Or you can reflect the figure first and then slide it the result is the same either way. What comes first in a glide reflection? – Related Questions How do you know if its a glide reflection?Ī glide reflection is just what it sounds like: You glide a figure (that’s just another way of saying slide or translate) and then reflect it over a reflecting line. Which two transformations are involved in a glide reflection? Iin fact it is a combination of two transformations: translation and reflection. What is the order of a glide reflection? Glide reflection occurs when you perform translation (glide) on a figure and followed by a reflection across a line parallel to the direction of translation.

It does not matter whether you glide first and then reflect, or reflect first and then glide. Reversing the direction of the composition will not affect the outcome. Therefore, the only required information is the translation rule and a line to reflect over.ĭoes order matter in a glide reflection? A glide reflection is commutative. This then carries on glide reflecting in a similar manner.What comes first in a glide reflection? A glide reflection is a composition of transformations.In a glide reflection, a translation is first performed on the figure, then it is reflected over a line. Here is the evolution of the lightweight spaceship: o Every two iterations these spaceships appear to have been glide reflected to their new positions. The four most common spaceships: the glider, LWSS, MWSS and HWSS all move using glide reflection. Glide reflection is common in Conway's Game of Life. Glide symmetry can be observed in nature among certain fossils of the Ediacara biota the machaeridians and certain palaeoscolecid worms. It is a reflection in a plane combined with a translation parallel to the plane. In 3D the glide reflection is called a glide plane. With additional symmetry it occurs also in cmm, p3m1, p31m, p4m and p6m. The translational symmetry is given by oblique translation vectors from one point on a true reflection line to two points on the next, supporting a rhombus with the true reflection line as one of the diagonals.

A glide reflection line parallel to a true reflection line already implies this situation. If there are also true reflection lines in the same direction then they are evenly spaced between the glide reflection lines. This corresponds to wallpaper group pg with additional symmetry it occurs also in pmg, pgg and p4g. Glide reflection symmetry with respect to two parallel lines with the same translation implies that there is also translational symmetry in the direction perpendicular to these lines, with a translation distance which is twice the distance between glide reflection lines. If the translation vector of a glide reflection is itself an element of the translation group, then the corresponding glide reflection symmetry reduces to a combination of reflection symmetry and translational symmetry. It is isomorphic to a semi-direct product of Z and C 2.įor any symmetry group containing some glide reflection symmetry, the translation vector of any glide reflection is one half of an element of the translation group. 6 (glide-reflections, translations and rotations) is generated by a glide reflection and a rotation about a point on the line of reflection. 2.Įxample pattern with this symmetry group:įrieze group nr. If that is all it contains, this type is frieze group nr. In the case of glide reflection symmetry, the symmetry group of an object contains a glide reflection, and hence the group generated by it.

The isometry group generated by just a glide reflection is an infinite cyclic group.Ĭombining two equal glide reflections gives a pure translation with a translation vector that is twice that of the glide reflection, so the even powers of the glide reflection form a translation group. In coordinates, it takes ( x, y) to ( x + 1, − y). These are the two kinds of indirect isometries in 2D.įor example, there is an isometry consisting of the reflection on the x-axis, followed by translation of one unit parallel to it.

Thus the effect of a reflection combined with any translation is a glide reflection, with as special case just a reflection. However, a glide reflection cannot be reduced like that. The combination of a reflection in a line and a translation in a perpendicular direction is a reflection in a parallel line. Depending on context, we may consider a reflection a special case, where the translation vector is the zero vector. Reversing the order of combining gives the same result. In geometry, a glide reflection is a type of isometry of the Euclidean plane: the combination of a reflection in a line and a translation along that line.