his heart, when the earth’s far away and there isn’t any more fear. It’s 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 (1905–1976)