L’algèbre n’est qu’une géométrie écrite; la géométrie n’est qu’une algèbre figurée.
—Sophie Germain
19.1
Affine Spaces
Geometrically, curves and surfaces are usually considered to be sets of points with some
special properties, living in a space consisting of “points.” Typically, one is also interested
in geometric properties invariant under certain transformations, for example, translations,
rotations, projections, etc. One could model the space of points as a vector space, but this is
not very satisfactory for a number of reasons. One reason is that the point corresponding to
the zero vector (0), called the origin, plays a special role, when there is really no reason to have
a privileged origin. Another reason is that certain notions, such as parallelism, are handled
in an awkward manner. But the deeper reason is that vector spaces and affine spaces really
have different geometries. The geometric properties of a vector space are invariant under
the group of bijective linear maps, whereas the geometric properties of an affine space are
invariant under the group of bijective affine maps, and these two groups are not isomorphic.
Roughly speaking, there are more affine maps than linear maps.
Affine spaces provide a better framework for doing geometry. In particular, it is possible
to deal with points, curves, surfaces, etc., in an intrinsic manner, that is, independently
of any specific choice of a coordinate system. As in physics, this is highly desirable to
really understand what is going on. Of course, coordinate systems have to be chosen to
finally carry out computations, but one should learn to resist the temptation to resort to
coordinate systems until it is really necessary.
Affine spaces are the right framework for dealing with motions, trajectories, and physical
forces, among other things. Thus, affine geometry is crucial to a clean presentation of
kinematics, dynamics, and other parts of physics (for example, elasticity). After all, a rigid
motion is an affine map, but not a linear map in general. Also, given an m × n matrix A
and a vector b ∈ m
n
R , the set U = {x ∈ R | Ax = b} of solutions of the system Ax = b is an
477
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CHAPTER 19. BASICS OF AFFINE GEOMETRY
affine space, but not a vector space (linear space) in general.
Use coordinate systems only when needed!
This chapter proceeds as follows. We take advantage of the fact that almost every affine
concept is the counterpart of some concept in linear algebra. We begin by defining affine
spaces, stressing the physical interpretation of the definition in terms of points (particles)
and vectors (forces). Corresponding to linear combinations of vectors, we define affine com-
binations of points (barycenters), realizing that we are forced to restrict our attention to
families of scalars adding up to 1. Corresponding to linear subspaces, we introduce affine
subspaces as subsets closed under affine combinations. Then, we characterize affine sub-
spaces in terms of certain vector spaces called their directions. This allows us to define a
clean notion of parallelism. Next, corresponding to linear independence and bases, we define
affine independence and affine frames. We also define convexity. Corresponding to linear
maps, we define affine maps as maps preserving affine combinations. We show that every
affine map is completely defined by the image of one point and a linear map. Then, we
investigate briefly some simple affine maps, the translations and the central dilatations. At
this point, we give a glimpse of affine geometry. We prove the theorems of Thales, Pappus,
and Desargues. After this, the definition of affine hyperplanes in terms of affine forms is
reviewed. The section ends with a closer look at the intersection of affine subspaces.
Our presentation of affine geometry is far from being comprehensive, and it is biased
toward the algorithmic geometry of curves and surfaces. For more details, the reader is
referred to Pedoe [85], Snapper and Troyer [95], Berger [6, 7], Coxeter [24], Samuel [87],
Tisseron [105], and Hilbert and Cohn-Vossen [54].
Suppose we have a particle moving in 3D space and that we want to describe the trajectory
of this particle. If one looks up a good textbook on dynamics, such as Greenwood [49], one
finds out that the particle is modeled as a point, and that the position of this point x is
determined with respect to a “frame” in
3
R by a vector.
Curiously, the notion of a frame is
rarely defined precisely, but it is easy to infer that a frame is a pair (O, (e1, e2, e3)) consisting
of an origin O (which is a point) together with a basis of three vectors (e1, e2, e3). For
example, the standard frame in
3
R has origin O = (0, 0, 0) and the basis of three vectors
e1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1). The position of a point x is then defined by
the “unique vector” from O to x.
But wait a minute, this definition seems to be defining frames and the position of a point
without defining what a point is! Well, let us identify points with elements of 3
R . If so, given
any two points a = (a1, a2, a3) and b = (b1, b2, b3), there is a unique free vector , denoted by
−
→
−
→
ab, from a to b, the vector ab = (b1 − a1, b2 − a2, b3 − a3). Note that
−
→
b = a + ab,
addition being understood as addition in
3
R . Then, in the standard frame, given a point
−→
x = (x1, x2, x3), the position of x is the vector Ox = (x1, x2, x3), which coincides with the
point itself. In the standard frame, points and vectors are identified. Points and free vectors
are illustrated in Figure 19.1.
19.1. AFFINE SPACES
479
b
−
→
ab
a
O
Figure 19.1: Points and free vectors
What if we pick a frame with a different origin, say Ω = (ω1, ω2, ω3), but the same basis
vectors (e1, e2, e3)? This time, the point x = (x1, x2, x3) is defined by two position vectors:
−→
Ox = (x1, x2, x3)
in the frame (O, (e1, e2, e3)) and
−→
Ωx = (x1 − ω1, x2 − ω2, x3 − ω3)
in the frame (Ω, (e1, e2, e3)).
This is because
−→
−→
−→
−→
Ox = OΩ + Ωx and OΩ = (ω1, ω2, ω3).
We note that in the second frame (Ω, (e1, e2, e3)), points and position vectors are no longer
identified. This gives us evidence that points are not vectors. It may be computationally
convenient to deal with points using position vectors, but such a treatment is not frame
invariant, which has undesirable effets.
Inspired by physics, we deem it important to define points and properties of points that
are frame invariant. An undesirable side effect of the present approach shows up if we attempt
to define linear combinations of points. First, let us review the notion of linear combination
of vectors. Given two vectors u and v of coordinates (u1, u2, u3) and (v1, v2, v3) with respect
to the basis (e1, e2, e3), for any two scalars λ, µ, we can define the linear combination λu + µv
as the vector of coordinates
(λu1 + µv1, λu2 + µv2, λu3 + µv3).
If we choose a different basis (e1, e2, e3) and if the matrix P expressing the vectors (e1, e2, e3)
over the basis (e1, e2, e3) is
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CHAPTER 19. BASICS OF AFFINE GEOMETRY
a
1
b1 c1
P =
a
,
2
b2 c2
a3 b3 c3
which means that the columns of P are the coordinates of the ej over the basis (e1, e2, e3),
since
u1e1 + u2e2 + u3e3 = u1e1 + u2e2 + u3e3
and
v1e1 + v2e2 + v3e3 = v1e1 + v2e2 + v3e3,
it is easy to see that the coordinates (u1, u2, u3) and (v1, v2, v3) of u and v with respect to
the basis (e1, e2, e3) are given in terms of the coordinates (u1, u2, u3) and (v1, v2, v3) of u and
v with respect to the basis (e1, e2, e3) by the matrix equations
u
1
u1
v1
v1
u
u
v
v
.
2 = P 2
and
2 = P 2
u3
u3
v3
v3
From the above, we get
u
1
u1
v1
v1
u
u
v
v
2 = P −1 2
and
2 = P −1 2 ,
u3
u3
v3
v3
and by linearity, the coordinates
(λu1 + µv1, λu2 + µv2, λu3 + µv3)
of λu + µv with respect to the basis (e1, e2, e3) are given by
λu
1 + µv1
u1
v1
λu1 + µv1
λu
u
v
λu
2 + µv2 = λP −1 2 + µP −1 2 = P −1
2 + µv2 .
λu3 + µv3
u3
v3
λu3 + µv3
Everything worked out because the change of basis does not involve a change of origin. On the
other hand, if we consider the change of frame from the frame (O, (e1, e2, e3)) to the frame
−→
(Ω, (e1, e2, e3)), where OΩ = (ω1, ω2, ω3), given two points a, b of coordinates (a1, a2, a3)
and (b1, b2, b3) with respect to the frame (O, (e1, e2, e3)) and of coordinates (a1, a2, a3) and
(b1, b2, b3) with respect to the frame (Ω, (e1, e2, e3)), since
(a1, a2, a3) = (a1 − ω1, a2 − ω2, a3 − ω3)
and
(b1, b2, b3) = (b1 − ω1, b2 − ω2, b3 − ω3),
19.1. AFFINE SPACES
481
the coordinates of λa + µb with respect to the frame (O, (e1, e2, e3)) are
(λa1 + µb1, λa2 + µb2, λa3 + µb3),
but the coordinates
(λa1 + µb1, λa2 + µb2, λa3 + µb3)
of λa + µb with respect to the frame (Ω, (e1, e2, e3)) are
(λa1 + µb1 − (λ + µ)ω1, λa2 + µb2 − (λ + µ)ω2, λa3 + µb3 − (λ + µ)ω3),
which are different from
(λa1 + µb1 − ω1, λa2 + µb2 − ω2, λa3 + µb3 − ω3),
unless λ + µ = 1.
Thus, we have discovered a major difference between vectors and points: The notion of
linear combination of vectors is basis independent, but the notion of linear combination of
points is frame dependent. In order to salvage the notion of linear combination of points,
some restriction is needed: The scalar coefficients must add up to 1.
A clean way to handle the problem of frame invariance and to deal with points in a more
intrinsic manner is to make a clearer distinction between points and vectors. We duplicate
3
R into two copies, the first copy corresponding to points, where we forget the vector space
structure, and the second copy corresponding to free vectors, where the vector space structure
is important. Furthermore, we make explicit the important fact that the vector space 3
R acts
on the set of points
3
R : Given any point a = (a1, a2, a3) and any vector v = (v1, v2, v3),
we obtain the point
a + v = (a1 + v1, a2 + v2, a3 + v3),
which can be thought of as the result of translating a to b using the vector v. We can imagine
that v is placed such that its origin coincides with a and that its tip coincides with b. This
action + :
3
3
3
R × R → R satisfies some crucial properties. For example,
a + 0 = a,
(a + u) + v = a + (u + v),
−
→
and for any two points a, b, there is a unique free vector ab such that
−
→
b = a + ab.
It turns out that the above properties, although trivial in the case of
3
R , are all that is
needed to define the abstract notion of affine space (or affine structure). The basic idea is
to consider two (distinct) sets E and E, where E is a set of points (with no structure) and
E is a vector space (of free vectors) acting on the set E.
Did you say “A fine space”?
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CHAPTER 19. BASICS OF AFFINE GEOMETRY
Intuitively, we can think of the elements of E as forces moving the points in E, considered
as physical particles. The effect of applying a force (free vector) u ∈ E to a point a ∈ E is
a translation. By this, we mean that for every force u ∈ E, the action of the force u is to
“move” every point a ∈ E to the point a + u ∈ E obtained by the translation corresponding
to u viewed as a vector. Since translations can be composed, it is natural that E is a vector
space.
For simplicity, it is assumed that all vector spaces under consideration are defined over
the field R of real numbers. Most of the definitions and results also hold for an arbitrary field
K, although some care is needed when dealing with fields of characteristic different from zero
(see the problems). It is also assumed that all families (λi)i∈I of scalars have finite support.
Recall that a family (λi)i∈I of scalars has finite support if λi = 0 for all i ∈ I − J, where
J is a finite subset of I. Obviously, finite families of scalars have finite support, and for
simplicity, the reader may assume that all families of scalars are finite. The formal definition
of an affine space is as follows.
Definition 19.1. An affine space is either the degenerate space reduced to the empty set,
or a triple E, E, + consisting of a nonempty set E (of points), a vector space E (of trans-
lations, or free vectors), and an action + : E × E → E, satisfying the following conditions.
(A1) a + 0 = a, for every a ∈ E.
(A2) (a + u) + v = a + (u + v), for every a ∈ E, and every u, v ∈ E.
(A3) For any two points a, b ∈ E, there is a unique u ∈ E such that a + u = b.
−
→
The unique vector u ∈ E such that a + u = b is denoted by ab, or sometimes by ab, or
even by b − a. Thus, we also write
−
→
b = a + ab
(or b = a + ab, or even b = a + (b − a)).
The dimension of the affine space E, E, + is the dimension dim(E) of the vector space
E. For simplicity, it is denoted by dim(E).
Conditions (A1) and (A2) say that the (abelian) group E acts on E, and condition (A3)
says that E acts transitively and faithfully on E. Note that
−−−−−→
a(a + v) = v
−−−−−→
−−−−−→
for all a ∈ E and all v ∈ E, since a(a + v) is the unique vector such that a+v = a+a(a + v).
−
→
Thus, b = a + v is equivalent to ab = v. Figure 19.2 gives an intuitive picture of an affine
space. It is natural to think of all vectors as having the same origin, the null vector.
The axioms defining an affine space E, E, + can be interpreted intuitively as saying
that E and E are two different ways of looking at the same object, but wearing different
sets of glasses, the second set of glasses depending on the choice of an “origin” in E. Indeed,
19.1. AFFINE SPACES
483
−
→
E
E
b = a + u
u
a
c = a + w
w
v
Figure 19.2: Intuitive picture of an affine space
we can choose to look at the points in E, forgetting that every pair (a, b) of points defines a
−
→
unique vector ab in E, or we can choose to look at the vectors u in E, forgetting the points
in E. Furthermore, if we also pick any point a in E, a point that can be viewed as an origin
in E, then we can recover all the points in E as the translated points a + u for all u ∈ E.
This can be formalized by defining two maps between E and E.
For every a ∈ E, consider the mapping from E to E given by
u → a + u,
where u ∈ E, and consider the mapping from E to E given by
−
→
b → ab,
where b ∈ E. The composition of the first mapping with the second is
−−−−−→
u → a + u → a(a + u),
which, in view of (A3), yields u. The composition of the second with the first mapping is
−
→
−
→
b → ab → a + ab,
which, in view of (A3), yields b. Thus, these compositions are the identity from E to E and
the identity from E to E, and the mappings are both bijections.
−
→
When we identify E with E via the mapping b → ab, we say that we consider E as the
vector space obtained by taking a as the origin in E, and we denote it by Ea. Because Ea is
a vector space, to be consistent with our notational conventions we should use the notation
Ea (using an arrow), instead of Ea. However, for simplicity, we stick to the notation Ea.
Thus, an affine space E, E, + is a way of defining a vector space structure on a set of
points E, without making a commitment to a fixed origin in E. Nevertheless, as soon as
484
CHAPTER 19. BASICS OF AFFINE GEOMETRY
we commit to an origin a in E, we can view E as the vector space Ea. However, we urge
the reader to think of E as a physical set of points and of E as a set of forces acting on E,
rather than reducing E to some isomorphic copy of
n
R . After all, points are points, and not
vectors! For notational simplicity, we will often denote an affine space E, E, + by (E, E),
or even by E. The vector space E is called the vector space associated with E.
One should be careful about the overloading of the addition symbol +. Addition
is well-defined on vectors, as in u + v; the translate a + u of a point a ∈ E by a
vector u ∈ E is also well-defined, but addition of points a + b does not make sense. In
this respect, the notation b − a for the unique vector u such that b = a + u is somewhat
confusing, since it suggests that points can be subtracted (but not added!).
Any vector space E has an affine space structure specified by choosing E = E, and letting
+ be addition in the vector space E. We will refer to the affine structure E, E, + on a
vector space E as the canonical (or natural) affine structure on E. In particular, the vector
space
n
n
n
n
R
can be viewed as the affine space R , R , + , denoted by A . In general, if K is
any field, the affine space Kn, Kn, + is denoted by
n
A . In order to distinguish between
K
the double role played by members of
n
R , points and vectors, we will denote points by row
vectors, and vectors by column vectors. Thus, the action of the vector space
n
R over the set
n
R simply viewed as a set of points is given by
u
1
(a
.
.
1, . . . , an) +
.
= (a1 + u1, . . . , an + un).
un
We will also use the convention that if x = (x
n
1, . . . , xn) ∈ R , then the column vector
associated with x is denoted by x (in boldface notation). Abusing the notation slightly, if
a ∈ n
n
n
R is a point, we also write a ∈ A . The affine space A is called the real affine space of
dimension n. In most cases, we will consider n = 1, 2, 3.
19.2
Examples of Affine Spaces
Let us now give an example of an affine space that is not given as a vector space (at least, not
in an obvious fashion). Consider the subset L of
2
A consisting of all points (x, y) satisfying
the equation
x + y − 1 = 0.
The set L is the line of slope −1 passing through the points (1, 0) and (0, 1) shown in Figure
19.3.
The line L can be made into an official affine space by defining the action + : L × R → L
of R on L defined such that for every point (x, 1 − x) on L and any u ∈ R,
(x, 1 − x) + u = (x + u, 1 − x − u).
19.2. EXAMPLES OF AFFINE SPACES
485
L
Figure 19.3: An affine space: the line of equation x + y − 1 = 0
It is immediately verified that this action makes L into an affine space. For example, for any
two points a = (a1, 1 − a1) and b = (b1, 1 − b1) on L, the unique (vector) u ∈ R such that
b = a + u is u = b1 − a1. Note that the vector space R is isomorphic to the line of equation
x + y = 0 passing through the origin.
Similarly, consider the subset H of
3
A
consisting of all points (x, y, z) satisfying the
equation
x + y + z − 1 = 0.
The set H is the plane passing through the points (1, 0, 0), (0, 1, 0), and (0, 0, 1). The plane
H can be made into an official affine space by defining the action + : H × 2
2
R → H of R on
u
H defined such that for every point (x, y, 1 − x − y) on H and any
∈ 2,
v
R
u
(x, y, 1 − x − y) +
= (x + u, y + v, 1 − x − u − y − v).
v
For a slightly wilder example, consider the subset P of
3
A consisting of all points (x, y, z)
satisfying the equation
x2 + y2 − z = 0.
The set P is a paraboloid of revolution, with axis Oz. The surface P can be made into an
official affine space by defining the action + : P × 2
2
R → P of R on P defined such that for
u
every point (x, y, x2 + y2) on P and any
∈ 2,
v
R
u
(x, y, x2 + y2) +
= (x + u, y + v, (x + u)2 + (y + v)2).
v
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CHAPTER 19. BASICS OF AFFINE GEOMETRY
−
→
E
E
b
−
→
ab
a
c
−
→
ac
−
→
bc
Figure 19.4: Points and corresponding vectors in affine geometry
This should dispell any idea that affine spaces are dull. Affine spaces not already equipped
with an obvious vector space structure arise in projective geometry.
19.3
Chasles’s Identity
−
→
−
→
Given any three points a, b, c ∈ E, since c = a + −
→
ac, b = a + ab, and c = b + bc, we get
−
→
−
→
−
→
−
→
−
→
c = b + bc = (a + ab) + bc = a + (ab + bc)