Geometry: A Comprehensive Course by Dan Pedoe

Author:Dan Pedoe [Pedoe, Dan]
Language: eng
Format: epub, mobi, pdf
Tags: Mathematics, Geometry, General
ISBN: 9781306340557
Google: s7DDxuoNr_0C
Amazon: 1306340551
Publisher: Dover Publications
Published: 2013-01-01T22:00:00+00:00

60.2The principle of duality in the projective plane

The symmetrical nature of the incidence relation

u0y0 + u1y1 + u2y2 = 0.

which expresses the fact that the line [u0, u1, u2] contains the point (y0, y1, y2), or equivalently, that the point (y0, y1, y2) lies on the line [u0, u1, u2] gives rise to the Principle of Duality in the projective plane. This asserts that by an automatic interchange of the terms point and line, lying on and passing through, join and intersection, collinear and concurrent, and so on, any theorem in the projective plane which involves only incidence properties of points and lines becomes, on transliteration with the help of the dictionary of interchanges we have just listed, a theorem involving lines and points.

To give an immediate example, the theorem of Desargues, which we have already encountered (§5.2) and which is essentially a theorem of the projective plane, says that if ABC, A′B′C′ are two triangles in the projective plane which are such that the lines AA′, BB′ and CC′ are concurrent (that is, pass through the same point), then the three points of intersection of corresponding sides, BC ∩ B′C′,CA ∩ C′A ′ and AB ∩ A′B′are collinear (lie on a line).

The theorem obtained by applying the Principle of Duality says that if abc, a′b′c′ are triangles (we should change the term triangle, which refers to a configuration formed by three points, to triline, to be precise, since we now consider two configurations formed by triads of lines, a, b and c and a′, b′ and c′, these not being concurrent: but we shall not do this) in the projective plane which are such that the points aa′, bb′ and cc′ are collinear (lie on a line) then the lines joining corresponding vertices, bc ∪ b′c′,ca ∪ c′aand ab ∪ a′b′ are concurrent (pass through a point).

We shall see, in fact, that exactly the same algebra which we shall use to establish the Desargues Theorem will establish the dual theorem, which is the converse of the Desargues Theorem (§62.2).

It will be noted that we are using the symbol AB to denote the line joining two distinct points A and B, and dually the symbol ab to denote the point of intersection of the two distinct lines a and b. Since we use the symbol ∩ for the intersection of two point-sets, the point of intersection of two lines AB and A′B′ is written as AB ∩ A′B′, and therefore dually we write ab ∪ a′b′ for the line joining the points ab and a′b′, although the ∪ symbol here does not stand for the join of two point-sets. Another symbol, if used, would also create difficulties. Some authors use the symbol A + B for the set of points on the line joining A to B. We shall not do this, and we do not think that there will be any misunderstanding in interpreting our symbolism.

The reader may wonder why we talk of the Principle of Duality, when our statement describing it amounts to a proof.

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