The following videos are a companion to the paper Morphing Schnyder drawings of planar triangulations by Fidel Barrera-Cruz, Penny Haxell and Anna Lubiw on morphing Schnyder drawings of planar triangulations.
What is a morph?
Given a graph \(G\) and two drawings of it, \(\Gamma_1\) and \(\Gamma_2\), a morph between \(\Gamma_1\) and \(\Gamma_2\) is a continuous family of drawings that starts at \(\Gamma_1\) and ends at \(\Gamma_2\) or, intuitively speaking, we may think of a morph as an animation between the two drawings.
Linear morphs are a special type of morph where the vertices move from their initial position to their final position along a straight line and at constant speed. However, as seen below not all such morphs are planar, even if both initial and final drawings are planar. The morph shown below goes forward and backward.
We now present some videos showing the linear morphs obtained for Schnyder drawings. Schnyder drawings are obtained from a Schnyder wood and an assignment of positive weights to the interior faces of a triangulation.
Flipping a facial triangle
After analyzing how a facial flip in a Schnyder wood affects the positions of the vertices of a planar triangulation we show that the resulting linear morph is planar. Each coloured set of vertices moves in a direction that is parallel to some exterior edge.
Weight redistribution is a useful tool to preprocess the drawing so that we can flip a separating triangle to morph. In the video below we show a morph resulting from shifting weight between the top two faces that change size.
Flipping a separating triangle
Once the weights are appropriately distributed, we may proceed to flip a separating triangle and the resulting linear morph will be planar.
Morphing through the Schnyder lattice
To conclude, we present a video where we morph between two Schnyder drawings of the icosahedron. The source and target drawings are shown at the bottom left and right respectively. Located at the bottom center is the Schnyder lattice of the icosahedron. The red dot in the lattice indicates our current position in the lattice.
Trying to flip separating triangles directly
When trying to directly flip a separating triangle without redistributing weights, some edge crossings may arise.