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)
“Only very slowly and late have men come to realize that unless freedom is universal it is only extended privilege.”
—Christopher Hill (b. 1912)
“All of life and human relations have become so incomprehensibly complex that, when you think about it, it becomes terrifying and your heart stands still.”
—Anton Pavlovich Chekhov (1860–1904)
“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)