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An exploration of transformation geometry often begins with a study of reflection symmetry as found in daily life. The first real transformation is reflection in a line or reflection against an axis. The composition of two reflections results in a rotation when the lines intersect, or a translation when they are parallel.
A typical example of glide reflection in everyday life would be the track of footprints left in the sand by a person walking on a beach. Frieze group nr. 6 (glide-reflections, translations and rotations) is generated by a glide reflection and a rotation about a point on the line of reflection. It is isomorphic to a semi-direct product of Z and C 2.
Point Q is the reflection of point P through the line AB. In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop a perpendicular from the point to the line (plane) used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure.
L is a 2-reflection and is a 3-reflection, so taking their geometric product PL in some sense produces a 5-reflection; however, as in the picture below, two of these reflections cancel, leaving a 3-reflection (sometimes known as a rotoreflection). In the plane-based geometric algebra notation, this rotoreflection can be thought of as a planar ...
Reflections are transformations that reverse the direction front to back, orthogonal to the mirror plane, like (real-world) mirrors do. The matrices corresponding to proper rotations (without reflection) have a determinant of +1. Transformations with reflection are represented by matrices with a determinant of −1.
Call the images of p 2 and p 3 under this reflection p 2 ′ and p 3 ′. If q 2 is distinct from p 2 ′, bisect the angle at q 1 with a new mirror. With p 1 and p 2 now in place, p 3 is at p 3 ″; and if it is not in place, a final mirror through q 1 and q 2 will flip it to q 3. Thus at most three reflections suffice to reproduce any plane ...
Rotations and reflections are represented as elements. Unlike a vector algebra, a geometric algebra naturally accommodates any number of dimensions and any quadratic form such as in relativity. Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space).
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. [1] [self-published source] [2] [3] The rigid transformations include rotations, translations, reflections, or any