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:
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)
“I have been accustomed to make excursions to the ponds within ten miles of Concord, but latterly I have extended my excursions to the seashore.”
—Henry David Thoreau (1817–1862)
“When distant and unfamiliar and complex things are communicated to great masses of people, the truth suffers a considerable and often a radical distortion. The complex is made over into the simple, the hypothetical into the dogmatic, and the relative into an absolute.”
—Walter Lippmann (1889–1974)
“It was the most ungrateful and unjust act ever perpetrated by a republic upon a class of citizens who had worked and sacrificed and suffered as did the women of this nation in the struggle of the Civil War only to be rewarded at its close by such unspeakable degradation as to be reduced to the plane of subjects to enfranchised slaves.”
—Anna Howard Shaw (1847–1919)