|
In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction: :Given three collinear points ''A, B, C,'' let ''L'' be a point not lying on their join and let any line through ''C'' meet ''LA, LB'' at ''M, N'' respectively. If ''AN'' and ''BM'' meet at ''K'', and ''LK'' meets ''AB'' at ''D'', then ''D'' is called the harmonic conjugate of ''C'' with respect to ''A, B''.〔R. L. Goodstein & E. J. F. Primrose (1953) ''Axiomatic Projective Geometry'', University College Leicester (publisher). This text follows synthetic geometry. Harmonic construction on page 11〕 What is remarkable is that the point ''D'' does not depend on what point ''L'' is taken initially, nor upon what line through ''C'' is used to find ''M'' and ''N''. This fact follows from Desargues theorem; it can also be defined in terms of the cross-ratio as (''A'', ''B''; ''C'', ''D'') = −1. ==Cross-ratio criterion== The four points are sometimes called a harmonic range (on the real projective line) as it is found that ''D'' always divides the segment ''AB'' ''internally'' in the same proportion as ''C'' divides ''AB'' ''externally''. That is: : If these segments are now endowed with the ordinary metric interpretation of real numbers they will be ''signed'' and form a double proportion known as the cross ratio (sometimes ''double ratio'') : for which a harmonic range is characterized by a value of -1, We therefore write: : The value of a cross ratio in general is not unique, as it depends on the order of selection of segments (and there are six such selections possible). But for a harmonic range in particular there are just three values of cross ratio: since -1 is self-inverse - so exchanging the last two points merely reciprocates each of these values but produces no new value, and is known classically as the harmonic cross-ratio. In terms of a double ratio, given points ''a'' and ''b'' on an affine line, the division ratio〔Dirk Struik (1953) ''Lectures on Analytic and Projective Geometry'', page 7〕 of a point ''x'' is : Note that when ''a'' < ''x'' < ''b'', then ''t''(''x'') is negative, and that it is positive outside of the interval. The cross-ratio (''c'',''d'';''a'',''b'') = ''t''(''c'')/''t''(''d'') is a ratio of division ratios, or a double ratio. Setting the double ratio to minus one means that when , then ''c'' and ''d'' are projective harmonic conjugates with respect to ''a'' and ''b''. So the division ratio criterion is that they be additive inverses. In some school studies the configuration of a harmonic range is called harmonic division. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「projective harmonic conjugate」の詳細全文を読む スポンサード リンク
|