A Treatise On the Circle And the Sphere
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xvii. 1SS0.
and Swift, ' On the Conditions that a Point Transformation of the Plane be a Projective Transformation', Bulletin American Math. Soc. vol. x, 190i.
310 CIRCLE TRANSFORMATIONS ch.
ABGD in place. Now if a square be invariant, every square contiguous, that is, sharing one of its sides, is invariant.
Moreover, if a square be invariant, the four equal contiguous squares into which it can be divided will be invariant, Hence the plane is covered with an everywhere dense network of invarian...t squares of sides as small as we please, and as betweenness is invariant every point is invariant. Hence T" is the identical transformation, T' is a conformal collineation, and T is the product of such a collineation and inversions.
As a matter of fact, if T be not itself a conformal collineation, it will carry straight lines into circles meeting in only one point not exceptional for the transformation, i.e. into circles through a singular point, and may be factored into the product of a conformal collineation and an inversion with the singular point as centre.
What to read after A Treatise On the Circle And the Sphere?
You can find similar books in the "Read Also" column, or choose other free books by Coolidge, Julian Lowell, 1873-1954 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
xvii. 1SS0.
and Swift, ' On the Conditions that a Point Transformation of the Plane be a Projective Transformation', Bulletin American Math. Soc. vol. x, 190i.
310 CIRCLE TRANSFORMATIONS ch.
ABGD in place. Now if a square be invariant, every square contiguous, that is, sharing one of its sides, is invariant.
Moreover, if a square be invariant, the four equal contiguous squares into which it can be divided will be invariant, Hence the plane is covered with an everywhere dense network of invarian...t squares of sides as small as we please, and as betweenness is invariant every point is invariant. Hence T" is the identical transformation, T' is a conformal collineation, and T is the product of such a collineation and inversions.
As a matter of fact, if T be not itself a conformal collineation, it will carry straight lines into circles meeting in only one point not exceptional for the transformation, i.e. into circles through a singular point, and may be factored into the product of a conformal collineation and an inversion with the singular point as centre.
What to read after A Treatise On the Circle And the Sphere?
You can find similar books in the "Read Also" column, or choose other free books by Coolidge, Julian Lowell, 1873-1954 to read onlineMoreLess
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