Reflection Across any Line - Online Math Tutor Juan Castaneda, MS

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(1) Reflection Across the X-axis
The Axis of Reflection is the X-axis in the applet below

Move around any red point, to see the effect of its reflection across the X-axis on the corresponding blue point:

Each blue point is the reflected, mirror image of the same-letter red point across the X-axis.
The blue points (A2, B2, C2, D2, E2, F2) automatically react to the movements of its corresponding red point
(A1, B1, C1, D1, E1, F1).


In plane geometry, Reflection across any line is defined by moving each point perpendicularly across a straight line (known as the Axis of Reflection), as far into the other side of the given axis, as the distance from the original point to the axis of reflection.


For example, when the axis of reflection is horizontal (like the X-axis), points originally above it move onto the lower side while points originally below it move onto the upper side. On the other hand, when the axis of reflection is vertical (like the Y-axis), points originally to its right move onto the left-hand side while points originally to its left move onto the right-hand side.


Formula for Reflection across the X-axis

If point P has coordinates (x, y), then point R = RX(P), its reflected image across the X-axis, will be defined by

R = (x, -y)

For example, the reflected image (across the X-axis) of point (3, 1), is point (3, -1)


Formula for Reflection across any Horizontal line { y = k }

If point P has coordinates (x, y), then point R = RH(P), its reflected image across the horizontal line H = { y = k }, will be defined by

R = (x, 2k-y)

For example, the reflected image of point (3, 1), across line { y = 5 }, is point (3, 9)




(2) Reflection Across the Y-axis
The Axis of Reflection is the Y-axis in the applet below

Move around any red point, to see the effect of reflection across the Y-axis on the blue points:

Each blue point is the reflected, mirror image of its corresponding red point across the Y-axis
The blue points (A2, B2, C2, D2, E2, F2) automatically react to the movements of its corresponding red point
(A1, B1, C1, D1, E1, F1).


You may want to try this little activity:

  1. In the applet above, where the Axis of Reflection is the Y-axis, do the following:
  2. Place red point D1 at coordinates (0, 2)
  3. Place red point E1 at coordinates (-3, -4)
  4. Place red point F1 at coordinates (3, -4)
  5. Now notice how the blue triangle D2E2F2, ended up right on top of the red triangle D1F1E1, both showing as the same triangle, a triangle that is symmetric with respect to the Y-axis (the axis of reflection in the second applet)
This activity illustrates an interesting property of the reflection across any line. That is, when a polygon is symmetric with respect to the axis of reflection, its reflected image ends up being the same polygon (with a different ordering of the vertices but still the same subset of the XY-plane).

Formula for Reflection across the Y-axis

If point P has coordinates (x, y), then point R = RY(P), its reflected image across the Y-axis, will be defined by

R = (-x, y)

For example, the reflected image (across the Y-axis) of point (3, 4), is point (-3, 4)


Formula for Reflection across any Vertical line { x = h }

If point P has coordinates (x, y), then point R = RV(P), its reflected image across the vertical line V = { x = h }, will be defined by

R = (2h-x, y)

For example, the reflected image of point (3, 4), across line { x = 1 } is point (-1, 4)




(3) Reflection Across the line { y = x }
The Axis of Reflection is shown in green in the applet below

Move around any red point, to see the effect of its reflection across the line y=x on the the corresponding blue point:

Each blue point is the reflected, mirror image of the corresponding red point, across the line y = x
The blue points (A2, B2, C2, D2, E2, F2) automatically react to the movements of its corresponding red point
(A1, B1, C1, D1, E1, F1).


The line y=x is sometimes called 'The Main Diagonal of the XY-plane'
This is the line going through the origin (0,0), with an inclination angle of 45 degrees.
That means the slope of the line is 1, and its Y-intercept has a value of 0.


Formula for Reflection across the Main Diagonal { y = x }

Algebraically, in terms of coordinates, you can get the reflected image of a point across the main diagonal simply by switching the x and y values in the coordinates of that point.

If point P has coordinates (x, y), then point R = RD(P), its reflected image across the line { y = x ), will be defined by

R = (y, x)

For example, if point P has coordinates (5, 2), then its reflected image across the line y = x
will have coordinates (2, 5).




(4) Reflection Across Any Line
The Axis of Reflection is the line PQ shown in green in the applet below
(You can place this green line anywhere you want for your homework)

Move around any red point, or the green points P and Q, to see the reflection effect on the blue points:

Each blue point (A2, B2, C2, D2) is the reflected, mirror image of its corresponding red point (A1, B1, C1, D1) across the line PQ.
The blue points automatically react to the movements of their corresponding red point, and/or to those of the axis of reflection (the green line).


In the interactive Geogebra worksheet above, by individually moving the green points P and/or Q, you can move the Axis of Reflection -green solid line- into any position (vertical, horizontal, or diagonally slanted). This allows you to see the effects of the reflection in each case, because the blue shape is always presented as the reflected image of the red shape, across the axis of reflexion PQ.


Formulas for Reflection Across Any Line L = { Ax + By + C = 0 }
(This is line L's general form equation)

If point P has coordinates (x, y), then point R = RL(P), its reflected image across line L = { Ax + By + C = 0 },
will be defined by

R = (u, v) = ( u(x,y), v(x,y) ), where

Formula for u, the first coordinate of the reflected image (u,v) of point (x,y) across line L = { Ax + By + C = 0 }

Formula for v, the second coordinate of the reflected image (u,v) of point (x,y) across line L = { Ax + By + C = 0 }

For example, the reflected image of point (1, -2) across line L = { 2x - y + 1 = 0 }, is point (-3, 0)

You will see this in the applet above if you place green point P on (-2, -3), green point Q on (1, 3), and
red point A1 on (1, -2), since then green line PQ (the axis of reflection) will have for equation: 2x - y + 1 = 0

giving values: A = 2; B = -1; C = 1; x = 1; y = -2
so when you apply the above formulas for u and v, you get u = -3 and v = 0.


By moving the red points, you will soon notice this property of reflections: the farther apart from the axis the original points are, the farther apart from the axis their reflected images will be but on the other side of the line. And vice-versa, the closer the original points are to the axis of reflection, the closer their reflected images will be to the same axis, just on the other side of it.


Another property of reflections is that the axis of reflection itself remains fixed under the reflection it defines. You will notice this not by moving the axis PQ but by leaving it fixed, while bringing onto it any one of the red points. When you place a red point exactly on the green line (the axis of reflection) you will see the corresponding blue point simultaneously moving onto the very same spot.


Formulas for Reflection Across Any Line L = { y = mx + b }
(This is line L's slope-intercept equation)

If point P has coordinates (x, y), then point R = RL(P), its reflected image across line L = { y = mx + b },
will be defined by

R = (u, v) = ( u(x,y), v(x,y) ), where

Formula for u, the first coordinate of the reflected image (u,v) of point (x,y) across line L = { y = mx + b }

Formula for v, the second coordinate of the reflected image (u,v) of point (x,y) across line L = { y = mx + b }

For example, the reflected image of point (1, 0) across line L = { y = (1/2)x + 2 }, is point (-1, 4)

You will see this in the applet above if you place green point P on (-4, 0), green point Q on (2, 3), and
red point A1 on (1, 0), since then green line PQ (the axis of reflection) will have for equation: y = (1/2)x + 2

giving values: m = 1/2; b = 2; x = 1; y = 0
so when you apply the above formulas for u and v, you get u = -1 and v = 4.


You may also notice an interesting property that somehow connects reflections with Rotation. When you move around one of the green points (either P or Q), the axis of reflection rotates around the other point, the one that remains fixed. At the same time the blue shape, being the reflected image of the red shape across the green line, automatically rotates around the same fixed point (P or Q) the axis of reflection is rotating around.


All reflections are examples of geometrical transformations called isometries, because they not only preserve the shapes of geometric figures but also their size.


Some reflections are also Linear Transformations. Not all reflections are linear transformations but only those where the axis of reflection goes through the origin (0,0) of the XY-plane. This is because Linear Transformations need to leave the origin fixed. Any geometric transformation of the XY-plane that moves the origin (0,0) to somewhere else, is not a linear transformation. However, when the axis of reflection goes through the origin (0,0) of the XY-plane, that particular reflection will be a linear transformation.




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Last review: April 7, 2017