Geodesic Polar Coordinates on Polygonal Meshes

Abstract

Geodesic Polar Coordinates (GPCs) on a smooth surface S are local surface coordinates that relates a surface point to a planar parameter point by the length and direction of a corresponding geodesic curve onS. They are intrinsic to the surface and represent a natural local parameterization with useful properties. We present a simple and efficient algorithm to approximate GPCs on both triangle and general polygonal meshes. Our approach, named DGPC, is based on extending an existing algorithm for computing geodesic distance. We compare our approach with previous methods, both with respect to efficiency, accuracy and visual qualities when used for local mesh texturing. As a further application we show how the resulting coordinates can be used for vector space methods like local remeshing at interactive frame‐rates even for large meshes.

Interactive shape modeling using a skeleton-mesh co-representation

“…When a sufficient number of smoothing iterations have been performed, we convert the vertex positions back to 3D. A similar smoothing operation is also discussed in [Melvaer and Reimers 2012]. The main difference is that we do not convert vertex positions back into 3D between iterations.…”

“…We initially project a PAM vertex, v onto T by finding its closest face and the barycentric coordinates of v in this face. Subsequently, for each PAM vertex, v we create a local uv map using the geodesic polar coordinates method of [Melvaer and Reimers 2012] and compute the positions of the neighbors of v. We smooth the position of v in uv coordinates and update its face id and barycentric coordinates to reflect the new position. When a sufficient number of smoothing iterations have been performed, we convert the vertex positions back to 3D.…”

Weighted linde-buzo-gray stippling

“…To approximate the centroid of each cell, we use the method proposed by Wang et al [2015] and iteratively move a base point to the centroid. In each iteration, we first project the cell to the planar tangent region of the current base point using geodesic polar coordinates [Melvaer and Reimers 2012], and then update the point to the centroid of the projection. Once we found the centroid, we project the cell to the planar region for new splitting points.…”

Stroke Parameterization

“…Many works have considered the problem of embedding a disc-shaped region of a surface in the plane [FH05, SPR06], while more recently there has been interest in application-specific parameterizations such as the integral-coordinate embeddings used in quadrangulation [BZK09] and decal parameterizations used for local texture mapping [LHN05,SGW06,SGW09,CK11,MR12].…”

“…Although various techniques are available to approximate the geodesic distance from S [CK11,CWW12], back-tracing through this distance field for each surface point x we wish to parameterize is prohibitively expensive [MR12]. Instead we propose a single-pass forward propagation to directly estimate the map P(x) = (tx, dx) (to simplify exposition we will now use dx as a shorthand for s(tx)dg(x, S(tx)) ).…”