on the opposite side of lsuch that the segment connect-
ing Pto P′is
PERPENDICULAR
to land bisected by it.
The points on litself are left unmoved. Any reflection is
an isometry that transforms geometric figures to their
mirror images. The line used in performing the reflec-
tion is called the line of reflection.
In a C
ARTESIAN COORDINATE
system, a reflection
about the x-axis takes a point with coordinates (x,y) to
the point (x,–y), and a refection about the y-axis
changes the sign of the x-coordinate: (x,y) becomes
(–x,y). The analog of a reflection in a line in three-
dimensional space is a reflection in a plane.
Translation
A geometric transformation that moves all points in the
plane a fixed distance in a fixed direction is called a
translation. No points are left unmoved by a transla-
tion. A translation is an isometry, and all geometric fig-
ures are transformed to new figures with the same size,
shape, and orientation as the originals.
If l1and l2are two
PARALLEL
lines in the plane, d
units apart, a reflection in the first line, followed by a
reflection in the second, has the same effect as translat-
ing all points in the plane a distance of 2dunits in a
fixed direction perpendicular to the two lines. Thus
every translation is equivalent to the
COMPOSITION
of
two reflections.
In a Cartesian coordinate system, a translation
takes a point with coordinates (x,y) to the point (x+ a,
y+ b) for some fixed values aand b.
Rotation
A rotation about a point Othrough an angle θis the
geometric transformation that maps a point Pin the
plane to the point P′such that Pand P′are the same
distance from O, and the angle POP′has measure θ. (A
counterclockwise turn is applied if θis positive; a
clockwise turn of θis negative.) Only the location of
the point Oremains unchanged under a rotation,
unless the angle θis a multiple of 360°, in which case
all points are fixed.
A rotation is equivalent to two reflections about
lines that intersect at Omaking an angle of between
them. Every rotation is an isometry. The analog of a
rotation about a point in three-dimensional space is a
rotation about a line.
Glide Reflection
A reflection in a line followed by a translation in a
direction parallel to that line is called a glide reflection.
Reflection in a Point
A reflection in a point Oin the plane is the isometry
that takes a point Pin the plane to the corresponding
point P′such that Olies at the
MIDPOINT
of the line
segment connecting Pand P′. A reflection in a point is
equivalent to a rotation of 180°about that point.
Dilation
A dilation with center Oand dilation factor k> 1 is the
geometric transformation that leaves Ofixed, and
moves any point Pfurther away from O, by a factor k,
along the ray from Othrough P. Thus a dilation
stretches figures uniformly outward from O. It is possi-
ble, for example, to convert a square into a rectangle
via a dilation. (A dilation with dilation factor k
between O and 1 “shrinks” all points closer to O.) A
dilation is not an isometry.
Circular Inversion
Also called an “inversion in a circle” or a “reflection in
a circle,” a circular inversion in a circle, with center O
and radius r, takes a point Pin the plane a distance d
from O, and maps it to the point P′a distance r2/d
from Oalong the same ray from Othrough P. Thus
points inside the circle are taken outside, and vice
versa. Points on the circle itself are left unmoved by the
transformation. The image of the center Ounder a cir-
cle inversion is undefined.
A circular inversion is not an isometry but proves to
be a useful mapping in the study of
GEOMETRY
. It has
the property that circles and straight lines in the plane
are converted to new circles and new straight lines.
See also
FRIEZE PATTERN
;
FUNDAMENTAL THEO
-
REM OF ISOMETRIES
;
SYMMETRY
;
TRANSFORMATION OF
COORDINATES
.
geometry The branch of mathematics concerned with
the properties of space and of figures, lines, curves, and
points drawn in space is called geometry. Plane geome-
try examines objects drawn in a plane (lines, circles,
polygons, and the like), solid geometry deals with fig-
ures in three-dimensional space (polyhedra, lines,
planes, and surfaces), and
SPHERICAL GEOMETRY
studies
θ
––
2
geometry 225