In The Elements, Euclid considers two figures to be
the same if they have the same size and shape.
That is, the triangles below are not equal because they are not the same
set of points.
But they are congruent— essentially
for Euclid's purposes— because we can imagine
picking the plane up,
sliding it over and rotating it a bit,
although not warping or stretching it,
and then putting it back down, to superimpose the first figure on
(Euclid never explicitly states this principle
but he uses it often (Casey 1890).)
In modern terminology, "picking the plane up ..."
means considering a
map from the plane to itself.
Euclid has limited consideration to only certain
transformations of the plane, ones
that may possibly slide or turn the plane but not bend or stretch it.
Accordingly, we define a map to be
distance-preserving or a rigid motion or an
if for all points ,
the distance from to equals the distance from
We also define a plane figure
to be a set of points in the plane and we say that two figures are
congruent if there is a distance-preserving map from the plane
to itself that carries one figure onto the other.
Many statements from Euclidean geometry
follow easily from these definitions.
Some are: (i) collinearity is invariant under any distance-preserving map
(that is, if , , and are collinear then so are
, , and ),
(ii) betweeness is invariant under any distance-preserving map
(if is between and then so is
between and ),
(iii) the property of being a triangle is invariant under
any distance-preserving map
(if a figure is a triangle then the image of that figure is also a triangle),
(iv) and the property of being a circle is invariant under any
In 1872, F. Klein suggested that
Euclidean geometry can be characterized as the study of properties that
are invariant under these maps.
(This forms part of Klein's Erlanger Program,
which proposes the organizing principle that each kind of
geometry— Euclidean, projective, etc.— can be described
as the study of the properties that are
invariant under some group of transformations.
The word "group" here means more than just "collection",
but that lies outside of our scope.)
We can use linear algebra to characterize the distance-preserving maps
of the plane.
First, there are distance-preserving transformations of the plane that
are not linear. The obvious example is this translation.
this example turns out to be the only example, in the
sense that if is distance-preserving and sends to
then the map is linear.
That will follow immediately from this statement: a map that is
distance-preserving and sends to itself is linear.
To prove this equivalent statement, let
for some .
Then to show that is linear, we can show that
it can be represented by a matrix, that is, that acts in this way
for all .
Recall that if we fix three non-collinear points
then any point
in the plane can be described by giving its distance from those three.
So any point in the domain is determined by its distance from
the three fixed points , , and .
Similarly, any point
in the codomain is determined by its distance from
the three fixed points , , and
(these three are not collinear because, as mentioned above,
collinearity is invariant and
, , and are not collinear).
In fact, because is distance-preserving, we can say more: for the
point in the plane that is determined by being
the distance from ,
the distance from , and the distance from ,
its image must be the unique point in the codomain
that is determined by being from ,
and from .
Because of the uniqueness,
checking that the action in () works in the
, , and cases
( is assumed to send to itself)
suffices to show that () describes .
Those checks are routine.
Thus, any distance-preserving can be written
for some constant vector
and linear map that is distance-preserving.
Not every linear map is distance-preserving, for example,
does not preserve distances.
But there is a neat characterization: a linear transformation of the
plane is distance-preserving if and only if both
and is orthogonal to .
The "only if" half of that statement is easy— because is
distance-preserving it must preserve the lengths of vectors,
and because is distance-preserving the Pythagorean theorem shows
that it must preserve orthogonality.
For the "if" half, it suffices to check that the map preserves lengths
of vectors, because then for all
and the distance between the two is preserved
For that check, let
and, with the "if" assumptions that
and we have this.
One thing that is neat about this characterization is that we can
easily recognize matrices that represent such a map with respect to
the standard bases. Those matrices have that when the columns are
written as vectors then they are of length one and are mutually
orthogonal. Such a matrix is called an
orthonormal matrix or orthogonal matrix (the first term
is commonly used to mean not just that the columns are orthogonal, but
also that they have length one).
We can use this insight to delimit the geometric actions possible in
distance-preserving maps. Because
, any is
mapped by to lie somewhere on the circle about the
origin that has radius equal to the length of .
In particular, and are
mapped to the unit circle. What's more, once we fix the unit vector
as mapped to the vector with components
and then there are only two places where
can be mapped if that image is to be
perpendicular to the first vector: one where
maintains its position a quarter circle clockwise from
and one where is is mapped a quarter circle counterclockwise.
We can geometrically describe these two cases.
Let be the angle between the -axis and the image of ,
The first matrix above represents, with respect to the standard bases,
of the plane by radians.
The second matrix above represents a reflection of the plane through the line
bisecting the angle between and .
(This picture shows reflected up into the first
quadrant and reflected down into the fourth quadrant.)
the angle between and runs counterclockwise,
and in the first map above the angle from
to is also counterclockwise,
so the orientation of the angle is preserved.
But in the second map the orientation is reversed.
A distance-preserving map is
direct if it preserves orientations and opposite
if it reverses orientation.
So, we have characterized the Euclidean study of congruence:
it considers, for plane figures, the properties that are invariant
under combinations of (i) a rotation followed by a translation,
or (ii) a reflection followed by a translation
(a reflection followed by a non-trivial
translation is a glide reflection).
Another idea, besides congruence of figures,
encountered in elementary geometry is that figures are similar
if they are congruent after a change of scale. These two triangles
are similar since the second is the same shape as the first, but
-ths the size.
From the above work, we have that figures are similar if there
is an orthonormal matrix such that the points
on one are derived from
the points by
for some nonzero real number and constant vector .
Although many of these ideas were first
explored by Euclid, mathematics is timeless and
they are very much in use today.
One application of the maps studied above is in computer graphics.
We can, for example, animate this top view of a cube
by putting together film frames of it rotating; that's a rigid motion.
We could also make the cube appear to be moving away from us by producing film
frames of it shrinking, which gives us figures that are similar.
| || ||
| Frame 1: || Frame 2: ||Frame 3:
Computer graphics incorporates techniques
from linear algebra in many other ways (see Problem 4).
So the analysis above of distance-preserving maps
is useful as well as interesting.
A beautiful book that explores some of this area is (Weyl 1952).
More on groups, of transformations and otherwise, can be found in any book
on Modern Algebra, for instance (Birkhoff & MacLane 1965).
More on Klein and the Erlanger Program is in (Yaglom 1988).
- Problem 1
Decide if each of these is an orthonormal matrix.
- Problem 2
Write down the formula for each of these distance-preserving maps.
- the map that rotates radians, and then
- the map that reflects about the line
- the map that reflects about and translates over
- Problem 3
- The proof that a map that is distance-preserving and
sends the zero vector to itself incidentally shows that
such a map is one-to-one and onto
(the point in the domain determined by , , and
corresponds to the point in the codomain determined by those
Therefore any distance-preserving map has an inverse.
Show that the inverse is also distance-preserving.
- Prove that congruence is an equivalence relation
between plane figures.
- Problem 4
In practice the matrix for the distance-preserving linear transformation
and the translation are often combined into one.
Check that these two computations yield the same
first two components.
(These are homogeneous coordinates;
see the Topic on Projective Geometry).
- Problem 5
- Verify that the properties described
in the second paragraph of this Topic as invariant
under distance-preserving maps are indeed so.
- Give two more properties that are of interest
in Euclidean geometry from your experience in studying that
subject that are also invariant under distance-preserving maps.
- Give a property that is not of interest in Euclidean
geometry and is not invariant under distance-preserving maps.
- Birkhoff, Garrett; MacLane, Saunders (1965), Survey of Modern Algebra, Macmillan .
- Casey, John (1890), The Elements of Euclid, Books I to VI and XI (9th ed.), Hodges, Figgis, and Co. .
- Weyl, Hermann (1952), Symmetry, Princeton University Press .
- Yaglom, I. M. (1988), Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Century, Birkhäuser .