In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a rotation of axes in two dimensions is a
mapping from an ''xy''-
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
to an ''x′y′''-Cartesian coordinate system in which the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
is kept fixed and the ''x′'' and ''y′'' axes are obtained by rotating the ''x'' and ''y'' axes counterclockwise through an angle
. A point ''P'' has coordinates (''x'', ''y'') with respect to the original system and coordinates (''x′'', ''y′'') with respect to the new system. In the new coordinate system, the point ''P'' will appear to have been rotated in the opposite direction, that is, clockwise through the angle
. A rotation of axes in more than two dimensions is defined similarly. A rotation of axes is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
and a
rigid transformation
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.
The rigid transformation ...
.
Motivation
Coordinate systems are essential for studying the equations of
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s using the methods of
analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and enginee ...
. To use the method of coordinate geometry, the axes are placed at a convenient position with respect to the curve under consideration. For example, to study the equations of
ellipse
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...
s and
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
s, the
foci are usually located on one of the axes and are situated symmetrically with respect to the origin. If the curve (hyperbola,
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descri ...
, ellipse, etc.) is ''not'' situated conveniently with respect to the axes, the coordinate system should be changed to place the curve at a convenient and familiar location and orientation. The process of making this change is called a
transformation of coordinates.
The solutions to many problems can be simplified by rotating the coordinate axes to obtain new axes through the same origin.
Derivation
The equations defining the transformation in two dimensions, which rotates the ''xy'' axes counterclockwise through an angle
into the ''x′y′'' axes, are derived as follows.
In the ''xy'' system, let the point ''P'' have
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
. Then, in the ''x′y′'' system, ''P'' will have polar coordinates
.
Using
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
, we have
and using the standard
trigonometric formulae for differences, we have
Substituting equations () and () into equations () and (), we obtain
Equations () and () can be represented in matrix form as
which is the standard matrix equation of a rotation of axes in two dimensions.
The inverse transformation is
or
Examples in two dimensions
Example 1
Find the coordinates of the point
after the axes have been rotated through the angle
, or 30°.
Solution:
The axes have been rotated counterclockwise through an angle of
and the new coordinates are
. Note that the point appears to have been rotated clockwise through
with respect to fixed axes so it now coincides with the (new) ''x′'' axis.
Example 2
Find the coordinates of the point
after the axes have been rotated clockwise 90°, that is, through the angle
, or −90°.
Solution:
The axes have been rotated through an angle of
, which is in the clockwise direction and the new coordinates are
. Again, note that the point appears to have been rotated counterclockwise through
with respect to fixed axes.
Rotation of conic sections
The most general equation of the second degree has the form
Through a change of coordinates (a rotation of axes and a
translation of axes), equation () can be put into a
standard form, which is usually easier to work with. It is always possible to rotate the coordinates at a specific angle so as to eliminate the ''x′y′'' term. Substituting equations () and () into equation (), we obtain
where
If
is selected so that
we will have
and the ''x′y′'' term in equation () will vanish.
When a problem arises with ''B'', ''D'' and ''E'' all different from zero, they can be eliminated by performing in succession a rotation (eliminating ''B'') and a translation (eliminating the ''D'' and ''E'' terms).
Identifying rotated conic sections
A non-degenerate conic section given by equation () can be identified by evaluating
. The conic section is:
*an ellipse or a circle, if
;
*a parabola, if
;
*a hyperbola, if
.
Generalization to several dimensions
Suppose a rectangular ''xyz''-coordinate system is rotated around its ''z'' axis counterclockwise (looking down the positive ''z'' axis) through an angle
, that is, the positive ''x'' axis is rotated immediately into the positive ''y'' axis. The ''z'' coordinate of each point is unchanged and the ''x'' and ''y'' coordinates transform as above. The old coordinates (''x'', ''y'', ''z'') of a point ''Q'' are related to its new coordinates (''x′'', ''y′'', ''z′'') by
Generalizing to any finite number of dimensions, a
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
:R = \begin
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\ ...
is an
orthogonal matrix that differs from the
identity matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
in at most four elements. These four elements are of the form
:
and
for some
and some ''i'' ≠ ''j''.
Example in several dimensions
Example 3
Find the coordinates of the point
after the positive ''w'' axis has been rotated through the angle
, or 15°, into the positive ''z'' axis.
Solution:
See also
*
Rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
*
Rotation (mathematics)
Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have sig ...
Notes
References
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Functions and mappings
Euclidean geometry
Linear algebra
Transformation (function)