Geometric distance fields of plane curves

This paper introduces a geometric generalization of signed distance fields for plane curves. We propose to store simplified geometric proxies to the curve at every sample. These proxies are constructed based on the differential geometric quantities of the represented curve and are used for queries such as closest point and distance calculations. We investigate the theoretical approximation order of these constructs and provide empirical comparisons between geometric and algebraic distance fields of higher order. We apply our results to font representation and rendering

Employing Pythagorean Hodograph curves for artistic patterns

In this paper we present a novel design element creator tool for the digital artist. The purpose of our tool is to support the creation of vines, swirls, swooshes and floral components. To create visually pleasing and gentle curves we employ Pythagorean Hodograph quintic spiral curves to join a hierarchy of control circles defined by the user. The control circles are joined by spiral segments with at least G2 continuity, ensuring smooth and seamless transitions. The control circles give the user a fast and intuitive way to define the desired curve. The resulting curves can be exported as cubic Bezier curves for further use in vector graphics applications

Operations on Signed Distance Functions

We present a theoretical overview of signed distance functions and analyze how this representation changes when applying an offset transformation. First, we analyze the properties of signed distance and the sets they describe. Second, we introduce our main theorem regarding the distance to an offset set in (X, || · ||) strictly normed Banach spaces. An offset set of D ⊆ X is the set of points equidistant to D. We show when such a set can be represented by f(x) − c = 0, where c 6= 0 denotes the radius of the offset. Finally, we apply these results to gain a deeper insight into offsetting surfaces defined by signed distance functions

Footvector representation of curves and surfaces

This paper proposes a foot mapping-based representation of curves and surfaces which is a geometric generalization of signed distance functions. We present a first-order characterization of the footvector mapping in terms of the differential geometric invariants of the represented shape and quantify the dependence of the spatial partial derivatives of the footvector mapping with respect to the principal curvatures at the footpoint. The practical applicability of foot mapping representations is highlighted by several fast iterative methods to compute the exact footvector mapping of the offset surface of CSG trees. The set operations for footpoint mappings are higher-order functions that map a tuple of functions to a single function, which poses a challenge for GPU implementations. We propose a code generation framework to overcome this that transforms CSG trees to the GLSL shader code

Generating Distance Fields from Parametric Plane Curves

Distance fields have been presented as a general representation for both curves and surfaces [4]. Using space partitioning, adaptive distance fields (ADF) found their way into various applications, such as real-time font rendering [5]. Computing approximate distance fields for implicit representations and mesh objects received much attention. Parametric curves and surfaces, however, are usually not part of the discussion directly. There are several algorithms that can be used for their conversion into distance fields. However, most of these are based converting parametric representations to piecewise linear approximations [7]. This paper presents two algorithms to directly compute distance fields from arbitrary parametric plane curves. One method is based on the rasterization of general parametric curves, followed by a distance propagation using fast marching. The second proposed algorithm uses the differential geometric properties of the curve to generate simple geometric proxies, segments of osculating circles, that are used to approximate the distance from the original curve

Operations on signed distance functions

We present a theoretical overview of signed distance functions and analyze how sphere tracing algorithms slow down in the presence of set operations on said implicit representations

Interactive Rendering Framework for Distance Function Representations

Sphere tracing, introduced by Hart in [5], is an efficient method to find ray- surface intersections, provided the surface is represented by a signed distance function (SDF) or a lower estimate of it. This paper presents an interactive rendering framework for visualising exact and estimate SDF representations. We demonstrate the performance of the system by visualising 3D fractals and its modularity by rendering algebraic and meta surfaces. In addition, we discuss SDF estimation of algebraic surfaces

Unbounding discrete oriented polytopes

We propose an efficient algorithm to compute k-sided unbounding discrete oriented polytopes (k-UDOPs) in arbitrary dimensions. These convex polytopes are constructed for a fixed set of directions and a given center point. The interior of k-UDOPs does not intersect the scene geometry. We discuss several types of general geometric queries on these constructs, such as intersection with rays, and provide an empirical investigation on the limit of these shapes as the number of sides increases. In the 2D case, we extend our construction to planar shapes enclosed by arbitrary parametric boundaries with known derivative bounds