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A reflection through an axis. In mathematics, a reflection (also spelled reflexion) [1] is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in ...
, = [ ] []. which is a reflection in the x-axis followed by a rotation by an angle θ, or equivalently, a reflection in a line making an angle of θ/2 with the x-axis. Reflection in a parallel line corresponds to adding a vector perpendicular to it.
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. The isometry group generated by just a glide reflection is an infinite cyclic group. [1]
Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the x-axis). In more generality, reflection across a line through the origin making an angle with the x-axis, is equivalent to replacing every point with coordinates (x, y) by the point with coordinates (x′,y′), where
Here we take , which is a planar reflection in the = plane, and , which is a 180-degree rotation around the x-axis. Their geometric product is e 123 {\displaystyle {\boldsymbol {e}}_{123}} , which is a point reflection in the origin, because that is the transformation that results from a 180-degree rotation followed by a planar reflection in a ...
An xy-Cartesian coordinate system rotated through an angle to an x′y′-Cartesian coordinate system In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and ...
We only consider stretches along the x-axis and y-axis. A stretch along the x-axis has the form x' = kx; y' = y for some positive constant k. (Note that if k > 1, then this really is a "stretch"; if k < 1, it is technically a "compression", but we still call it a stretch. Also, if k = 1, then the transformation is an identity, i.e. it has no ...
In the complex reflection coefficient plane the Smith chart occupies a circle of unity radius centred at the origin. In cartesian coordinates therefore the circle would pass through the points (+1,0) and (−1,0) on the x-axis and the points (0,+1) and (0,−1) on the y-axis.