Swirling-sweepers: Constant-volume modeling

Abstract

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.

As-rigid-as-possible shape manipulation

“…Volume preservation is another feature that the current formulation cannot achieve [Angelidis et al 2004]. With volume preservation, an object is squashed vertically when it is stretched horizontally.…”

WarpCurves: A tool for explicit manipulation of implicit surfaces

“…Swirling Sweepers [1] and zero-divergence vector field deformation [36] are efficient and highly intuitive tools for direct manipulation of discrete 3D surfaces. These techniques can be applied to functional surfaces, but involve a relatively expensive path integration for each point.…”

Fast and Rigid 3D Shape Deformation Based on Moving Least Squares

“…A skeleton structure can be exploited for deforming articulated shapes [10,20]. Twisters [13] and swirl sweepers [3] enable extremely large deformations by controlling the position and orientation of the handles. Deformation methods using radial basis functions can edit the arbitrary region of the space with control points and curves [5].…”