on the opposite side of l such that the segment connecting P to P′ is PERPENDICULAR to l and bisected by it.
The points on l itself are left unmoved. Any reflection is
an isometry that transforms geometric figures to their
mirror images. The line used in performing the reflection is called the line of reflection.
In a CARTESIAN 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 threedimensional 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 translation. A translation is an isometry, and all geometric figures are transformed to new figures with the same size,
shape, and orientation as the originals.
If l 1 and l 2 are 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 translating all points in the plane a distance of 2d units 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 a and b.
Rotation
A rotation about a point O through an angle θ is the
geometric transformation that maps a point P in the
plane to the point P′ such that P and P′ are the same
distance from O, and the angle POP′ has measure θ. (A
counterclockwise turn is applied if θ is positive; a
clockwise turn of θ is negative.) Only the location of
the point O remains unchanged under a rotation,
unless the angle θ is a multiple of 360°, in which case
all points are fixed.
A rotation is equivalent to two reflections about
lines that intersect at O making 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 O in the plane is the isometry
that takes a point P in the plane to the corresponding
point P′ such that O lies at the MIDPOINT of the line
segment connecting P and P′. A reflection in a point is
equivalent to a rotation of 180° about that point.
Dilation
A dilation with center O and dilation factor k > 1 is the
geometric transformation that leaves O fixed, and
moves any point P further away from O, by a factor k,
along the ray from O through P. Thus a dilation
stretches figures uniformly outward from O. It is possible, 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 P in the plane a distance d
from O, and maps it to the point P′ a distance r
2
/d
from O along the same ray from O through 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 O under a circle 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 THEOREM 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 geometry examines objects drawn in a plane (lines, circles,
polygons, and the like), solid geometry deals with figures in three-dimensional space (polyhedra, lines,
planes, and surfaces), and SPHERICAL GEOMETRY studies
θ
––
2
geometry 225
The points on l itself are left unmoved. Any reflection is
an isometry that transforms geometric figures to their
mirror images. The line used in performing the reflection is called the line of reflection.
In a CARTESIAN 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 threedimensional 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 translation. A translation is an isometry, and all geometric figures are transformed to new figures with the same size,
shape, and orientation as the originals.
If l 1 and l 2 are 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 translating all points in the plane a distance of 2d units 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 a and b.
Rotation
A rotation about a point O through an angle θ is the
geometric transformation that maps a point P in the
plane to the point P′ such that P and P′ are the same
distance from O, and the angle POP′ has measure θ. (A
counterclockwise turn is applied if θ is positive; a
clockwise turn of θ is negative.) Only the location of
the point O remains unchanged under a rotation,
unless the angle θ is a multiple of 360°, in which case
all points are fixed.
A rotation is equivalent to two reflections about
lines that intersect at O making 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 O in the plane is the isometry
that takes a point P in the plane to the corresponding
point P′ such that O lies at the MIDPOINT of the line
segment connecting P and P′. A reflection in a point is
equivalent to a rotation of 180° about that point.
Dilation
A dilation with center O and dilation factor k > 1 is the
geometric transformation that leaves O fixed, and
moves any point P further away from O, by a factor k,
along the ray from O through P. Thus a dilation
stretches figures uniformly outward from O. It is possible, 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 P in the plane a distance d
from O, and maps it to the point P′ a distance r
2
/d
from O along the same ray from O through 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 O under a circle 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 THEOREM 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 geometry examines objects drawn in a plane (lines, circles,
polygons, and the like), solid geometry deals with figures in three-dimensional space (polyhedra, lines,
planes, and surfaces), and SPHERICAL GEOMETRY studies
θ
––
2
geometry 225