Inversive Geometry (Dover Books on Mathematics)
Frank Morley
Language: English
Pages: 288
ISBN: B00I17XW1I
Format: PDF / Kindle (mobi) / ePub
The two-part treatment begins with the applications of numbers to Euclid's planar geometry, covering inversions; quadratics; the inversive group of the plane; finite inversive groups; parabolic, hyperbolic, and elliptic geometries; the celestial sphere; flow; and differential geometry. The second part addresses the line and the circle; regular polygons; motions; the triangle; invariants under homologies; rational curves; conics; the cardioid and the deltoid; Cremona transformations; and the n-line.
Mathematical Methods for Business and Economics
A Readable Introduction to Real Mathematics (Undergraduate Texts in Mathematics)
Discrete Dynamical Models (UNITEXT, Volume 76)
The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry
the line PI, Pa. Hence for an acute-angled triangle the pitch, namely, I plW -PI I is the perimeter of the triangle PI> P2, Pa. Exercise 9 - For a triangle with an obtuse angle at ~ the pitch is I~-AI-IA-AI+IA-~I If now we regard an acute-angled triangle as an instrument, and overturn it first with the side 1 fixed, then with the side 2 fixed, and then with the side 3 fixed, we get fig. 17. The points Pa, PI of the first position (marked I) are in line with the points PI, P2 of the second
together. Thus with the general inversion is associated the selfconjugate form pxx -ax -ai+a INVERSIONS 42 This defines a curve-a bilinear curve-which is a general word for the three types~ircle, extra pair, double point. In general we associate with any self-conjugate form in x and x the word curve. H we replace x by fi (or x by y) we get the transformation associated with the curve-a generalisation of inversion. § 22. Inversors - An appropriate instrument for inversion is a linkwork or
mean point or centroid g(I-A)=hl -Ah2 or A= (g - h 1 )/(g - h 2 ) The solutions are then (x - h 1 )/(x - h 2 ) =a cube root of A Thus 6 82 FINITE INVERSIVE GROUPS This gives the circumcircle of the three points Xi; and the angle hlXh2 is a third of the angle h 1 yh 2. This gives three FIG. 28 arcs from ~ to h2' which cut the circumcircle at the points Xi (fig. 28). § 43. The Groups of the Rectangle and Rhombus - In § 26 we had the four inversions: x=y X= -Y xy=1 xy= - I These form a
then represents the motion H, and we call the circle also H. The motion is hyperbolic, parabolic, or elliptic as the circle cuts, touches, or does not meet O. For two motions HI, H2 we have two directed circles. When negative common tangent arcs exist (fig. 31 (a» they 94 PARABOLIC, HYPERBOLIC, ELLIPTIC GEOMETRIES have as their end-points the fixed points of HIH2 and those of H 2H 1 . When positive common tangent arcs exist (fig. 31 (b», they give the common pairs of HI' H 2-that is, the fixed
a and b, we take ap on the ray to a, bp on the ray to b. Euclid's theory states that a - band ap - bp are parallel. OPERATIONS OF ELEMENTARY GEOMETRY 9 If we join (fig. 4) b to ap, the parallel on bp gives ap2. If we join a to bp, the parallel on b gives alp or ap-l. We construct thus on either ray a geometrical progression with positive ratio or radial scale. The point 0 is inaccessible. Equally if we take the line on 0 and a, any point on it is ap where p is a real. And taking a second line
