Rotation of Axes
Encyclopedia
A rotation of axes is a form of Euclidean transformation in which the entire xy-coordinate system is rotated in the counter-clockwise direction with respect to the origin
(0, 0) through a scalar
quantity denoted by θ.
With the exception of the degenerate cases, if a general second-degree equation has a
term, then 
represents one of the 3 conic section
s, namely, an ellipse
, hyperbola
, or parabola
.
is defined on the xy-coordinate system as
, then it is denoted as 
on the rotated x'y'-coordinate system.
Likewise, if a locus is defined on the x'y'-coordinate system as
, then it is denoted as 
on the "un-rotated" xy-coordinate system.
, the 
term can be removed by rotating the xy-coordinate system by an angle 
, where
or 
,
i.e.:
.
If
then 
.
.
Now, the equation is rotated by a quantity
, hence
Expanding, the equation becomes
Collecting like terms,
In order to eliminate the x'y'-term, the coefficient of the x'y'-term must be set equal to 0.
If
If
can be identified by evaluating the value of 
:
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
(0, 0) through a scalar
Scalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...
quantity denoted by θ.
With the exception of the degenerate cases, if a general second-degree equation has a
term, then represents one of the 3 conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s, namely, an ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
, hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
, or parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
.
Rotation of loci
If a locusLocus (mathematics)
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....
is defined on the xy-coordinate system as
, then it is denoted as on the rotated x'y'-coordinate system.Likewise, if a locus is defined on the x'y'-coordinate system as
, then it is denoted as on the "un-rotated" xy-coordinate system.Elimination of the xy term by the rotation formula
For a general, non-degenerate second-degree equation, the term can be removed by rotating the xy-coordinate system by an angle , where or ,i.e.:
.If
then .Derivation of the rotation formula
.Now, the equation is rotated by a quantity
, hence
Expanding, the equation becomes
Collecting like terms,
In order to eliminate the x'y'-term, the coefficient of the x'y'-term must be set equal to 0.
If
If
Identifying rotated conic sections
A non-degenerate conic section with the equation can be identified by evaluating the value of :