Complex Conics And Their Real Representation
Complex Conics And Their Real Representation
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So the real axis of r maps into the real branch of the hyperbola.
It can be shown^ that to concentric circles about the origin in (r') correspond the pencil of circles having ± 1 for their limiting points, and to the radial lines in the r'-plane correspond the pencil of circles with vertices ±1. So we have the case corresponding to that of the ellipse: the hyperbolic pencil of circles in the r-plane with the common points ± 1 maps into confocal hyperbolas in the w- and s-surfaces; in particular..., the radical axis of the pencil going into the real branch of the hyper- bola. The orthogonal pencil with ± 1 as limiting points goes into the confocal ellipses.
6. The Real Parabola. — We take for the equation of the para- bola (1) V' = 4p(^ + P), ^ See Holzmiiller, p. 63.
IG COMPLEX CONICS AND THEIR REAL REPRESENTATION.
with the finite focus at the origin. Form the pencil, 77 = 2Ta + p).
Then, and, 1 — T- p 2p , 1 + 2ir, 1 + 2ir As in the previous cases we simplify these expressions by a linear transformation of the parameter (2) r = .
What to read after Complex Conics And Their Real Representation?
You can find similar books in the "Read Also" column, or choose other free books by Benjamin Ernest Mitchell to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
So the real axis of r maps into the real branch of the hyperbola.
It can be shown^ that to concentric circles about the origin in (r') correspond the pencil of circles having ± 1 for their limiting points, and to the radial lines in the r'-plane correspond the pencil of circles with vertices ±1. So we have the case corresponding to that of the ellipse: the hyperbolic pencil of circles in the r-plane with the common points ± 1 maps into confocal hyperbolas in the w- and s-surfaces; in particular..., the radical axis of the pencil going into the real branch of the hyper- bola. The orthogonal pencil with ± 1 as limiting points goes into the confocal ellipses.
6. The Real Parabola. — We take for the equation of the para- bola (1) V' = 4p(^ + P), ^ See Holzmiiller, p. 63.
IG COMPLEX CONICS AND THEIR REAL REPRESENTATION.
with the finite focus at the origin. Form the pencil, 77 = 2Ta + p).
Then, and, 1 — T- p 2p , 1 + 2ir, 1 + 2ir As in the previous cases we simplify these expressions by a linear transformation of the parameter (2) r = .
What to read after Complex Conics And Their Real Representation?
You can find similar books in the "Read Also" column, or choose other free books by Benjamin Ernest Mitchell to read onlineMoreLess
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