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 Rn 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]
^Mortenson, Michael E. (2007), Geometric Transformations for 3D Modeling (2nd ed.), Industrial Press, p. 211, ISBN 9780831192419.
geometry, obliquereflections generalize ordinary reflections by not requiring that reflection be done using perpendiculars. If two points are oblique reflections...
including cavalier and cabinet projection Obliquereflection, in Euclidean geometry Oblique shock, in gas dynamics Oblique type, a form of type that slants slightly...
more general types of reflection symmetry. For example: with respect to a non-isometric affine involution (an obliquereflection in a line, plane, etc...
3D are: the identity the obliquereflection in respect to a plane the obliquereflection in respect to a line the reflection in respect to a point. In...
oblique muscle, also internal oblique muscle or interior oblique, is an abdominal muscle in the abdominal wall that lies below the external oblique muscle...
binoculars. Although total internal reflection can occur with any kind of wave that can be said to have oblique incidence, including (e.g.) microwaves...
gloss maps (also known as shinemaps), transparency maps, reflection maps, obliquereflection maps, hole maps, and bump maps, as well as leaving the map...
Young Urban Professionals. As a result, the film can be seen "as an obliquereflection on what might happen when the counterculture becomes the dominant...
a true reflection line already implies this situation. This corresponds to wallpaper group cm. The translational symmetry is given by oblique translation...
An oblique shock wave is a shock wave that, unlike a normal shock, is inclined with respect to the direction of incoming air. It occurs when a supersonic...
cannot be assessed in electrophysiology procedures. The oblique sinus is an inverted J-shaped reflection of the venae cavae and pulmonary veins. It lies behind...
The oblique order (also known as the 'declined flank') is a military tactic whereby an attacking army focuses its forces to attack a single enemy flank...
starry effect, etc. Linear and circular polarising filters reduce obliquereflections from non-metallic surfaces. Many filters absorb part of the light...
base-centered monoclinic lattice, the primitive cell has the shape of an oblique rhombic prism; it can be constructed because the two-dimensional centered...
larger class of oblique shocks. Further, the name "normal" is with respect to geometry rather than frequency of occurrence. Oblique shocks are much more...
contains translations and may contain glide reflections, reflections along the long axis of the strip, reflections along the narrow axis of the strip, and...
[∞]+×[∞]+ Lattice: oblique Point group: C1 The group p1 contains only translations; there are no rotations, reflections, or glide reflections. Examples of group...
to the mirror surface. As an optical effect, it results from specular reflection off from surfaces of lustrous materials, especially a mirror or water...
light. They are used as polarizing filters in photography to reduce obliquereflections from non-metallic surfaces, and are the lenses of the 3D glasses...
Oblique subduction is a form of subduction (i.e. a tectonic process involving the convergence of two plates where the denser plate descends into Earth's...
needed] Optical microscopy and electron microscopy involve the diffraction, reflection, or refraction of electromagnetic radiation/electron beams interacting...
extraocular muscles, the four recti muscles, and the superior and inferior oblique muscles, control movement of the eye. The other muscle, the levator palpebrae...
transition to Mach reflection occurs. In a steady flow situation, if a wedge is placed into a steady supersonic flow in such a way that its oblique attached shock...
from the lower part of the common aponeurosis of the abdominal internal oblique muscle and the transversus abdominis muscle, joining the muscle to the...
internal reflection requires medium 2 to have a lower refractive index than medium 1, and for the waves must be incident at sufficiently oblique angles...
Global Information
