### Abstract

We study the feasibility of design for a layer-deposition manufacturing process called stereolithography which works by controlling a vertical laser beam which when targeted on a photocurable liquid causes the liquid to harden. In order to understand the power as well as the limitations of this manufacturing process better, we define a mathematical model of stereolithography (referred to as vertical stereolithography) and analyze the class of objects that can be constructed under the assumptions of the model. Given an object (modeled as a polygon or a polyhedron), we give algorithms that decide in O(n) time (where n is the number of vertices in the polygon or polyhedron) whether or not the object can be constructed by vertical stereolithography. If the answer is in the affirmative, the algorithm reports a description of all the orientations in which the object can be made. We also show that the objects built with vertical stereolithography are precisely those that can be made with a 3-axis NC machine. We then define a more flexible model that more accurately reflects the actual capabilities of stereolithography (referred to as variable-angle stereolithography) and again study the class of feasible objects for this model. We give an O(n)-time algorithm for polygons and O(n log n)- as well as O(n)-time algorithms for polyhedra. We show that objects formed with variable-angle stereolithography can also be constructed using another manufacturing process known as gravity casting. Furthermore, we show that the polyhedral objects formed by vertical stereolithography are closely related to polyhedral terrains which are important structures in geographic information systems (GIS) and computational geometry. In fact, an object built with variable-angle stereolithography resembles a terrain with overhangs, thus initiating the study of more realistic terrains than the standard ones considered in geographic information systems. Finally, we relate our results to the area of grasping in robotics by showing that the polygonal and polyhedral objects that can be built by vertical stereolithography can be clamped by parallel jaw grippers with any positive-sized gripper.

Original language | English (US) |
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Pages (from-to) | 61-83 |

Number of pages | 23 |

Journal | Algorithmica (New York) |

Volume | 19 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 1997 |

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### Keywords

- 3-Axis NC-machine
- Algorithms
- Computational geometry
- Computer-aided design
- Computer-aided manufacturing
- Geographic information systems
- Grasping
- Gravity casting
- Manufacturing
- Polygon
- Polyhedra
- Polyhedral terrains
- Robotics
- Stereolithography

### ASJC Scopus subject areas

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics

### Cite this

*Algorithmica (New York)*,

*19*(1), 61-83. https://doi.org/10.1007/PL00014421