Diagrams
for Exercise 2.10. Sketch and describe a circle for each model.
I am interested in mathematics only as
a creative art.
Godfrey H. Hardy (18771947)
The equation and at least one general diagram is given for each circle. Try to derive the equations for each circle.
Taxicab Circle
|x a| + |y b| = r
Euclidean Circle
(x a)2 + (y b)2
= r2
Max-distance Circle
max{|x a| , |y b|} = r
Poincarι Half-plane Circle
(x a)2 + (y bcosh
r)2 = (bsinh r)2
Missing-Strip Circles
For 0 <= a 1 < r and x < 0, (Euclidean otherwise)
(Note the division, /, in each equation.)
(x a)2 + (y b)2
= r2[1
(2(x a) + 1)/(x a + 1)2]
or
For r < a < 0 and x >= 1, (Euclidean otherwise)
(x a)2 + (y b)2
= r2[1
+ (2(x a) 1)/(x a 1)2]
or
For a <= r or a 1 >= r,
(x a)2 + (y b)2
= r2
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© Copyright 2005, 2006 - Timothy Peil |
