Projective harmonic conjugate

The harmonious division referred to in the geometry of a particular location ratio of four points on a line. A harmonic division occurs when a distance ( for example, [AB] ) by two other points (such as T and U ) is internally and externally divided so that the two dividing ratios (here (ATB ) and ( AUB) ) have the same have amount. In other words, this means that the cross-ratio is -1.

Finding the points of division

If the segment [ AB ] and the division ratio given λ, as can be the inner and the outer part point by the methods given in " split ratio " find.

Often, however, a stretch and a division point are given, and to find the fourth harmonic point (more precisely, the fourth point, which adds up with these 3 points a harmonious division ). These help a construction according to the adjacent drawing: The point C is chosen arbitrarily, the straight lines AC and BD are parallel. Point D (or D ' ) is obtained by the combination of C and the given division point. D to D 'is transmitted ( or vice versa), the lines [BD ] and [ BD '] are equal. The missing part is due to the connection point of C with D '( or D).

Mathematically, there is the length of the segment [ AU ] if the length of [ AB ] is given ( and part of the point T), from the formula:

• If, in this formula, a negative distance, the point U is entered on the side of the line that does not contain the point B.
• The formula can also be used, if T is the outer and the inner part of the U point. If then T of A as seen on the side of the line that does not contain the point B, it must be registered for a negative number.

Relations and special cases

If two points T and U is a segment [ AB ] share harmoniously, so the points A and B are also harmonic component points of the segment [ TU ]. The harmonious division is therefore a mutual relationship of two on the same straight lines lying. Not quite right, it is, however, to say that " four points are harmonious with each other ." Taking namely [ AT ] or [ AU ] as the starting line, thus generating the two remaining points no harmonic division. Rather, the double ratios ( ATBU ) = 2 and ( AUTB ) = 1/2.

The fourth harmonic point to the midpoint of a segment is the far point of the corresponding straight line, and vice versa.

The fourth harmonic point to the end point of a route is the point itself

Functional relationship

Given two points A and B and a third variable point T on the line AB the assignment of to T ( with respect to the segment [ AB ] ) fourth harmonic point U is a reversible one mapping of the line AB ( including their far point! ) Is on itself

Is used as the segment [ AB ] the interval [ 0, 1 ] - which by suitable coordinates always goes without limiting the generality of the points option - and let t be the coordinate of the point T, then the coordinate is u of U from the functional equation