Equation in The Extended Complex Plane
The extended plane of inversive geometry can be identified with the extended complex plane, so that equations of complex numbers can be used to describe lines, circles and inversions.
A circle Γ is the set of points z in a plane that lie at radius r from a center point γ.
Using the complex plane, we can treat γ as a complex number and circle Γ as a set of complex numbers.
Using the property that a complex number multiplied by its conjugate gives us the square of the modulus of the number, and that its modulus is its Euclidean distance from the origin, we can express the equation for Γ as follows:
We can multiply this by a real constant A to get an equation of the form
where A and D are real, and B and C are complex conjugates. Reversing the steps, we see that in order for this to be a circle, the radius squared must be equal to BC/A^2 - D/A > 0. So the above equation defines a generalized circle whenever AD < BC. Note that when A is zero, this equation defines a straight line.
Read more about this topic: Generalised Circle
Famous quotes containing the words equation, extended, complex and/or plane:
“Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.”
—Anna Quindlen (b. 1952)
“No: until I want the protection of Massachusetts to be extended to me in some distant Southern port, where my liberty is endangered, or until I am bent solely on building up an estate at home by peaceful enterprise, I can afford to refuse allegiance to Massachusetts, and her right to my property and life. It costs me less in every sense to incur the penalty of disobedience to the State than it would to obey. I should feel as if I were worth less in that case.”
—Henry David Thoreau (1817–1862)
“The human mind is so complex and things are so tangled up with each other that, to explain a blade of straw, one would have to take to pieces an entire universe.... A definition is a sack of flour compressed into a thimble.”
—Rémy De Gourmont (1858–1915)
“Even though I had let them choose their own socks since babyhood, I was only beginning to learn to trust their adult judgment.. . . I had a sensation very much like the moment in an airplane when you realize that even if you stop holding the plane up by gripping the arms of your seat until your knuckles show white, the plane will stay up by itself. . . . To detach myself from my children . . . I had to achieve a condition which might be called loving objectivity.”
—Anonymous Parent of Adult Children. Ourselves and Our Children, by Boston Women’s Health Book Collective, ch. 5 (1978)