Efficient C2 Continuous Surface Creation Technique Based on Ordinary Differential Equation

Abstract

In order to reduce the data size and simplify the process of creating characters' 3D models, a new and interactive ordinary differential equation (ODE)-based C2 continuous surface creation algorithm is introduced in this paper. With this approach, the creation of a three-dimensional surface is transformed into generating two boundary curves plus four control curves and solving a vector-valued sixth order ordinary differential equation subjected to boundary constraints consisting of boundary curves, and first and second partial derivatives at the boundary curves. Unlike the existing patch modeling approaches which require tedious and time-consuming manual operations to stitch two separate patches together to achieve continuity between two stitched patches, the proposed technique maintains the C2 continuity between adjacent surface patches naturally, which avoids manual stitching operations. Besides, compared with polygon surface modeling, our ODE C2 surface creation method can significantly reduce and compress the data size, deform the surface easily by simply changing the first and second partial derivatives, and shape control parameters instead of manipulating loads of polygon points.

Character Modelling with Sketches and ODE-Based Shape Creation

“…A fourth-order ordinary differential equation was introduced in [47] to develop an analytical solution, which is combined with C 1 continuous boundary constraints to develop C 1 continuous sweeping surfaces. The work was extended to create C 2 continuous sweeping surfaces by using C 2 continuous boundary constraints to determine the unknown constants involved in the analytical solution to a sixth-order ordinary differential equation [7]. Here, C 1 and C 2 continuous boundary constraints mean the first and second partial derivatives at the interface between two adjacent sweeping surfaces are identical, respectively.…”

PDE Surface-Represented Facial Blendshapes

“…Differential equation-based modelling can be divided into Ordinary Differential Equation (ODE)-based geometric modelling [19,20] and Partial Differential Equation (PDE)based geometric modelling [21]. ODE-based modelling is used in C2 continuous surface creation [22] and dynamic skin deformations [23,24]. Since ODE modelling involves one parametric variable, how to manipulate surfaces in two-dimensional regions requires further investigations.…”

Efficient and Physics-based Facial Blendshapes based on ODE sweeping Surface and Newton’s second law

“…MODELS Creation of ODE sweeping surface-represented 3D models is inspired by [14]. For completeness, this section briefly introduces the theory and method, including the mathematical model in Subsection A, accurate closed form solution in Subsection B, continuities between two adjacent patches in Subsection C, and the creation process of ODE sweeping surface-represented 3D models in Subsection D.…”

“…When curvature continuity is not required, curvature can be used to manipulate surfaces efficiently. Based on these considerations, this paper uses the following vector-valued sixth-order ordinary differential equation proposed in [14]:…”

“…1, we can draw two control curves next to each of the boundary curves. From the boundary curve and two control curves, we can use the method given in [14] to obtain 𝒄 2 (𝑣), 𝒄 5 (𝑣), 𝒄 3 (𝑣) and 𝒄 6 (𝑣).…”