Smooth Curves on Smooth Surfaces

1
Smooth Curves on Smooth Surfaces
Advisor Professor Carlo H. Séquin
Undergraduate Researcher Ling Xiao
Abstract The general goal of my research is to
generate smooth curves on smooth surfaces, with
the special focus to demonstrate how graphs of
relatively high interconnectivity can be mapped
crossing free onto surfaces of suitable genus.
As an example, we will show how to map Dycks
graph with 12 vertices and 48 edges onto a
symmetrical genus 3 surface (Fig1, Fig 2).
Multi-resolution Greedy Refinement Algorithm At
each level of the subdivision process we perform
greedy optimization. A gradient descent algorithm
moves all intermediate vertices of a path along
the polyhedron edges that it crosses, using a
cost function designed to even-out the curvature
of the neighboring vertices along the path (Fig
4).
Fig 4. Path refining at subdivision level 2
Fig 1. Dyke Graph on Tetra-frame by Prof. Séquin
When the process exhibits minimal changes in
energy, we increase the level of the subdivision
surface, project the path onto the new finer
mesh, and repeat the refinement algorithm on the
higher resolution path. The process is then
repeated until the desired level of smoothness
is achieved (Fig 5a,b).
Introduction To map the graph, we first place
the nodes of the graph in symmetrical locations
on the surface, and then connect these nodes
according to the graph edges. The goal of this
research is to create smooth, crossing-free
curves on the surface. We decided to use curves
with geodesic curvature that vary linearly with
arc length (LVC). These (LVC) curves are smooth,
graceful and give us some control over the
starting and ending directions, which allow us to
distribute the curves evenly on the surface in an
aesthetically pleasing fashion.
Fig 6c. After mesh split and coloring
Split and Color Once all nodes of the graph are
connected with LVCs on the surface, we split the
surface into regions that are bounded by the
edges of the graph, and color these with
procedurally assigned colors. To obtain an
arbitrary number of hues, smoothly distributed
over the full visible spectrum, we use cyclic
sampling around the color wheel inspired by the
phyllotaxis process that places the individual
blossoms in a composite flower (Fig 6a,b,c).
These colors can then be reassigned by the user
to different schemes, possibly exhibiting the
symmetry of the whole object.
Fig 5a. Path refining at subdivision level 3
The algorithm exhibits rather fast convergence,
since the level of the path resolution is
inversely proportional to the amount of
refinement needed, i.e. the major correction
steps occur at low resolutions with few points to
move.
Fig 2. The genus 6 surface onto which we will map
the graph
Coarse Mesh and Connected Path To find an LVC,
we first start with a coarse polygonal path
between the two points we wish to connect on a
rough polyhedral model of the surface (Fig 3).
Such a path is easy to specify on a coarse
subdivision mesh, as the mesh, as well as the
path are defined by only a few vertices. The
surface and the path are then smoothed and
refined together.
Future Work High-quality rendering of this
object by making the surface semi-transparent and
using a ray-tracing algorithm (such as Radiance)
to give the appearance of a Tiffany lamp or
stained glass window. Produce a physical model
of the design using a rapid prototyping
machine Use the same approach to map other
complex symmetrical graphs onto surfaces of
suitable genus, in particular, the fully
connected graph of 12 vertices onto a genus 6
surface.
Fig 5b. Path refining at subdivision level 3
Fig 3. Vertices of the coarse path specified on
the surface
Fig 6a. After mesh split and coloring
Fig 6b. After mesh split and coloring