Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

, oblique reflections generalize ordinary reflections by not requiring that reflection be done using

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

s. If two points are oblique reflections of each other, they will still stay so under

affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generall ...

s. Consider a plane ''P'' in the three-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...

. The usual reflection of a point ''A'' in space in respect to the plane ''P'' is another point ''B'' in space, such that the midpoint of the segment ''AB'' is in the plane, and ''AB'' is perpendicular to the plane. For an ''oblique reflection'', one requires instead of perpendicularity that ''AB'' be parallel to a given reference line. Formally, let there be a plane ''P'' in the three-dimensional space, and a line ''L'' in space not parallel to ''P''. To obtain the oblique reflection of a point ''A'' in space in respect to the plane ''P'', one draws through ''A'' a line parallel to ''L'', and lets the oblique reflection of ''A'' be the point ''B'' on that line on the other side of the plane such that the midpoint of ''AB'' is in ''P''. If the reference line ''L'' is perpendicular to the plane, one obtains the usual reflection. For example, consider the plane ''P'' to be the ''xy'' plane, that is, the plane given by the equation ''z''=0 in

Cartesian coordinates 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 i ...

. Let the direction of the reference line ''L'' be given by the vector (''a'', ''b'', ''c''), with ''c''≠0 (that is, ''L'' is not parallel to ''P''). The oblique reflection of a point (''x'', ''y'', ''z'') will then be :

\left(x-\frac, y-\frac, -z\right).

The concept of oblique reflection is easily generalizable to oblique reflection in respect to an affine hyperplane in R^''n'' with a line again serving as a reference, or even more generally, oblique reflection in respect to a ''k''-dimensional affine subspace, with a ''n''−''k''-dimensional affine subspace serving as a reference. Back to three dimensions, one can then define oblique reflection in respect to a line, with a plane serving as a reference. An oblique reflection is an

, and it is an involution, meaning that the reflection of the reflection of a point is the point itself..

References

{{DEFAULTSORT:Oblique Reflection Affine geometry Functions and mappings