# Technical Reports

## HPL-2009-144

**Orienting Transverse Fiber Products**

* Ramshaw, Lyle; Basch, Julien*

HP Laboratories

HPL-2009-144

**Keyword(s):** Boundary, convolution, fiber product, Minkowski sum, mixed associativity, offset, orientation, pullback, quiver

**Abstract:**
Direct products (a.k.a. Cartesian products) are familiar: Given linear spaces *A* and *B* of dimensions *a* and *b*, their direct product *A X B* is the linear space of dimension *a + b* consisting of all ordered pairs (**a,b**), for ** a** in

*A*and

**in**

*b**B*; and direct products of smooth manifolds are an analogous story. Fiber products (a.k.a. pullbacks) are a less familiar generalization. Given three linear spaces

*A, B,*and

*S*of dimensions

*a, b*, and

*s*and given linear maps

*f:A -> S*and

*g:B -> S*, their fiber product

*A X*is that subspace of the direct product

_{s}B*A X B*on which the maps

*f*and

*g*agree, that is, the set of ordered pairs (

**) with**

*a,b**f(*in

**a**)=g(**b**)*S*. When the image spaces

*f(A)*and

*g(B)*together span all of

*S*, the maps

*f*and

*g*are said to be

*transverse*, and the dimension of the fiber product is then

*a + b - s*. Fiber products make sense also for manifolds: Given smooth manifolds

*A, B*, and

*S*of dimensions

*a, b*, and

*s*and given smooth maps

*f:A -> S*and

*g:B -> S*that are transverse in the appropriate sense, it is a standard result that the fiber product

*A X*is itself a smooth manifold of dimension

_{s}B*a + b - s*.

But what if the input manifolds *A, B*, and *S* are oriented? Is there then some natural rule for orienting the fiber-product manifold *A X _{s} B*? Such a rule is needed for an application of fiber products in computer-aided geometric design and robotics --- in particular, for computing the boundary of a Minkowski sum from the boundaries of its summands. We show that there is a unique rule for orienting transverse fiber products that satisfies the

*Axiom of Mixed Associativity: (A X*.

_{s}B) X_{T}C = A X_{S}(B X_{T}C)173 Pages

External Posting Date: June 21, 2009 [Fulltext]. Approved for External Publication

Internal Posting Date: June 21, 2009 [Fulltext]