Circular Points
The homogeneous form for the equation of a circle is x2 + y2 + 2axz + 2byz + cz2. The intersection of this curve with the line at infinity can be found by setting z = 0. This produces the equation x2 + y2 = 0 which has two solutions in the complex projective plane, (1, i, 0) and (1, −i, 0). These points are called the circular points at infinity and can be regarded as the common points of intersection of all circles. This can be generalized to curves of higher order as circular algebraic curves. A commonly known type of homogeneous coordinates are trilinear coordinates.
Read more about this topic: Homogeneous Coordinates
Famous quotes containing the words circular and/or points:
“‘A thing is called by a certain name because it instantiates a certain universal’ is obviously circular when particularized, but it looks imposing when left in this general form. And it looks imposing in this general form largely because of the inveterate philosophical habit of treating the shadows cast by words and sentences as if they were separately identifiable. Universals, like facts and propositions, are such shadows.”
—David Pears (b. 1921)
“The three main medieval points of view regarding universals are designated by historians as realism, conceptualism, and nominalism. Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics under the new names logicism, intuitionism, and formalism.”
—Willard Van Orman Quine (b. 1908)