Radical Axis and Central Line
Except for the two special cases of a pencil of concentric circles and a pencil of coincident lines, any two circles within a pencil have the same radical axis, and all circles in the pencil have collinear centers. Any three or more circles from the same family are called coaxal circles or coaxial circles.
The elliptic pencil of circles passing through the two points C and D (the set of red circles, in the figure) has the line CD as its radical axis. The centers of the circles in this pencil lie on the perpendicular bisector of CD. The hyperbolic pencil defined by points C and D (the blue circles) has its radical axis on the perpendicular bisector of line CD, and all its circle centers on line CD.
The radical axis of any pencil of circles, interpreted as an infinite-radius circle, belongs to the pencil. Any three circles belong to a common pencil whenever all three pairs share the same radical axis and their centers are collinear.
Read more about this topic: Apollonian Circles
Famous quotes containing the words radical, axis, central and/or line:
“The radical invents the views. When he has worn them out the conservative adopts them.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“A book is not an autonomous entity: it is a relation, an axis of innumerable relations. One literature differs from another, be it earlier or later, not because of the texts but because of the way they are read: if I could read any page from the present time—this one, for instance—as it will be read in the year 2000, I would know what the literature of the year 2000 would be like.”
—Jorge Luis Borges (1899–1986)
“Et in Arcadia ego.
[I too am in Arcadia.]”
—Anonymous, Anonymous.
Tomb inscription, appearing in classical paintings by Guercino and Poussin, among others. The words probably mean that even the most ideal earthly lives are mortal. Arcadia, a mountainous region in the central Peloponnese, Greece, was the rustic abode of Pan, depicted in literature and art as a land of innocence and ease, and was the title of Sir Philip Sidney’s pastoral romance (1590)
“The line of separation was very distinct, and the Indian immediately remarked, “I guess you and I go there,—I guess there’s room for my canoe there.” This was his common expression instead of saying “we.” He never addressed us by our names, though curious to know how they were spelled and what they meant, while we called him Polis. He had already guessed very accurately at our ages, and said that he was forty-eight.”
—Henry David Thoreau (1817–1862)