New Shape Discovered!

I discovered a new shape:-

Meet the Dimkpa Automaton:





 Fig: 80-non-repeating Dimkpa Automaton (k=1.06) - smooth



Fig: 80-non-repeating Dimkpa Automaton (k=1.02) (smoothed out)

I know, it looks like a football!

Definition

The Dimkpa Automaton is the locus of a point moving in turns within a square such that it starts from the origin and is always projected at a constant angle.

Equation

The equation governing the Dimkpa Automaton is:



Turns

A turn is a unit of motion of the automaton; where it is initially projected from the origin or somewhere along the base of the square (along the x-axis) to “bounce off” the opposite square edge before bouncing off other edges to return to the base.

For K>1 it is called a non-repeating Dimkpa Automaton and it does not leave the imaginary square onto which it is projected. If K < 1 it becomes a repeating Dimkpa Automaton  and may leave the boundaries of the square.

Relationship to other Known Shapes

If K = 0 it is a square. If K>>1 or K<<1 it is a spiral.

The (beautiful) examples above are non-repeating Dimkpa Automatons. The figure below shows what a 50-turn non-repeating Dimkpa Automaton looks like if it is not smoothed out (Its true shape):








Fig: 50-non-repeating Dimkpa Automaton (plain)











The expression above gives the Area of the Dimkpa Automaton.




 I leave you with a picture of the first Dimkpa Automation ever created! It's a K=1 DA