# Recursive Calculation of Relative Convex Hulls

• Gisela Klette
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6607)

## Abstract

The relative convex hull of a simple polygon A, contained in a second simple polygon B, is known to be the minimum perimeter polygon (MPP). Digital geometry studies a special case: A is the inner and B the outer polygon of a component in an image, and the MPP is called minimum length polygon (MLP). The MPP or MLP, or the relative convex hull, are uniquely defined. The paper recalls properties and algorithms related to the relative convex hull, and proposes a (recursive) algorithm for calculating the relative convex hull. The input may be simple polygons A and B in general, or inner and outer polygonal shapes in 2D digital imaging. The new algorithm is easy to understand, and is explained here for the general case. Let N be the number of vertices of A and B; the worst case time complexity is $${\cal O}(N^2)$$, but it runs for “typical” (as in image analysis) inputs in linear time.

## Keywords

relative convex hull minimum perimeter polygon minimum length polygon shortest path path planning

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