Reflection symmetry on graph1/13/2024 The set E ϕ denotes the set of ϕ-fixed edges. The sets E L and E R denote the edges on the left and right sides of the reflection line ℓ, respectively. Assuming that the line of reflection is vertical, the drawing of G gives rise to a partition of the edges E of G into three blocks: E= E L∪ E ϕ∪ E R. This involution will always be denoted ϕ, and is a map V→ V that induces a map E→ E.Ĭondition (2) above means that no edge of G crosses the axis of symmetry. Reflection about a line ℓ takes the drawing into itself, andĮvery edge that is fixed by this reflection about ℓ is fixed point-wise.įor a graph with reflective symmetry, the reflection of the distinguished drawing gives rise to an involution. There are many other ways through which one can take reflections of graph but they are beyond this topic.A graph with reflective symmetry is a graph G=( V, E) with a distinguished, non-degenerate drawing in R 2 such that Thus, all the points that lie below the x- axis are reflected back to above the x- axis and hence the required graph is as shown above. For instance, consider the function f ( x ) = x 4 − 3 x 2 + 1 f\left( x \right)= x = ± 3 , the graph goes down to negative y axis, and it is already given that function has to generate a positive value no matter what the Such function and that is, if the given function is an even function, then their reflections will be symmetric about y − y - y − axis, i.e., the image of y = f ( x ) y=f\left( x \right) y = f ( x ) will be same as the pre – image while the graphs of odd functions are symmetric about the origin. There is an important property associated with If it becomes − f ( x ) -f\left( x \right) − f ( x ), then the function is an odd function, and if it becomes f ( x ) f\left( x \right) f ( x ), then the function is an even function. So, to check whether a given real valued function is an even or odd, simply replace x x x to − x - x − x in y = f ( x ) y=f\left( x \right) y = f ( x ) and see if the function turns out to be − f ( x ) -f\left( x \right) − f ( x ) or f ( x ) f\left( x \right) f ( x ). This is where the concept of even and odd function becomes important. Sometimes, one may find that, even after taking a reflection, the image is still the same. But before discussing the reflection in coordinate planes, one shouldĬlearly understand the concept of even and odd functions and how to determine the axis of symmetry, which is discussed in the following section. The same analogy governs the reflection of the graphs of functions in a cartesian or in a coordinate plane. Such reflections are very common in day – to – day life, for instance, reflections over the water or any shiny surface, in mirrors, etc., and to get a glimpse of what reflection looks like, one can observe the following the following image: Is towards the observer and the point A ′ A' A ′ is away from the observer. In geometry, the reflected image is labeled through prime symbol, i.e., suppose there is a triangle A B C ABC A B C lying in a horizontal plane, then its reflection will be denoted by A ′ B ′ C ′ A'B'C' A ′ B ′ C ′ and the y − y - y − axis here is the axis of symmetry as shown below:įrom the figure, one can observe that, upon reflecting the tringle A B C ABC A B C, its size and shape remains same, and the only change appears is in its direction of facing, i.e., point A A A Once the reflection is taken, this reflection or reflected object is known as the “image” and the original object is known as the “pre – image” of the given object. The axis of symmetry may be any line such as x − x - x − axis, y − y - y −axis, z − z - z −axis, or any line passing through the origin in a coordinate system. This line of reflection is called as the “axis of symmetry” in mathematics and physics. A reflection in general is nothing but the “flipping” or the “folding” of a geometrical object about the line of reflection.
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