This paper introduces two new abstract morphs for two 2-dimensional shapes. The intermediate shapes gradually reduce the Hausdorff distance to the goal shape and increase the Hausdorff distance to the initial shape. The morphs are conceptually simple and apply to shapes with multiple components and/or holes. We prove some basic properties relating to continuity, containment, and area. Then we give an experimental analysis that includes the two new morphs and a recently introduced abstract morph that is also based on the Hausdorff distance [Van Kreveld et al., 2022]. We show results on the area and perimeter development throughout the morph, and also the number of components and holes. A visual comparison shows that one of the new morphs appears most attractive.

Given two shapes A and B in the plane with Hausdorff distance 1, is there a shape S with Hausdorff distance 1/2 to and from A and B? The answer is always yes, and depending on convexity of A and/or B, S may be convex, connected, or disconnected. We show that our result can be generalized to give an interpolated shape between A and B for any interpolation variable α between 0 and 1, and prove that the resulting morph has a bounded rate of change with respect to α. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two input sets. We show how to approximate or compute this middle shape, and that the properties relating to the connectedness of the Hausdorff middle extend from the case with two input sets. We also give bounds on the Hausdorff distance between the middle set and the input.

We solve an open problem posed by Michael Biro at CCCG 2013 that was inspired by his and others' work on beacon-based routing. Consider a human and a puppy on a simple closed curve in the plane. The human can walk along the curve at bounded speed and change direction as desired. The puppy runs with unbounded speed along the curve as long as the Euclidean straight-line distance to the human is decreasing, so that it is always at a point on the curve where the distance is locally minimal. Assuming that the curve is smooth (with some mild genericity constraints) or a simple polygon, we prove that the human can always catch the puppy in finite time.

Autonomous agents typically need a navigation mesh of a 3D virtual environment to allow efficient path planning. This mesh needs as input a continuous representation of the walkable areas. However, the walkable environment (WE), i.e. the parts of the 3D environment that an agent can walk on, may contain gaps. These may be due to the filtering steps performed to compute the WE, because of modelling errors in the 3D model, or simply be part of the geometry of the environment. We provide an algorithm that identifies and fills these gaps. We detect gaps, up to a given distance, between pairs of boundary edges of the walkable environment, and fill them with polygons. We employ a heuristic for choosing which pairs of edges should be connected. We compare our algorithm to Recast [10], a voxel-based method for navigation mesh generation. We find that our method gives more accurate results in many environments: it retains the exact representation of the walkable environment, semantically separates the gaps from the walkable areas, and requires no tweaking of parameters to obtain good results. However, our method is currently slower than Recast, and requires more memory.

We consider the problem of testing, for a given set of planar regions R and an integer k, whether there exists a convex shape whose boundary intersects at least k regions of R. We provide polynomial-time algorithms for the case where the regions are disjoint axis-aligned rectangles or disjoint line segments with a constant number of orientations. On the other hand, we show that the problem is NP-hard when the regions are intersecting axis-aligned rectangles or 3-oriented line segments. For several natural intermediate classes of shapes (arbitrary disjoint segments, intersecting 2-oriented segments) the problem remains open.

Abstract Thek‐nearest neighbour (kNN) problem appears in many different fields of computer science, such as computer animation and robotics. In crowd simulation,kNN queries are typically used by a collision‐avoidance method to prevent unnecessary computations. Many different methods for finding these neighbours exist, but it is unclear which will work best in crowd simulations, an application which is characterised by low dimensionality and frequent change of the data points. We therefore compare several data structures for performingkNN queries. We find that the nanoflann implementation of ak‐d tree offers the best performance by far on many different scenarios, processing 100,000 agents in about 35 ms on a fast consumer PC.