This is the first part of a possibly continuous series of pieces on, well, chess pieces. Future posts might also deal with other aspects of chess variants or generalized chess, but all these will simply attempt to create a taxonomy and/or set definitions regarding some concepts of generalized chess.
For this first essay on pieces, we will look at a basic kind of piece, the Leaper. A leaper is a piece that moves from one space (its origin) to another space a set distance and direction away (its destination). The move is instantaneous and is not impeded by intervening pieces.
Leap Distances
There are two ways to define the distance of a piece's move: destination coordinates and length of leap. Coordinates measure a (x,y) distance from origin to destination on a lattice grid. The simplest way to put it is to imagine the piece leaping x squares in one direction then y squares perpendicular. Length of leap measures the straight line drawn from the origin to the destination in this lattice grid (On a chessboard the points will be the centers of the squares).
Leapers are either simple or composite based on how many unique coordinates it has.
For example, a (1,1) leaper has a leap distance of square root of 2, while a (1, 2) leaper, a knight, has a leap distance of square root of 5. Compounding coordinates and lengths will give more types of leapers. These distance systems do not take into account direction and therefore cannot define leapers with distance constraints.
A well-known piece named from its leap distance is the root-fifty leaper, whose leap distance of sqrt-50 has two coordinates (5,5) and (1,7)
Leap Directions
While the methods of the previous section help with defining the distance of a leap, pieces can also be restrained by direction. Directions of limited leapers are defined relative to the perspective of the mover.
Shogi variants are peppered with examples of limited hoppers, usually relative to a king's movement, e.g. gold and silver general.
Sometimes this leads to a piece that can only move in one direction and might require some extra provisions exclusive to them lest they become deadweight.
Board Range
A chess knight can visit all 64 squares of a chess board once, but other leapers are limited in where they can go. A way for me to gauge a piece's range is to start with a piece on a random square on the board and color the squares based on the least number of leaps needed to reach a square. The board will either be filled with color or contain untouched spots.
Whether a piece should have full range or not is within the decision of the designer, but the utility of pieces that cannot traverse the whole board is a topic usually glossed over. For these series, pieces that have full board range are "free-moving", otherwise they are "constrained."
Constraints
The most common piece constraint is that of colorboundedness, the state of a piece that can only traverse one color on a checkered board. For the purposes of this essay any piece more constrained than this is seen as heavily constrained.
From one color to another
A chess knight can only leap to squares that are not the same as the square it is on, while a king can move to a square of either color. While the consequences of the properties of these moves are sometimes mentioned in chess study (e.g. the knight can never lose tempo), I am yet to see any further talk in regards to chess variants (geometries involved in these sorts of leaps may be nontrivial).
This alternating leaping is different from a colorbound one only in the sense of destination squares, as colorbound leapers practically move on a board of their own and can be subject to the same tempo issues.
The question then, if a leaper can always go to either color square, can it always lose tempo?
The Rider
Riders are pieces that move continuously through unoccupied squares. A rider is blocked by a friendly piece and stops moving when it captures. The rider is constructed as making successive hops, with pieces in their trajectory able to intervene. A rook and a bishop are prime examples of simple riders, the queen a compound rider.
To make sense of a rider being made of successive hops, let our example be the knightrider, which makes continuous knight hops. Just like an actual knight it hops over pieces not within its ride, i. e. only pieces on squares within a successive line of knight hops will matter in its trajectory.
If a piece can be blocked on its way to a square it can go to by virtue of a piece getting in the way of its path, it's a rider. A leaper can only be blocked by friendly pieces on its destination squares.
Riders long and short
A long rider can travel to its full extent in any direction it goes, blocked only by friendly pieces, captures and the edge of the board. All orthodox chess riders ride long. A short rider has a finite range, i.e. riders with any limited velocity are short may it be constant or maximum.
In Chess with Different Armies, one of the armies features a short rook, in this case a rook that can only move up to four squares when it moves.
A change of trajectory
A rider is bent if it requires that in the middle of its move it has to change direction. Bent riders are rarely long, although notable are riders that can diagonally bounce against walls.
A short bent rider has a finite ride that cannot be extended and must traverse through all the squares or else it is blocked. The knight in Chinese chess is short and bent in this regard. Long bent riders are possible in practice by requiring every step of the ride to bend a certain way, either making the ride a zigzag or a loop. Other regular patterns can also be done
As a more general extension of the short bent rider concept, let's look at a piece called the Sissa. The sissa moves by first riding a set number of squares as either a bishop or a rook, then changing direction and moving the same number of squares in a non-orthogonal direction. If a piece is in the way of the sissa that cannot be captured, that move is blocked.
To give an example on the necessity of limiting long bent riders, consider a piece that can make multiple knight moves in succession, regardless of direction as long as it doesn't eventually land on the same square and it captures at the end of its turn. Considering that a knight can visit an entire chessboard, how different is thus souped-up knight from a piece that is defined by leaping to any other arbitrary square?
