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NOTES ON HISTORY OF GEOMETRY
We consider only the plane under different geometries. We use PG(F) for the
projective plane over a field F, and AG(F) for the affine plane over F.
Projective plane: In homogeneous coordinates, the transformations are
x’1 = a11x1 + a12x2 + a13x3
x’2 = a21x1 + a22x2 + a23x3
x’3 = a31x1 + a32x2 + a33x3
In non-homogeneous coordinates:
x’ = (a11x1 + a12y2 + a13)/(a31x1 + a32y2 + a33)
y’ = (a21x1 + a22y2 + a23)/(a31x1 + a32y2 + a33)
In any plane, a continuous collineation is always a projective transformation.
In the real plane, a projective transformation always has at least one fixed point and one
fixed line.
Invariants include: linearity, collinearity, cross ratio, harmonic sets, property of being a
conic section
Examples of theorems of projective geometry: Desargues’ Thm (and its dual, which
turns out to be its converse), Pascal’s Thm (and its dual Brianchon’s Thm),
and Pappus’s Thm
A projective plane PG(F) can be defined over any skew field F. Then Pappus’s Theorem
holds in PG(F) iff the skew field F is a field (i.e. multiplication is
commutative) iff the Fund. Thm. holds in PG(F) iff the Corollary of the
Fund. Thm. holds in PG(F).
Fundamental Theorem of (Plane) Projective Geometry: There is a unique projective
transformation that sends 3 given collinear points A, B, C into 3 given
collinear points A’, B’, C’ respectively. [Seidenberg, 25]
The Fund. Thm. can be expressed for the n-dimensional projective geometry over F as
follows: There is a unique projective transformation T that sends n+2
given points into n+2 given points, where no n+1 of the given points and
no n+1 of the image points lie in a hyperplane. [Pedoe, 295]
Thus, in the plane, this says that there is a unique projective transformation that sends 4
given points (no 3 collinear) into 4 given points (no 3 collinear).
Corollary of Plane F.T.: Any projective transformation between two given distinct lines
that has a fixed point is a perspectivity.
Def. If Desargues’ Theorem holds in PG(F), it is called a Desarguesian projective plane.
If the Fundamental Theorem holds in PG(F), it is called a classical
projective plane. (PG(F) is classical if F is a field; otherwise F is a division
ring and PG(F) is called a non-classical projective plane.)
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Pappus’s Thm is a special case of Pascal’s Thm, which is true in every classical
projective plane.
In PG(F), Desargues’ Thm. can hold but the Fundamental Thm fail. Desargues’ Thm is
not a consequence of the axioms of proj. geometry. But every plane in a
3-dimensional projective geometry is Desarguesian.
In a classical projective plane, any 5 points (no 3 of which are collinear) lie on a unique
conic.
Affine plane: In non-homogeneous coordinates, the transformations are
x’ = ax + by + j
y’ = cx + dy + k
where ad – bc ≠ 0 (all of a,b,c,d,j,k are
constant)
This means that a31 = a32 = 0 and that a33 ≠ 0 and that a = a11/a33 and b = a12/a33 and c
= a21/a33 and d = a22/a33
(Note that such a transformation preserves area, according to Coxeter p.206, iff
ad – bc = 1.)
In effect, we have fixed any line in the projective plane and called it the “line at
infinity”. The affine group leaves the line at infinity fixed (though the
affine group may interchange points on this line).
Invariants include: all those of the projective group as well as parallelism (also
betweenness of pts. on a line and the midpoint of a segment; also, for
collinear pts. A, B, C with B between A and C, the ratio AB/AC, though
not the distances AB or AC; also the notions of segment, half line, and half
plane)
[Note that two pts on an affine line are parallel if they have the same ideal
pt I, which is not in the affine plane; the midpoint M of the segment AB is
defined as the harmonic conjugate of I with respect to A and B, i.e. A, B, I,
M is a harmonic set.]
Note that dilations are examples of affine transformations that are not Euclidean
transformations.
Affine geometry was first noted by Euler and later by Möbius.
Some theorems of affine geometry that are not theorems of projective geometry:
The medians of a triangle are concurrent.
Playfair’s axiom: Given a line l and a point P not on l, there is
exactly one line through P parallel to l.
Theorem: There is a unique affine transformation that sends 3 given non-collinear
points A, B, C into 3 given non-collinear points A’, B’, C’ respectively
(i.e. sends a given triangle to a given triangle). [Coxeter, 203] Notice that
this theorem fails for existence in similarity geometry and for
uniqueness in projective geometry.
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Similarity Geometry: A subgroup of the affine group in the plane is the group
of similarity transformations. (Such transformations are those that,
when considered in the complex projective plane, leave the circular
points at infinity fixed. These are the points (1,i,0) and (1,–i,0)
which belong to every circle in the complex projective plane.)
It seems that as a consequence of this, in the real plane there is a unique
similarity transformation which maps two given points to two
given points. Contrast this with affine and projective maps.
The group of plane similarity geometry: In non-homogeneous coordinates, the
transformations are
x’ = ax – by + j
y’ = bcx + acy + k where a2 + b2 ≠ 0 and c2 = 1
(all of a,b,c,j,k are constant)
Invariants include: All those of affine geometry as well as angle measure, but not
distance.
Examples of theorems of similarity geometry that are not theorems of affine
geometry:
(Euclid I, 27) If a straight line falling on two straight lines makes
the alternate angles equal to each other, then the two straight lines
are parallel.
(Euclid I, 32) In any triangle, if one of the sides is extended, then
the exterior angle is equal to the sum of the two interior and
opposite angles, and the three interior angles of the triangle have a
sum equal to two right angles [i.e. 180 degrees].
Equiareal Geometry: A different subgroup of the affine group (in the plane) is
the group of equiareal transformations. These preserve the area of
a triangle. According to Adler (1966, 348), the Euclidean group is
the intersection of the similarity group and the equiareal group.
The equiareal transformations have determinant = 1, according to the comment
of Coxeter above.
Inversive Geometry: Any projective involution T (i.e. T is of order 2) of a line
(in the real projective plane) onto itself has exactly 2 real fixed
points (and no imaginary ones) or else exactly 2 imaginary fixed
points (and no real ones).
The (real) inversive plane is the Euclidean plane with a point at infinity
appended (not a line at infinity).
The real inversive plane can be thought of as the complex projective line. In nonhomogeneous coordinates, its transformations (sometimes called
Moebius transformations) are the complex projective
transformations on a complex projective line:
z’ = (a11z + a12)/(a21z + a22)
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with det(aij) ≠ 0
In the (real) inversive plane, an inversive transformation maps a circle or line to a
circle or line (i.e. is circle-preserving in that lines can be thought of
as circles of infinite radius). These transformations are also
conformal (i.e. angle-preserving).
An inversive transformation can have at most two fixed points, or else it is the
identity mapping.
Consider the subgroups of these transformations which preserve the unit circle C
and map points inside it to points inside it. The interior of this circle is
then the Poincaré model of hyperbolic geometry (with “lines” as
segments of circles or straight lines that are orthogonal to C).
Metric planes (Euclidean, hyperbolic, elliptic, etc.): In non-homogeneous
coordinates, the transformations are
x’ = ax + by + j
y’ = cx + dy + k where ad – bc = ±1
(a,b,c,d,j,k are constant)
These are what is obtained when the transformations fix a given conic (Cayley’s
absolute) in the projective plane. If the conic is real, imaginary, or
degenerate, we obtain hyperbolic, elliptic, or parabolic (Euclidean)
geometry respectively [Bell, Development, 353].
The Euclidean group consists of the rotations, translations, and reflections.
Invariants include: all of the above, and distance.
The transformations of the Euclidean plane are
x’ = rx cos θ – ry sin θ + j
y’ = rx sin θ + ry cos θ + k
where r =±1
(θ,r,j,k are constant)
Example of a theorem of Euclidean geometry that is not a theorem of similarity
geometry or of equiareal geometry:
The Pythagorean Theorem
In the Euclidean plane, a non-degenerate conic intersects the line at infinity in two
distinct points (a hyperbola), in exactly one point (a parabola), or in
two imaginary points (an ellipse).
The points of the absolute are at an infinite distance from all other points.
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Absolute geometry (i.e. Euclidean geometry with the Parallel Postulate omitted, and
so the common part of both Euclidean and hyperbolic geometry): What
does its group of transformations look like algebraically?
Examples of propositions that, in absolute geometry, are equivalent to the
Parallel Postulate:
1. (Playfair’s Axiom) Given a line and a point not on the line, there is only
one line not meeting the given line.
2. (Euclid I, 29) A line falling on two lines that do not meet makes the
alternate interior angles equal.
3. (Euclid I, 32) The sum of the angles of any triangle is two right angles.
4. There is a triangle whose angle sum is 180 degres.
5. (Euclid I, 33, corollary) Two non-intersecting lines are everywhere equidistant.
6. There exists a rectangle (i.e. a quadrilateral with four right angles).
7. There exist two triangles that are similar but not congruent.
8. There exists a pair of lines that are everywhere equidistant.
9. Given any three non-collinear points, there is a circle that passes through them.
10. Given any area, there is a triangle whose area is greater than the given area.
To characterize hyperbolic geometry, return to projective geometry (i.e., we cannot
merely go back to affine geometry) and consider a definite but arbitrary
real, non-degenerate conic (the absolute). The subgroup of projectivities
which leave this conic invariant (though not necessarily pointwise) is
called the hyperbolic metric group. (Points on the conic are then
considered to be on the line at infinity.)
Invariants: those associated with congruence.
The points of the absolute are at an infinite distance from all other points.
In hyperbolic geometry, there are various theorems that are false in Euclidean
geometry, such as
1. There is a least upper bound to the area that a triangle may have. This
l.u.b. is approached as the sum of the angles approaches zero.
2. The area of a triangle is proportional to its defect d. If the angles of the
triangle are A, B, C, then the defect d = 180 – (A + B + C), i.e. the
difference between the angle sum and 180 degrees. So area = kd,
where k is a positive constant. (Cf. 10 above.)
Elliptic geometry is the subgroup of the projective group leaving a definite
imaginary conic (the absolute) of the projective plane invariant.
Invariants: those associated with congruence.
In elliptic geometry:
Every straight line is closed and of finite length.
A straight line does not divide the plane into two convex regions.
There is no (real) line at infinity.
The angles of a triangle add up to more than 180 degrees.
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