You'll notice that the "squares" on the board are not really square— they range from more or less square to semicircular. Even though the "squares" differ in shape, each "square" has four sides and four corners.
Even the two semicircular "squares" at the center of the board have four sides and four corners. These two "squares" share two sides and three corners in common. The corner between them which lies at the very center of the board is the singularity from which this chess variation takes its name.
Since the board layout is a "curved space," straight moves and diagonal moves have to be defined locally instead of globally. A straight move can be defined as a move which enters a "square" through one side, and may continue on to exit the "square" through the opposite (nonadjacent) side. A diagonal move can be defined as a move which enters a "square" through one corner, and may continue on to exit the "square" through the opposite (nonadjacent) corner.
This leads to some long-distance moves which are anything but straight as we think of straight. Notice how the rook's move can take it looping around the center of the board, and right back like a boomerang to the same side of the board it started from.
The bishop's move also loops around the singularity at the center of the board. Thus the same bishop can sometimes threaten a square from two different directions at once.
The knight's move likewise is distorted in an almost psychedelic fashion.
As a pawn reaches the semicircular "squares" at the center of the board, it makes sense to me to have it continue to move toward the far side of the board, even though technically this amounts to having the pawn move "sideways" instead of "forward." (Move "forward" indicated in white.) It looks "forward" to us, and it's the only way the pawn will ever make it to the far end where it can be promoted.
It would also make sense to have the pawn's "forward" diagonals for capturing be through the two corners on either side of its next "forward" move. (Indicated in yellow.) Though this would mean that a rook's pawn still on its original square could capture the opposing rook's pawn on its original square— not sure whether that should be permitted or disallowed.
One odd feature of the singularity, or "corner" at the center of the board, is that a piece which moves "diagonally" through the singularity will come out on squares of the opposite color. Thus the bishop above starts on a white square, but if it moves through the singularity, it will end up on a black square.