Abstract An algorithm for cutting solid objects in a topology‐controlled manner is presented. Concretely, given a loop on the object boundary, a disk‐topology cut surface bounded by the loop is constructed in the interior. In contrast to various previous approaches, both disk topology and conformance to the prescribed loop are ensured by construction, while supporting not only contractible but also incontractible loops on the boundaries of manifold objects of higher genus and arbitrary non‐trivial topology. We describe an implementation of this algorithm in the discrete setting, with triangle mesh cut surfaces embedded in tetrahedral mesh objects. Making use of this novel cutting algorithm, we describe a method for the reliable construction of bijective volumetric maps between solid objects, demonstrating the algorithm's utility. This mapping method overcomes restrictions of the state of the art to topological balls, extending coverage to objects of arbitrary genus, specifically so‐called 1 ‐handlebodies.

A method for the construction of bijective volumetric maps between 3D shapes is presented. Arbitrary shapes of ball-topology are supported, overcoming restrictions of previous methods to convex or star-shaped targets. In essence, the mapping problem is decomposed into a set of simpler mapping problems, each of which can be solved with previous methods for discrete star-shaped mapping problems. Addressing the key challenges in this endeavor, algorithms are described to reliably construct structurally compatible partitions of two shapes with constraints regarding star-shapedness and to compute a parsimonious common refinement of two triangulations.

A method is presented to compute volumetric maps and parametrizations of objects over 3D domains. As a key feature, continuity and bijectivity are ensured by construction. Arbitrary objects of ball topology, represented as tetrahedral meshes, are supported. Arbitrary convex as well as star-shaped domains are supported. Full control over the boundary mapping is provided. The method is based on the technique of simplicial foliations, generalized to a broader class of domain shapes and applied adaptively in a novel localized manner. This increases flexibility as well as efficiency over the state of the art, while maintaining reliability in guaranteeing map bijectivity.