# Oblique reflection

In Euclidean geometry, **oblique reflections** generalize ordinary reflections by not requiring that reflection be done using perpendiculars. If two points are oblique reflections of each other, they will still stay so under affine transformations.

Consider a plane *P* in the three-dimensional Euclidean space. 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.^{[1]}

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. 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

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 affine transformation, and it is an involution, meaning that the reflection of the reflection of a point is the point itself.^{[2]}

## References

- ↑ Mortenson, Michael E. (2007),
*Geometric Transformations for 3D Modeling*(2nd ed.), Industrial Press, p. 211, ISBN 9780831192419. - ↑ Kapur, Jagat Narain (1976),
*Transformation geometry*, Affiliated East-West Press Pvt., p. 124.