Coordinate transformation for point reflection matrix
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A vector could be represented by an ordered pair (x,y) but it could also be represented by a column matrix:
[xy]
[xy]
Polygons could also be represented in matrix form, we simply place all of the coordinates of the vertices into one matrix. This is called a vertex matrix.
Example
A square has its vertexes in the following coordinates (1,1), (-1,1), (-1,-1) and (1,-1). If we want to create our vertex matrix we plug each ordered pair into each column of a 4 column matrix:
[x1y1x2y2x3y3x4y4]=[11−11−1−11−1]
[x1x2x3x4y1y2y3y4]=[1−1−1111−1−1]
We can use matrices to translate our figure, if we want to translate the figure x+3 and y+2 we simply add 3 to each x-coordinate and 2 to each y-coordinate.
[x1+3y1+2x2+3y2+2x3+3y2+2x4+3y2+2]
[x1+3x2+3x3+3x4+3y1+2y2+2y2+2y2+2]
If we want to dilate a figure we simply multiply each x- and y-coordinate with the scale factor we want to dilate with.
3⋅[x1y1x2y2x3y3x4y4]
3⋅[x1x2x3x4y1y2y3y4]
When we want to create a reflection image we multiply the vertex matrix of our figure with what is called a reflection matrix. The most common reflection matrices are:
for a reflection in the x-axis
[100−1]
[100−1]
for a reflection in the y-axis
[−1001]
[−1001]
for a reflection in the origin
[−100−1]
[−100−1]
for a reflection in the line y=x
[0110
[xy]
[xy]
Polygons could also be represented in matrix form, we simply place all of the coordinates of the vertices into one matrix. This is called a vertex matrix.
Example
A square has its vertexes in the following coordinates (1,1), (-1,1), (-1,-1) and (1,-1). If we want to create our vertex matrix we plug each ordered pair into each column of a 4 column matrix:
[x1y1x2y2x3y3x4y4]=[11−11−1−11−1]
[x1x2x3x4y1y2y3y4]=[1−1−1111−1−1]
We can use matrices to translate our figure, if we want to translate the figure x+3 and y+2 we simply add 3 to each x-coordinate and 2 to each y-coordinate.
[x1+3y1+2x2+3y2+2x3+3y2+2x4+3y2+2]
[x1+3x2+3x3+3x4+3y1+2y2+2y2+2y2+2]
If we want to dilate a figure we simply multiply each x- and y-coordinate with the scale factor we want to dilate with.
3⋅[x1y1x2y2x3y3x4y4]
3⋅[x1x2x3x4y1y2y3y4]
When we want to create a reflection image we multiply the vertex matrix of our figure with what is called a reflection matrix. The most common reflection matrices are:
for a reflection in the x-axis
[100−1]
[100−1]
for a reflection in the y-axis
[−1001]
[−1001]
for a reflection in the origin
[−100−1]
[−100−1]
for a reflection in the line y=x
[0110
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