matrices transformations Flashcards
A linear transformation is a transformation in which the image (x’, y’) of a point (x, y) can be written as
x’=ax+by
y’=cx+dy
x’
y’
(a b)(x)
c d)(y
easy way to find the matrices representing simple transformations is to think about the images of the points
(1, 0) and (0, 1)
A stretch parallel to the x-axis with scale factor k maps the point (1, 0) to the point
(k, 0), but leaves the point (0, 1) unchanged
So the matrix representing this is
(k 0)
(0 1)
A stretch parallel to the y-axis with scale factor k maps the point (0, 1) to the point (0, k).
matrice:
(k 0)
(0 1)
An enlargement, centre the origin, with scale factor k
(k 0)
0 k
A rotation through 90° anticlockwise about the origin maps the point (1, 0)
to (0,1)
(0 -1)
(1 0)
A rotation through 180° about the origin maps the point (1, 0)
to -1,0
(-1 0)
(0 -1)
A rotation through 90° clockwise about the origin maps the point (1, 0)
to 0,-1
(0 1)
(-1 0)
anticlockwise rotation through an angle x maps the point (1, 0)
(cosx, sinx)
anticlockwise rotation through an angle x maps the point (0,1)
(-sinx, cost)
matrix represents a acw rotation through any angle x is given
(cosx -sinx)
sinx cosx
a clockwise rotation would have
the angle as a negative
Reflection in the x-axis
1 0
0 -1
Reflection in the y-axis
-1 0
0 1
