![]() ![]() If M is the intersection of m and c, then P and M are distinct points such that σ cσ bσ a( P)=σ cσ mσ l(P)=σ cσ m( P)= σ M( P)≠ P. Then there is a line m perpendicular to c such that σ b σ a = σ m σ l. Let line l be the perpendicular from P to c. To show this holds for the glide reflection, suppose P is any point. We might as well call line m the axis of σ m as the reflection and the glide reflection then share the property that the midpoint of any point P and its image under the isometry lies on the axis. In math, you can create mirror images of figures by reflecting them over a given line. If a and b are distinct lines perpendicular to line c, then σ c σ b σ a is called a glide reflection with axis c. What is a Reflection When you look in the mirror, you see your reflection. The intermediate step between reflection and translation can look different from the starting configuration, so objects with glide symmetry are in general, not symmetrical under reflection alone. When the line of reflection is the y-axis the (x, y) transforms into (-x, y), hence keeping y the same but flipping the x to a negative. ![]() Then σ b σ a is a translation or glide and σ c is, of course, a reflection. In 2-dimensional geometry, a glide reflection is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. However, we begin with the special case where a and b are perpendicular to c. This tutorial introduces you to reflections and shows you some examples of reflections. Glide reflection has a glide and the reflect effect when applied to any image. What is a Reflection When you look in the mirror, you see your reflection. Although it seems there might be many cases, depending on which of a, b, c intersect or are parallel to which, we shall see this turns out not to be the case. The glide reflection meaning is actually in its name. Only those odd isometries σ c σ b σ a where a, b, c are neither concurrent nor have a common perpendicular remain to be considered. An odd isometry is a reflection or a product of three reflections. Tessellating With Translations The simplest and most flexible tessellations are Escher's Type I systems, which can be based on a paralellogram, rhombus, rectangle or square. We have classified all the even isometries as translations or rotations. In fact, using the mathematical definition of glide reflection this sketch has two different types of glide reflection symmetry, both in the vertical direction. ![]()
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