Projectivities
Before we start, I should not that there are a lot of definitions in this installment. Here are the first few.
The range of a line is the set of all points incident to the line.
The pencil of a point is the set of all lines incident the point.
Both ranges and pencils are examples of forms.
Part 1
We start be proving 6.1 from the previous chapter, We first prove a weaker version of 6.2.
Theorem 1: Given a line and a point not incident to each other, the corresponding range and pencil have the same cardinality.
Proof: Let our point and line be \(P\) and \(l\) respectively. Then construct a function \(|f|\) from \(l\)’s range to \(P\)’s pencil so that for any point \(A\) on \(l\), \(PA = |f|(A)\). From Theorem 1 of the previous chapter, \(|f|\) is injective, as non-injectivity implies that a line through \(P\) is incident to two distinct points on \(l\). From Axiom 2, \(|f|\) is surjective, because any line \(m\) though \(P\) must meet \(l\), and then \(|f|(lm) = l\). Thus there is a bijection between the range and the pencil, and their cardinalities must be identical.
Lemma 1:
Given two distinct lines, there is a point which is incident to neither of them. (Proof left to reader.)
Corollary 1.1: Given any two distinct lines, their ranges have the same cardinality.
Proof: Starting with the lines \(a\) and \(b\), from Lemma 1 we obtain a point \(P\) not on \(a\) or \(b\). From Theorem 2, the range of \(a\) and pencil of \(P\) have the same cardinality. Again from Theorem 2, the pencil of \(P\) and the range of \(b\) have the same cardinality. Thus, the ranges of \(a\) and \(b\) have the same cardinality, as required.
The constructions used in the proof of Theorem 2 and implied in the proof of Corollary 1.1 are the subject of this particular section. First, we need to rigorously define these constructions.
Given a line \(a\) and a point \(T\), the half-perspectivity from \(a\) to \(T\) is the bijection from the range of \(a\) to the pencil of \(T\) such that for any point \(C\) on \(a\), its image is \(CT\).
The half-perspectivity from \(T\) to \(a\) also exists, and it is the bijection from the pencil to the range such that for any \(b\) on \(T\), its image is \(ab\). Note that this is the inverse of the half-perspectivity from \(a\) to \(T\).
Notation: The half-perspectivity from \(a\) to \(T\) shall be called \(a>T\), and the half-perspectivity from \(T\) to a shall be called \(T<a\).
Given two distinct lines \(a\), \(b\) and a point \(C\) not incident to either, the perspectivity from \(a\) to \(b\) with centre \(C\) is the function from the range of \(a\) to the range of \(b\) where for any point \(X\) on \(a\), its image \(Y\) under the perspectivity is \(X' = b[XC]\).
Given two distinct points \(P\), \(Q\) and a line \(a\) not incident to either, the perspectivity from \(P\) to \(Q\) with axis \(a\) is the function from the pencil of \(P\) to the pencil of \(Q\) where for any line \(x\) through \(P\), its image \(y\) under the perspectivity is \(x' = Q[xa]\)
Notation: The perspectivity from \(a\) to \(b\) with centre \(C\) can be written as \(a>C<b\). The perspectivity from \(P\) to \(Q\) with axis \(a\) can be written as \(P<a>Q\).
Note that both types of perspectivities can be considered a composition of two compatible half-perspectivities. We can generalize this idea as follows.
A projectivity is function between forms composed from a finite number of half-perspectivities.
Note that this definition encompasses both perspectivities and half-perspectivities. Also, the identity perspectivity, which takes a form to itself with all elements self-corresponding, is also a projectivity.
Notation: The name of a projectivity may be its presentation in half-perspectivities(e.g. \(a>B<c>D<e>F\)), a string of letters(and other notation) enclosed by vertical bars(e.g. \(|hKmN|\), \(|W/(xy)Z|\)) or any combination of the two (e.g. \(P<|abc|<i>J<k\), \(A<|x|<e\)). The composition of projectivities \(|a|\) and \(|b|\) is \(|ab|\), such that for any \(x\), \(|ab|(x) = |b|(|a|(x))\). The inverse of \(|a|\) is \(|/a|\). The identity projectivity is \(|1|\), regardless of domain.
More notation: If we wish to keep track of the images of particular points, the name of a line or point in a projectivity presentation can be suffixed by a tuple of elements from the form (e.g. \(l(A, B, C)>|R|>P(d, e, f)\)). The elements in each tuple in the presentation are associated in order, first to first and so on. If the tuples are present, then the line or point names may be suppressed (e.g. \(abc<|T|<DEF\)).
Given two tuples of the same size each with elements from a single form, we say the two tuples are projective if there exists some projectivity which takes one tuple to the other tuple.
Notation: We make note of a pair of projective tuples with the \(\sim\) symbol (e.g. \((U, V, W) \sim (x, y, z)\)).
We say that two projectivities are identical(or equal) when they have the same domain and co-domain and for each point in the domain, its image under either projectivity is the same.
Part 2
When compared to a generic projectivity, perspectivities and half-perspectivities have certain special properties. One is the restriction of the type of form the domain and co-domain can be. Half-perspectivities must have domain and co-domain of different types, while for perspectivities they must be of the same type. Furthermore, the domain and the co-domain for a perspectivity must be distinct, while there is no such restriction for projectivities.
If we look carefully at the definition of a perspectivity, then the following theorem is apparent.
Theorem 2: For any perspectivity, the common point of the domain and the co-domain is self-corresponding.
For a generic projectivity, even if the domain and co-domain are of the same type yet distinct, their common element may not be self-corresponding, as shown by the theorem below.
Theorem 3: Given any two distinct lines \(m\) and \(n\), and tuples of distinct points \((A, B, C)\) and \((A', B', C')\) with elements from the ranges of \(m\) and \(n\) respectively, then \((A, B, C) \sim (A', B', C')\).
(Note: On the term “tuples of distinct points”, only distinctness within a tuple is intended, as is the case most of the time. Exceptions will be noted on occurrence.)
The following lemma will be use in the proof below and many other proofs.
Lemma 2 (Collinearity Lemma(Col L)):
If the intersection of two collinear sets contains two distinct points, then the union of the two sets and any subset of it is also collinear. (Proof left to reader)
Naturally the dual Concurrency Lemma(Con L) also holds.
Proof of Theorem 3: Assume without loss of generality that neither \(A\) nor \(A'\) is the point \(mn\). Define the points \(B"\) and \(C"\) as \([[A'B][AB']]\) and \([[A'C][AC']]\) respectively. If \(B”\) and \(C”\) are identical, that would imply \(\{A, B', C'\}\) is a collinear set, which in turn implies that \(A\) lies on \(n\), contrary to the premise. This \(B”\) and \(C”\) are distinct and the line \(B”C” = o\) is well-defined.
If we assume the line \(o\) passes through \(A\), then the set \(\{A, B”, C”\}\) is collinear. The sets \(\{A, B”, B'\}\) and \(\{A, C”, C'\}\) are collinear by construction, thus by the Col. L. the set \(\{A, B', C'\}\) must be as well, which again implies \(A\) lies on \(n\), contrary to the premise. Therefore \(B”C”\) does not pass through \(A\). An analogous argument shows that \(o\) does not pass through \(A'\). Let \(A” = o[AA']\). We have now shown that the following projectivity exists.
\(m(A, B, C)>A'<o(A”, B”, C”)>A<n(A', B', C')\) (see pic 1)
This projectivity shows \((A, B, C) \~ (A', B', C')\), as required.
Looking closely at the proof, we that if, say, the \(B' = mn\), then all that implies is that \(om = B = B”\). If \(C = mn\) as well, then \(o = BC'\), \(C” = C'\) and the projectivity still exists (see pic 2). In these cases the point \(mn\) will definitely not be self-corresponding.
Part 3
This result still leaves open the possibility of a converse to Theorem 2 for ‘eligible’ projectivities. What remains to be seen is what counts as an eligible projectivity. In the next few sections our main focus will be to introduce new axioms that bear on this question. Right now, even with just the plane axioms, we prove some consequences of a converse of Theorem 2.
Lemma 3: Given two projectivities \(|a|\) and \(|b|\), \(|a|\) and \(|b|\) are identical iff \(|a/b| = |1|\)
Proof: Let \(x\) be an element in the common domain of \(|a|\) and \(|b|\). First we assume \(|a|\) and \(|b|\) are identical. Then \(|a|(x) = |b|(x) = y\). Thus, we have \(|/b|(y) = x\), and composing gives us \(|a/b|(x) = y\). Since \(x\) was chosen arbitrarily, \(|a/b| = |1|\) holds. Now assume \(|a/b| = |1|\). Let \(|a|(x) = z\). Therefore, \(|/b|(z) = x\), and thus, \(|b|(x) = z = |a|(x)\), as required.
Lemma 4: If \(|PQ| = |1|\), then so does \(|QP|\).
Proof: By the premise, for any \(X\) in the domain of \(|PQ|\), \(|PQ|(X) = X\).
Let \(Y = |P|(X)\). Then it must be the case that \(|Q|(Y) = X\). Therefore, \(|QP|(Y) = Y\). However, since \(|P|\) is a bijection and \(x\) was chosen arbitrarily, \(Y\) is also arbitrary. Thus, \(|QP| = |1|\).
Theorem 4: If the projectivity \(a>R<b>P<c\) is identical to the perspectivity \(a>Q<c\), then \(a>Q<c>P<b\) is also identical to \(a>R<b\).
Proof: The premise implies \(a>R<b>P<c>Q<a = |1|\) by Lemma 3, which in turn implies \(b>P<c>Q<a>R<b = |1|\) by Lemma 4. Then, by Lemma 3 again, \(b>P<c>Q<a = b>R<a\), and taking the inverse of both we have \(a>Q<c>P<b = a>R<b\), as required.#
Theorem 5: If the projectivity \(a>R<b>P<c\) is identical to the perspectivity \(a>Q<c\), then either \(\{a, b, c\}\) is concurrent, or \(\{ac, P, R\}\) is collinear.
Proof: Since the two projectivities are equal, then the images of one point under each are the same. In particular, the image of the point \(ac\) is the same for both. Since \(a>Q<c\) is a perspectivity, then \(ac\) is self-corresponding. If \(a\), \(b\), and \(c\) are concurrent, then this is the case for \(a>R<b>P<c\) anyways. If \(a\), \(b\), and \(c\) are not concurrent(see pic 1), then the component perspectivity \(a>R<b\) must send \(ac\) to a point not equal to \(ac\) on \(b\). Call this point \(E\). By construction, \(\{E, ac, R\}\) is collinear. Since under the full projectivity \(ac\) is self-corresponding, the second component perspectivity \(b>P<c\) must send \(E\) back to \(ac\), thus \(\{E, ac, P\}\) collinear. Therefore, \(\{P, R, ac\}\) is collinear, by Col L.
Theorem 6: If the projectivity \(a>R<b>P<c\) is identical to the perspectivity \(a>Q<c\) and \(\{a, b, c\}\) is concurrent, then \(\{P, Q, R\}\) is collinear.
Proof: First assume \(P\), \(Q\), and \(ac\) are not collinear. Let \(M = a[PR]\), \(M' = b[PR]\), and \(M” = c[PR]\)(see pic 2). By assumption \(M\), \(M'\) and \(M”\) are distinct. Note that \(M'\) is the image of \(M\) under \(a>R<b\), and \(M”\) is the image of \(M'\) under \(b>P<c\). By composition and the theorem premise, \(M”\) is the image of \(M\) under \(a>Q<c\). Thus, \(Q\) lies on \(MM”\), but \(MM” = PR\) by definition, proving the theorem. This proof breaks down if \(\{P, R, ac\}\) is collinear. In this case, from Theorem 4 we have \(b>P<c>Q<a\) identical to \(b>R<a\). If \(\{Q, P, ab\}\) is not collinear, \(R\) can be proven to lie on \(QP\) with the argument above. If they are collinear, that implies \(\{P, R, Q\}\) collinear by Col L, again satisfying the theorem.
Theorem 7: If the projectivity \(a>R<b>P<c\) is identical to the perspectivity \(a>Q<c\) and \(\{a, b, c\}\) is not concurrent, then \(Q = [R[bc]][P[ab]]\).
Proof: By the theorem premise, \(ab\) and \(bc\) are distinct. Note also that neither \(P\) or \(R\) can lie on \(b\). Let \(F\) be the image of \(ab\) under \(b>P<c\) and let \(D\) be the image of \(cb\) under \(b>R<a\)(see pic 3). By composition, \(a>R<b>P<c\), and thus \(a>Q<c\), must take \(ab\) to \(F\) and \(D\) to \(bc\). Thus \(\{ab, P, Q\}\) is collinear, as well as \(\{Q, R, bc\}\). Thus, \(Q\) lies on \(P[ab]\) and \(R[bc]\). Since \(P\) and \(R\) do not lie on \(b\) while \(ab\) and \(bc\) do, \(P[ab]\) and \(R[bc]\) are distinct lines, making \(Q\) the meet of \(P[ab]\) and \(R[bc]\).
One last theorem of note is the existence of a particular projectivity with identical domain and co-domain.
Theorem 8: Given any 4 distinct lines \(a, b, c, d\) in the pencil \(P\), \(P(a, b, c, d) \sim P(b, a, d, c)\).
Proof: Choose points \(P^*\),\(P''\) on lines \(c\), \(d\) respectively distinct from \(P\). Choose \(X\) on line a distinct from \(P\) and not collinear to \(P^*\) and \(P''\). Then label the following lines:
\[e = P^*P''\] \[a^* = XP^*\] \[a'' = XP''\]
\[b^* = P^*[a''b]\]
\[b'' = P''[ab^*]\]
(see pic 4 and 5)
Thus \(abcd<a''>a^*b^*ce<a>a''b''de<b^*>badc\) provides the required perspectivity.