Arbitrary V-Tiles

Constructing V-tiles with arbitrary curves

Instead of classic polygons or curved polygons, we can use nearly arbitrary curves (paths) to construct a V-tile.
We can choose a curve from R to T and another from T to Q. Reflect RT at R and TQ at Q.
Certain conditions have to be met:
The images of the curves V and TZ, wen rotated around Z by angle ε must not intersect the original curves.
The slopes of the curves at point T have to form an angle smaller than 180°-ε (there has to be a corner at T). Otherwise the transformations of the curves will intersect near the Point V.
In the figures, the arbitrarily chosen curves are colored red. The black curves are obtained through the necessary transformations described in the first chapters.

Correct construction

Incorrect construction

Correct V-tiles surrounding two copies

Note that the points Z and V belong to all 4 tiles.

Maximal internal angle at T

The internal angle at T has to be smaller than 180°-ε.
If this angle is equal to 180°-ε, then the point C ∈ VF', where C is the reflection of A at R and F' the reflection of F at S.