Reflection (mathematics)

In

Some mathematicians use "flip" as a synonym for "reflection".[2]^[3][4]

In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop a perpendicular from the point to the line (plane) used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure.

To reflect point P through the line AB using

• Step 1 (red): construct a circle with center at P and some fixed radius r to create points A′ and B′ on the line AB, which will be equidistant from P.
• Step 2 (green): construct circles centered at A′ and B′ having radius r. P and Q will be the points of intersection of these two circles.

Point Q is then the reflection of point P through line AB.

The

Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes. In general, a group generated by reflections in affine hyperplanes is known as a reflection group. The finite groups generated in this way are examples of Coxeter groups.

Reflection across an arbitrary line through the origin in

${\displaystyle \operatorname {Ref} _{l}(v)=2{\frac {v\cdot l}{l\cdot l}}l-v,}$

where ${\displaystyle v}$ denotes the vector being reflected, ${\displaystyle l}$ denotes any vector in the line across which the reflection is performed, and ${\displaystyle v\cdot l}$ denotes the dot product of ${\displaystyle v}$ with ${\displaystyle l}$. Note the formula above can also be written as

${\displaystyle \operatorname {Ref} _{l}(v)=2\operatorname {Proj} _{l}(v)-v,}$

saying that a reflection of ${\displaystyle v}$ across ${\displaystyle l}$ is equal to 2 times the projection of ${\displaystyle v}$ on ${\displaystyle l}$, minus the vector ${\displaystyle v}$. Reflections in a line have the eigenvalues of 1, and −1.

Given a vector ${\displaystyle v}$ in Euclidean space ${\displaystyle \mathbb {R} ^{n}}$, the formula for the reflection in the

${\displaystyle a}$

${\displaystyle \operatorname {Ref} _{a}(v)=v-2{\frac {v\cdot a}{a\cdot a}}a,}$

where ${\displaystyle v\cdot a}$ denotes the dot product of ${\displaystyle v}$ with ${\displaystyle a}$. Note that the second term in the above equation is just twice the vector projection of ${\displaystyle v}$ onto ${\displaystyle a}$. One can easily check that

• Ref_a(v) = −v, if ${\displaystyle v}$ is parallel to ${\displaystyle a}$, and
• Ref_a(v) = v, if ${\displaystyle v}$ is perpendicular to a.

Using the

${\displaystyle \operatorname {Ref} _{a}(v)=-{\frac {ava}{a^{2}}}.}$

Since these reflections are isometries of Euclidean space fixing the origin they may be represented by

where ${\displaystyle I}$ denotes the ${\displaystyle n\times n}$ identity matrix and ${\displaystyle a^{T}}$ is the transpose of a. Its entries are

${\displaystyle R_{ij}=\delta _{ij}-2{\frac {a_{i}a_{j}}{\left\|a\right\|^{2}}},}$

where δ_ij is the Kronecker delta.

The formula for the reflection in the affine hyperplane ${\displaystyle v\cdot a=c}$ not through the origin is

${\displaystyle \operatorname {Ref} _{a,c}(v)=v-2{\frac {v\cdot a-c}{a\cdot a}}a.}$

• Additive inverse
• Coordinate rotations and reflections
• Householder transformation
• Inversive geometry
• Plane of rotation
• Reflection mapping
• Reflection group
• Reflection symmetry