Casting is a manufacturing process where liquid material is poured into a mold having the shape of a desired product. After the material solidifies, the product is pulled out of the mold. We study the case in which the mold is made of a single part and the object to be produced is a three-dimensional polyhedron. Objects that can be produced this way are called castable with a single-part mold. A direction in which the object can be removed without breaking the mold is called a valid pull-out direction. We give an  algorithm that decides whether a given polyhedron with n facets is castable with a single-part mold , and if so indicates how to orient the polyhedron in the mold and a direction in which the product can be pulled out without breaking the mold. Our algorithm runs in O(n ) time. The best previously known algorithm for this problem runs in O(n^2) time .
Casting is a manufacturing process where liquid material is poured into a mold having the shape of a desired product. After the material solidifies, the product is removed from the mold. We study the case where the mold is made of a single part and the object to be produced is a three-dimensional polyhedron. Objects that can be produced this way are called castable with a single-part mold. A direction in which the object can be removed without breaking the mold is called a valid removal direction. We give an O(n)-time algorithm that decides whether a given polyhedron with n facets is castable with a single-part mold . When possible, our algorithm provides an orientation of the polyhedron in the mold and a direction in which the product can be removed without breaking the mold. Moreover, we provide an optimal \Theta(n \log n)-time algorithm to compute all valid removal directions for polyhdera that are castable with a single-part mold. Both algorithms are an improvement by a linear factor over the previously best known algorithms for both of these problems .
R
pulled out of
removed from
R
in which
where
R
pull-out
removal
A
O(n)-time
R
, and if so indicates how to orient
. When possible, our algorithm provides an orientation of
R
pulled out
removed
R
Our algorithm runs in O(n ) time. The best previously known algorithm for this problem runs in O(n^2) time
Moreover, we provide an optimal \Theta(n \log n)-time algorithm to compute all valid removal directions for polyhdera that are castable with a single-part mold. Both algorithms are an improvement by a linear factor over the previously best known algorithms for both of these problems