Glide Plane - Formal Treatment

Formal Treatment

In geometry, a glide plane operation is a type of isometry of the Euclidean space: the combination of a reflection in a plane and a translation in that plane. Reversing the order of combining gives the same result. Depending on context, we may consider a reflection a special case, where the translation vector is the zero vector.

The combination of a reflection in a plane and a translation in a perpendicular direction is a reflection in a parallel plane. However, a glide plane operation with a nonzero translation vector in the plane cannot be reduced like that. Thus the effect of a reflection combined with any translation is a glide plane operation in the general sense, with as special case just a reflection. The glide plane operation in the strict sense and the pure reflection are two of the four kinds of indirect isometries in 3D.

The isometry group generated by just a glide plane operation is an infinite cyclic group. Combining two equal glide plane operations gives a pure translation with a translation vector that is twice that of the glide plane operation, so the even powers of the glide plane operation form a translation group.

In the case of glide plane symmetry, the symmetry group of an object contains a glide plane operation, and hence the group generated by it. For any symmetry group containing glide plane symmetry, the translation vector of any glide plane operation is one half of an element of the translation group. If the translation vector of a glide plane operation is itself an element of the translation group, then the corresponding glide plane symmetry reduces to a combination of reflection symmetry and translational symmetry.

See also lattice.

Read more about this topic:  Glide Plane

Famous quotes containing the words formal and/or treatment:

    Then the justice,
    In fair round belly with good capon lined,
    With eyes severe and beard of formal cut,
    Full of wise saws and modern instances;
    And so he plays his part.
    William Shakespeare (1564–1616)

    Ambivalence reaches the level of schizophrenia in our treatment of violence among the young. Parents do not encourage violence, but neither do they take up arms against the industries which encourage it. Parents hide their eyes from the books and comics, slasher films, videos and lyrics which form the texture of an adolescent culture. While all successful societies have inhibited instinct, ours encourages it. Or at least we profess ourselves powerless to interfere with it.
    C. John Sommerville (20th century)