![]() ![]() In the case of glide-reflection symmetry, the symmetry group of an object contains a glide reflection, and hence the group generated by it. 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. ![]() The isometry group generated by just a glide reflection is an infinite cyclic group. This isometry maps the x-axis to itself any other line which is parallel to the x-axis gets reflected in the x-axis, so this system of parallel lines is left invariant. In the Euclidean plane, reflections and glide reflections are the only two kinds of indirect (orientation-reversing) isometries.įor example, there is an isometry consisting of the reflection on the x-axis, followed by translation of one unit parallel to it. It can also be given a Schoenflies notation as S 2∞, Coxeter notation as, and orbifold notation as ∞×. A glide reflection can be seen as a limiting rotoreflection, where the rotation becomes a translation. However, when a reflection is composed with a translation in any other direction, the composition of the two transformations is a glide reflection, which can be uniquely described as a reflection in a parallel hyperplane composed with a translation in a direction parallel to the hyperplane.Ī single glide is represented as frieze group p11g. When a reflection is composed with a translation in a direction perpendicular to the hyperplane of reflection, the composition of the two transformations is a reflection in a parallel hyperplane. Objects with glide-reflection symmetry are in general not symmetrical under reflection alone, but two applications of the same glide reflection result in a double translation, so objects with glide-reflection symmetry always also have a simple translational symmetry. Glide-reflection symmetry is seen in frieze groups (patterns which repeat in one dimension, often used in decorative borders), wallpaper groups (regular tessellations of the plane), and space groups (which describe e.g. When some geometrical object or configuration appears unchanged by a transformation, it is said to have symmetry, and the transformation is called a symmetry operation. The displacement vector of the translation is called the glide vector. When the context is three-dimensional space, the hyperplane of reflection is a plane called the glide plane. ![]() When the context is the two-dimensional Euclidean plane, the hyperplane of reflection is a straight line called the glide line or glide axis. In geometry, a glide reflection or transflection is a geometric transformation that consists of a reflection across a hyperplane and a translation ("glide") in a direction parallel to that hyperplane, combined into a single transformation. Applying the glide reflection maps each left footprint into a right footprint and vice versa. This footprint trail has glide-reflection symmetry. Geometric transformation combining reflection and translation A glide reflection is the composition of a reflection across a line and a translation parallel to the line.
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