Reflection: Meaning, Reflection Formula, Properties

The reflection meaning in mathematics, a reflection (sometimes spelt reflexion) is an isometric mapping from a Euclidean space to itself that uses a hyperplane as a collection of fixed points; this set is known as the axis (in dimension 2) or plane (in dimension 3). A figure's mirror image in the axis or plane of reflection is called a reflection image or reflection point. For example, for a reflection about a vertical axis, the mirror image of the minuscule Latin letter p would be q. Its reflection in a horizontal axis would produce picture b. A reflection is an involution in which every point returns to its original place and every geometrical object is returned to its original state when applied twice in succession.

One of the four types of transformations in geometry is reflection.

The four fundamental transformations are as follows:

• Reflection

• Rotation

• Dilation or Resizing

• Translation

Let's look at the definition of reflection transformation in math, reflection formula, reflections on the coordinate plane, and examples.

What is reflection meaning?

A flip is a term used in mathematical geometry to describe a reflection. A mirror image of a shape is called a reflection. The line of reflection is a line along which an image reflects. If a figure is stated to be a mirror of another figure, then each point in the first figure is equidistant from the corresponding point in the second figure. The reflected picture should have the same shape and size as the original, but it should face the opposite way. The translation may occur as a result of changes in position during reflection. The original image is referred to as a pre-image, and its reflection is referred to as an image.

The letters ABC and A'B'C' stand for pre-image and image, respectively. The term "reflection" is frequently used to refer to a broader class of mappings from a Euclidean space to itself, notably involutions, which are non-identity isometries. The collection of fixed points (the "mirror") in such isometries is an affine subspace that is possibly smaller than a hyperplane. A reflection through a point, for example, is an involutive isometry with only one fixed point; the image of the letter p beneath it would appear to be a point d.

Construction

Drop a 90 degree line from the point to the line (plane) used for reflection and extend it the same distance on the opposite side to determine the reflection of a point in a plane (or, equivalently, 3-dimensional) geometry. To locate the figure's reflection, reflect on each point in the figure.

Using a compass and a straightedge, reflect point P through the line AB as follows (see figure):

Step 1 (red): Create points A′ and B′ on line AB that are equidistant from P by constructing a circle with the centre at P and a fixed radius r.

Step 2 (green): Make circles with a radius of r that are centred at A′ and B′. The points of intersection of these two circles will be P and Q.

The reflection of point P through line AB is then pointed to Q.

Important Points Regarding the ‘Reflection’ Definition

We understood what a mathematical transformation. Now, let us learn the reflection definition by using the following pointers for a better understanding.

• Flipping an image is called a Reflection in geometry. So, the resulting image will be the mirror image to the origins structure.

• Only the direction of the resulting image is the opposite. However, the size and shape remain the same.

• Since the position is changed in this transformation, there are chances for Translation as well.

• Both the figures (before and after reflection) are equidistant from all their points over their surfaces.

• There is 1 line that helps in reflecting the object and this line is said to be the line of reflection.

Reflections in the Coordinate Plane

The reflection transformation might be performed to both the X and Y axes.

1. Reflection over X-axis

The x-coordinates of a point remains the same when it is reflected across the X-axis. The Y-coordinates, on the other hand, is changed to the opposite sign. As a result, the point (x, y) is reflected across the X-axis (x, -y).

2. Reflection over Y-axis

The Y-coordinates of a point remains the same when it is reflected across the Y-axis. The X-coordinates, on the other hand, is changed into their opposites. As a result, the point (x, y) is reflected across the Y-axis (-x, y).

3. Reflection over Y = X

The x-coordinates and y-coordinates change when a point is reflected across the line y = x. Similarly, when a point is reflected across the line y = -x, the x- and y-coordinates change points and then become negated.

Therefore,

The point (x, y) is reflected over the line y = x. (y, x).

The point (x, y) is reflected over the line y = – x. (-y, -x).

Reflection in a Point

A reflection point is formed when a figure is created around a single point known as the point of reflection or the figure's centre. On the opposite side of every point in the figure, another point is revealed. When the figure is reflected, its size and shape remain unchanged.

Reflection in Origin (0, 0)

Any point in the coordinate plane can be used as a reflection point. The term "origin" is the most widely used.

Reflection in Slope

When a figure is reflected, the reflecting line is the perpendicular bisector of all segments that connect pre-image locations to their corresponding image points.

Here's an example of a problem that makes use of this concept: Triangle J'K'L' is the reflection of triangle JKL over a reflecting line in the diagram below. Using the coordinates J and J', get the equation of the reflecting line. After that, double-check that this reflecting line is transmitting K to K' and L to L'.

The perpendicular bisector of segments connecting pre-image points to their image points is known as the reflection of a point about a line. Because a segment's perpendicular bisector passes through its midpoint, the first step in determining the equation of the reflecting line is to determine the midpoint of line segment JJ':

Now the reflection of a point about a line formula is given below,

$Midpoint = \left ( \frac{x_{1}+x_{2}}{2} ,\frac{y_{1}+y_{2}}{2} \right )$

$Midpoint = \left ( \frac{ - 1+ 15}{2} ,\frac{ 7 + 5}{2} \right )$

$Midpoint = \left ( 7 ,6 \right )$

Now reflection of a point about a line y=mx is as follows,

$Slope = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$Slope = \frac{5 - 7}{ 15 - (-1)}=\frac{-2}{16}=\frac{-1}{8}$

The slope (y=mx) of the perpendicular bisector of $\overline{JJ^{,}}$ is the opposite reciprocal of the slope (y=mx) $\overline{JJ^{,}}$.

The slope of $\overline{JJ^{,}}$ is -1/8, So the slope of the perpendicular bisector, and therefore the reflection of a point about a line is 8.

Reflection Formula List

Properties of Reflection

The reflection matrix is orthogonal, with a determinant of 1 and eigenvalues of 1, 1, 1,..., 1. A specific orthogonal matrix that describes a rotation is the product of two such matrices. Every proper rotation is produced by the even number of hyperplane reflections via the origin, but every improper rotation is produced by an odd number. The Cartan Dieudonné theorem is the result of reflections generating the orthogonal group. Similarly, reflections in affine hyperplanes form the Euclidean group, which contains all isometries of Euclidean space. A reflection group is a group formed by reflections in affine hyperplanes in general. Coxeter groups are instances of finite groups created in this fashion.

Solved Example

1. When the point P with coordinates (9, 11) is reflected in the x-axis, it is mapped onto P', and then P' is mapped onto P" when reflected in the y-axis. Determine the coordinates of P.

Solution:

P' denotes the coordinates (9, -11).

When P' is reflected in the y-axis and projected onto P", the coordinates of P" become (-9, -11).

1. When a point P is reflected in the y-axis, it is mapped onto P', while P' is mapped onto P" when reflected in the x-axis. Find the coordinates of P if the coordinates of P" are (-7, 4).

Solution:

Let P's coordinates be (x, y). The coordinates of P' are then calculated (-x, y). As a result, the coordinates of P" are (-x, -y).

The coordinates of P" are given in the question (-7, 4). As a result, –x = -7 and –y = 4.

As a result, x = 7 and y = -4.

As a result, P's coordinates are (7, -4).

Conclusion

A reflection is a form that is a mirror image of itself. The line of reflection is formed when an image reflects through a line. When a figure is stated to be a mirror of another figure, every point in that figure is equidistant from every corresponding point in another figure. The reflected image should be the same shape and size as the original, but it should be oriented in the opposite direction. In reflection, translation may occur as a result of changes in posture. The original picture is referred to as pre-image, and its reflection is referred to as image. The pre-image and image are represented by ABC and A'B'C', respectively. The coordinate system may be used to reference the reflection transformation (X and Y-axis).