Affine transformation pattern

Parametric equations

An affine map adds a translation to the linear part, so a shape can move and copy itself:

[x^{'} y^{'}] = [a c b d] [x y] + [t_{x} t_{y}]

x^{'} = (s cos θ) x - (s sin θ) y + t_{x}, y^{'} = (s sin θ) x + (s cos θ) y + t_{y}

θ is the rotation angle. s breathes slightly over time (about 0.72 ± 0.03). $t_{x}, t_{y}$ is the translation vector. The pattern splits and grows outward from the central parent shape. The same affine map acts on a base square, and a second layer, scaled and translated (factor 0.5), is stacked in each of four directions. The result is a symmetric, multiplying pattern.

Interaction

Rotation angle θ: change the rotation applied at each iteration, and with it the symmetry of the pattern
Translation e: control the spacing and density of the pattern in the plane
Evolution speed ω: drive the periodic change of the parameters so the pattern evolves continuously

What to notice

Repeating an affine map can produce periodic or quasi-periodic patterns.
The combination of translation and rotation sets the lattice structure of the pattern.
When the parameters change continuously, the pattern deforms smoothly instead of switching at once.

Affine transformation pattern

Parametric equations

Interaction

What to notice

Further reading

Comments

Parametric equations

Interaction

What to notice

Related

Further reading

Comments