## Topological Modeling for Vector Graphics

Boris Dalstein

PhD thesis, University of British Columbia, 2017

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### Implementation

Our open source implementation VPaint is available at www.vpaint.org

The successors of VPaint, VGC Illustration and VGC Animation, are being developed at VGC

### Thesis

Topological Modeling for Vector Graphics

Boris Dalstein

PhD thesis, University of British Columbia, 2017

[ PDF ] [ BibTeX ]

In recent years, with the development of mobile phones, tablets, and web technologies, we have
seen an ever-increasing need to generate vector graphics content, that is, resolution-independent
images that support sharp rendering across all devices, as well as interactivity and animation.
However, the tools and standards currently available to artists for authoring and distributing such
vector graphics content have many limitations. Importantly, basic topological modeling, such
as the ability to have several faces share a common edge, is largely absent from current vector
graphics technologies. In this thesis, we address this issue with three major contributions.

First, we develop theoretical foundations of vector graphics topology, grounded in algebraic topology.
More specifically, we introduce the concept of Point-Curve-Surface complex (PCS complex) as
a formal tool that allows us to interpret vector graphics illustrations as non-manifold, non-planar,
non-orientable topological spaces immersed in R^{2}, unlike planar maps which can only represent
embeddings.

Second, based on this theoretical understanding, we introduce the vector graphics complex (VGC)
as a simple data structure that supports fundamental topological modeling operations for vector
graphics illustrations. It allows for the direct representation of incidence relationships between
objects, while at the same time keeping the geometric flexibility of stacking-based systems, such
as the ability to have edges and faces overlap each others.

Third and last, based on the VGC, we introduce the vector animation complex (VAC), a data
structure for vector graphics animation, designed to support the modeling of time-continuous
topological events, which are common in 2D hand-drawn animation. This allows features of a connected
drawing to merge, split, appear, or disappear at desired times via keyframes that introduce the
desired topological change. Because the resulting space-time complex directly captures the time-
varying topological structure, features are readily edited in both space and time in a way that
reflects the intent of the drawing.