#LINE OF REFLECTION GEOMETRY HOW TO#
#LINE OF REFLECTION GEOMETRY CODE#

If you already have matrix multiplication routines available, because you're also doing rotations for example, this might be the most maintainable way of implementing reflections. The situation is very similar to rotation matrices in affine geometry. Where V.L and L.L are dot product and * is scalar multiple.įinally, shift everything back up by adding b, and the final answer is (x',y'+b).Īs an affine transformation you can write the above operation as the composition (product) of three matrices, first representing the shift y => y-b, then the reflection through the line through the origin, then the shift y => y+b: The final figure will be an equal distance. A reflection reverses the object’s orientation relative to the given line. The most frequently used lines are the y-axis, the x-axis, and the line y x, though any straight line will technically work. This is called the line of reflection or axis of. Now calculate the reflection by the line through the origin, (x',y') = 2(V.L)/(L.L) * L - V A reflection in geometry is a mirror image of a function or object over a given line in the plane. This question identifies something very important we have to know before we start reflecting our figures. Then a vector inside the line is L=(1,m). When a shape is reflected, its size does not change - the image just. First translate (shift) everything down by b units, so the point becomes V=(x,y-b) and the line becomes y=mx. To reflect an object, you need a mirror line. R(1)=2*a-p ! GIVE BACK THE REFLECTION POINTS COORDINATE M2=-1./M1 ! CALCULATE THE SLOPE OF THE LINE L2Ĭ2=S(3,1)-M2*S(3,2) ! CALCULATE THE 'C' OF THE LINE L2

#LINE OF REFLECTION GEOMETRY CODE#

I haven't checked the following code but the crude form of code in the FORTRAN is SUBROUTINE REFLECTION(R,p,q) The algorithm for finding the reflection of the point is as follows:ġ) Find slope of L2 is = -1/m, as L1 and L2 are perpendicularĢ) Using the m' and A(x,y) find c' using eqn of L2ģ) Find the intersection point 'M' of L1 anSd L2Ĥ) As now we have coordinate of A and M so coordinate of A' can be easily obtained using the relation /2 = M(a,b) The line joining the points A and A' is L2 with eqn: y= m'*x + c' We want to find the reflection of the point A(p,q) to line L1,eqn y = m*x + c. Then, pl = p + (d * / ), so pl_x = p_x + (b * p1_x) / (1 + m * m), pl_y = p_y + (m * p1_x) / (1 + m * m), and then use p2 = 2 * pl - p1 to get the final values. In this case, the x axis would be called the axis of reflection. A reflection of a point, a line, or a figure in the X axis involved reflecting the image over the x axis to create a mirror image. To find pl, we have to move the whole problem so that the line goes through the origin by subtracting p from p1, using the above formula, and moving it back. First, let’s start with a reflection geometry definition: Math Definition: Reflection Over the X Axis. It is given by d * / where is the dot product between two vectors. The formula you need here is the projection of a vector v onto a line which goes through the origin in direction d.

The projected point on the line will be pl and your output point p2, then, is p1 + 2 * (pl - p1) = 2 * pl - p1 Let's call this point p = (0, b), the vector d = (1, m), and your input point will be p1. Instead of y = m x + b, let's represent the line by a point p that is on the line and a vector d in the line's direction. The trick is to notice that what you want is to project that point orthogonally on the line, move it by that much, and then move it once again, in the same direction.įor these types of problems, it's easier to work with a slightly more redundant representation for a line. In this case, theY axis would be called the axis of reflection.This is a simple explanation of Il-Bhima's solution. The image of the reflection is congruent to the preimage, just flipped. Math Definition: Reflection Over the Y AxisĪ reflection of a point, a line, or a figure in the Y axis involved reflecting the image over the Y axis to create a mirror image. A reflection occurs when we flip an object or shape across a line, like in a mirror.

#LINE OF REFLECTION GEOMETRY HOW TO#

This complete guide to reflecting over the x axis and reflecting over the y axis will provide a step-by-step tutorial on how to perform these translations.įirst, let’s start with a reflection geometry definition: Math Definition: Reflection Over the X AxisĪ reflection of a point, a line, or a figure in the X axis involved reflecting the image over the x axis to create a mirror image. This idea of reflection correlating with a mirror image is similar in math. In real life, we think of a reflection as a mirror image, like when we look at own reflection in the mirror. Learning how to perform a reflection of a point, a line, or a figure across the x axis or across the y axis is an important skill that every geometry math student must learn.

Line of reflection geometry