Gaussian Curvature
To approximate Gaussian curvature at a vertex, we take the sum of the angles
incident to the vertex, and subtract this sum from 360 degrees. If the vertex
is in the bottom of a bowl shape or on top of a dome shape, the incident
angles will have a sum less than 360 degrees, resulting in positive Gaussian
curvature, and if the vertex is a saddle point, the incident angles will have
a sum greater than 360 degrees, resulting in negative Gaussian curvature.
Gaussian Curvature (Voronoi)
We use the above expression for AMixed in the below equation
and other approximations for curvature. AVoronoi for a given
triangle adjacent to the vertex is equal to the area of the quadrilateral
defined by the sides of the triangle and the circumcenter point if the triangle
is non-obtuse. If the triangle is obtuse, the midpoint of the edge opposite the
obtuse angle is used instead of the circumcenter.
Mean Curvature
To approximate mean curvature at a vertex v, we use the formula from the paper
by Hsu, Kusner, and Sullivan:
Mean Curvature (Voronoi)
Principal Curvatures
"Overall" Curvature
To measure "overall" curvature, we consider a combination of the Gaussian
curvature and the mean curvature. Some possible combinations:
(G = Gaussian curvature, H = mean curvature, k1 = maximum principal curvature,
k2 = minimum principal curvature, so G = k1*k2, H = (k1+k2)/2):
- (G+H)/2
Although this is easy to calculate, the units are not the same (G has units of
curvature squared, while H has units of curvature).
- (sqrt(G) + H)/2
This solves the unit problem; however, this will not work with saddle points
(G < 0).
- (G + H^2)/2
This also solves the unit problem; however, this value is the same if k1 is
replaced by -k1 and k2 is replaced by -k2, as squaring H loses the sign
information. This will cause "bulges" and "bowls" to have the same "overall"
curvature value.
-
Using the discrete curvature approximations from above, we must consider the
problem of finding a suitable functional whose minimization results in fairing
of a surface. To do this, we consider several "penalty" functions that measure
the difference between a vertex's curvature measure and its neighbors'. The
basic formula is to take the absolute value of the difference between a
vertex's curvature measure and its neighbors', but for this we must find a
suitable weighting scheme for the neighboring vertices' curvature values:
Equal Weights
In this scheme, to calculate the curvature "penalty" at a vertex v, we take
the simple average of the curvature measures of the neighbors of v and compare
the average with the curvature measure at v; the "penalty" is simply the
absolute value of the difference. Though this scheme is very simple, this
ignores the relative distances of the neighbors.
Weight by Distance
In this scheme, instead of taking the simple average, we give each neighbor of
v a weight inversely proportional to the length of the edge connecting v to the
neighbor, and normalize the weights so they add up to 1. The "penalty" is then
the absolute value of the weighted average and the curvature measure at v. This
scheme takes into account the relative distances of the neighbors, giving more
weight to neighbors that are closer to v. However, if one edge is nearly 0,
this will cause the weight of that vertex to be almost 1, causing the others to
have almost zero weight, which will obscure the effect they should have on the
curvature at v.
Weight by Distance, Gaussian
This scheme is the same as above, except that instead of giving each neighbor a
weight inversely proportional to the distance from the vertex, we fit the
weights to a Gaussian distribution. The distance from the vertex v to a
neighbor determines the x-coordinate of the weight for that vertex on the
Gaussian distribution; the curve is shifted so that the weights sum to 1. This
has the advantage of giving more weight to neighbors that are closer to v as in
the above scheme, but it is less likely to give weights extremely close to 1 to
any vertex. However, because the Gaussian distribution is defined by an
exponential function, it becomes difficult to find the scale factor to make the
weights sum to 1.
Weight by Angles
In this scheme, we give each neighbor of v a weight that is proportional to the
sum of the angle made at v by the edges to the two vertices on either side of
the neighbor and the angle made at the neighbor in the polygon connecting the
vertices adjacent to v. (I'll put a picture here which is much less confusing.)
This scheme takes into account the relative distances to each neighbor in
addition to their positions relative to each other, so that vertices that are
hidden behind others are given a smaller weight.
Given a suitable "penalty" function, we wish to move the vertices on a mesh in
such a way as to decrease the "penalty" at each vertex.