Subdivision Surfaces
Loop Subdivision
Algorithm Overview
- subdivision by 1-to-4 face split, topological operation update position of old (even) and new (odd) vertices, the smoothing step
The Subdivision Algorithm
The position of the even vertices is obtained as follows for regular vertices, left image and for irregular vertices, right image:
The weighting coefficient \(\beta\) for smoothing irregular vertices is given by: \[ \begin{align} &\text{Loop's formula}\\ &\beta = \frac{1}{n}\left(\frac{5}{8}-(\frac{3}{8}+\frac{1}{4}\cos \frac{2\pi}{n})^2 \right) \end{align} \] or \[ \begin{align} &\text{Warren's formula}\\ &\beta = \begin{cases} \frac{3}{8n} & \quad n > 3\\ \frac{5}{16} & \quad n = 3 \end{cases} \end{align} \]
The image below explains the smoothing step for odd vertices.
Smoothing boundary even and odd vertices works as follows for odd vertices (left) and even vertices (right):
Implementation Details
Loop subdivison is commonly performed in two steps:
- topological subdivision: 1-to-4 face split
- vertex smoothing: repositioning of even (old) and odd (new) vertices
After the topological subdivision the odd vertices have an initial position. Odd vertices are obtain by splitting an edge at the midpoint, that is the initial position of an edge vertex is the average of the position of the endpoints of the edge. This initial position is used to smooth odd vertices for convenience as follows:
-
compute the onering of the odd vertex in the subdivided mesh.
In the image below this are the vertices \(v_0, \dots , v_5\)
- label even vertices as 'vertex' and odd vertices as 'edge'
- smooth odd vertices as follows: \(v_{new} = \frac{2}{8}(v_{v0} + v_{v1}) + \frac{1}{8}(v_{e0}+v_{e2}+v_{e2}+v_{e3})\) where \(v_{vi}\) are the even vertices and \(v_{ei}\) are the odd vertices with the positions before smoothing. The image below shows the corresponding vertices.
- Remark: is much simpler to compute the one-ring of the odd vertex in the subdivided mesh than collecting the corresponding even vertices from the input mesh for smoothing.
