Brick product (InstanceTopic, 3)
From Hi.gher. Space
The brick product is a very generalized operation, written as Q = P{A1, A2, ..., An}. P is called the operator and Ai are called the operands. Their values must satisfy the following constraints:
- P must be a brick.
- The dimensionality of P must equal n.
The resultant shape Q will have the following properties:
- The dimensionality of Q will be the sum of the dimensionalities of Ai.
- Q will be a brick if and only if all Ai are bricks.
- Q will be convex if and only if both P and all Ai are convex.
Definition
For each point x = (x1, x2, ..., xn)/2 in P, the brick symmetry guarantees that (-x1, x2, ..., xn)/2 is also in P. This means the line segment between these two points has length x1. In fact, for any 1 ≤ i ≤ n, a line segment can be constructed between (x1, ..., xi, ..., xn)/2 and (x1, ..., -xi, ..., xn)/2 with both endpoints in P, by definition of brick symmetry of P, each with length xi.
Now, for each point x, we form the Cartesian product (x1A1) × (x2A2) × ... × (xnAn), where the notation xiAi means the operand Ai scaled by xi.
Then Q = P{A1, A2, ..., An} is the union over all x in P of the surfaces of the Cartesian products (x1A1) × (x2A2) × ... × (xnAn).
As a concise expression:
Q = ⋃{ surface( ∏{ xiAi | 1 ≤ i ≤ n } ) | x ∈ P }
where ⋃ represents the union of all members of a set, ∏ represents the Cartesian product of all members of a set, and surface(X) is a function mapping X to its preceding frame, i.e. reduces the net space of X by one while keeping the bounding space the same.