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
Some of these are nonlinear,
because they involve a nontrivial translation.
The line makes an angle of
with the -axis.
Thus and .
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.
Let be distance-preserving and consider .
Any two points in the codomain can be written as and
Because is distance-preserving, the distance from
to equals the distance from to .
But this is exactly what is required for to be
Any plane figure is congruent to itself via the
identity map , which is obviously
If is congruent to (via some ) then
is congruent to via , which is
distance-preserving by the prior item.
Finally, if is congruent to (via some ) and
is congruent to (via some ) then is
congruent to via , which is easily checked
to be distance-preserving.
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).
The first two components of each are and .
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.
The Pythagorean Theorem gives that
three points are
colinear if and only if
(for some ordering of them into , , and ),
Of course, where is distance-preserving, this holds
if and only if
which, again by Pythagoras, is true if and only if
, , and are colinear.
The argument for betweeness is similar (above, is
between and ).
If the figure is a triangle then it is the union of three
line segments , , and .
The prior two paragraphs together show that the property of
being a line segment is invariant.
So is the union of three line segments, and so is a
A circle centered at and of radius is the set of
all points such that .
Applying the distance-preserving map gives that the image
is the set of all subject to the condition that
Since , the set is also
a circle, with center and radius .
Here are two that are easy to verify: (i) the
property of being a right triangle, and (ii) the property of
two lines being parallel.
One that was mentioned in the section is the "sense" of
A triangle whose vertices read clockwise as , ,
may, under a distance-preserving map, be sent to a triangle
read , , counterclockwise.