Brick product (InstanceTopic, 3)

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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 ≤ in, 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 ≤ in } ) | 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.