Swirling-sweepers is a new method for modeling shapes while preserving volume. The artist describes a deformation by dragging a point along a path. The method is independent of the geometric representation of the shape. It preserves volume and avoids self-intersections, both local and global. It is capable of unlimited stretching and the deformation can be controlled to affect only a part of the model.
MotivationIn a virtual modeling context, there is no material. A challenge for computer graphics is to provide a modeling tool that convinces the artist that there is a material. Volume is one of the most important factors influencing the way an artist models with real materials.The limitation of existing volume-preserving methods is either that they only apply to a specific type of geometric representation, or they only apply to shapes whose volume can be computed, with the exception of [Decaudin 1996]. His technique does not always preserve volume, and is discontinuous at one point.
Principle of Swirling-SweepersA Swirling-Sweeper is a new space deformation based on our framework called Sweepers [Angelidis et al. 2004b]. It is a blend of simpler deformations that we call swirls. In Figure 1, we show that a swirl is a rotation whose magnitude decreases away from its center, c. We represent the magnitude of rotation by a C 2 monotonic scalar function, φ, which vanishes outside a neighborhood of radius λ around c. More formally, a swirl is a rotation matrix raised to the power φ f (p) = exp (φ(||p − c||) log M ) p (1) A swirl preserves volume since the determinant of its Jacobian is always equal to 1. There is a convenient way of blending n swirls to produce a more complex deformation From Swirls to Swirling-Sweepers: By specifying a single translation t, an artist can input n swirls. As shown in Figure 2, we place n swirl centers, ci, on the circle of center h, and radius r lying in a plane perpendicular to t. These points correspond to n consistently-oriented unit tangent vectors vi. Each pair, (ci, vi), together with an angle, θ, define a rotation. Along with radii of neighborhood λ = 2r, we define n swirls. The radius of the circle, r, is left to the user to choose. The following value for θ will transform h exactly into h + t, and preserves volume for sufficiently small t: Preserving coherency and volume: To preserve coherency and volume, it is necessary to subdivide input vector t into a series of smaller vectors. We use s = max(1, −4|| t ||/r ) sub-vectors. This decomposition is shown in Figure 3.Achieving Real-Time: We have a closed-form for the logarithm of a rotation matrix and also for computing (exp N )p, when N is the logarithm of an unknown rotation matrix.
