Arithmancy · bend the grid
experiment no. 007 · sketch
Bend the grid.
change one thing and see what moves. every point goes through the same two steps, slide sideways by the shear, then spin by the rotation
linear algebrainteractive
annotation
this was supposed to just be a static illustration, then it wasn't
matrix · R·S
[ 0.94 0.03 ] [ 0.34 1.08 ]
what is happening?
shear slides each row sideways. rotate spins the whole thing around after.
the solid arrow is where (1, 0) lands, the dashed one is where (0, 1) lands. every other point just follows those two.
S = [ 1 0.40 ] R = [ cos θ −sin θ ]
[ 0 1.00 ] [ sin θ cos θ ]
shear first, then rotate, so S is applied before R.✦ the chalkboard
the identity. nothing moves, so every direction is an eigenvector.
move your cursor to bend it, double-click to reset.drag across the grid to bend it. more like this →
What you are looking at
the faint square is the original 13 × 13 grid, spread out evenly from −1 to 1 in each direction. the bright points are those same 169 points after getting transformed.
the solid arrow shows where the point (1, 0) ends up, and the dashed one shows where (0, 1) ends up. since this transformation is linear, those two arrows decide where literally every other point goes too, a point at (x, y) lands at x times the first arrow plus y times the second.
shear happens first: u = x + k·y. then it rotates by θ: x′ = cos θ · u − sin θ · y and y′ = sin θ · u + cos θ · y. doing it in the other order gives you a totally different picture, which is a pretty good way to see that matrix multiplication actually cares about order.