2-Dimension Deformation Based On Scale-Invariant Intrinsic Variables

This paper presents a novel representation for planar geometric graphics based on scale-invariant intrinsic variables which include the length ratio and orientation angle of adjacent edges of vertices at its first order neighborhood. The geometric graphics is globally represented by a scale-invariant intrinsic matrix. The scale-invariant intrinsic variables and matrix respectively represent the local and global intrinsic geometry characteristic of the planar geometric graphics and they are invariant to geometric transformations such as translation, rotation and scaling. The shape can be reconstructed by the scale-invariant intrinsic matrix in the least square sense, which results in a sparse linear system. Any linear constrains can be easily appended to the linear system to meet the need of users.As the above virtues the scale-invariant intrinsic variables representation has, we apply it to planar polygon, skeleton and mesh deformation, such as editing and morphing.Applying this method to planar polygon deformation, the structural geometric details can be preserved while shapes being changed. To preserve the global characteristic of polygon and enhance the deformation effect, we propose an algorithm that preserves the perceptual feature. First extract the simple feature polygon of the original polygon, deformating the feature polygon based on scale-invariant intrinsic variables method, and then add the deformed feature polygon as constrains to scale-invariant intrinsic matrix, finally solve the linear system and obtain the target polygon. The artifacts such as local shrinkage, distortion and self intersections will be avoided efficiently by this arithmetic.Applying this method to planar skeleton deformation, it improves the deformation effect as we not only consider the boundary information but also the inner information. When in skeleton morphing, the original and target skeletons only need to have corresponding sets of vertices and edges, but the uniform topology in vertices is not needed, so it is available for complex geometric graphics deformation.Triangular mesh, a good measure to denote geometric graphics, is widely used in curve and surface deformation. Applying scale-invariant intrinsic variable to planar triangular mesh deformation, combine both the boundary information and interior information together into a whole, and represented by one matrix. Preserve the similarity of the corresponding triangular meshes when the global shape was changed, so the deformed geometric graphics is similar to the original one.The algorithm is linear and competent for complex deformation in real-time.