Ball-morph: definition, implementation, and comparative evaluation.

Area of Science:

• Computer Graphics
• Computational Geometry
• Differential Geometry

Background:

• Curve morphing is essential in computer graphics and geometric modeling.
• Existing morphing techniques often lack symmetry or introduce significant distortion.
• The concept of b-compatibility for curves is introduced to ensure well-behaved morphs.

Purpose of the Study:

• To define b-compatibility for planar curves.
• To propose and construct three novel symmetric ball morphing techniques.
• To compare these novel morphs against existing methods using various cost metrics.

Main Methods:

• Definition of b-compatibility for planar curves.
• Development of three ball morphing techniques based on automatic ball-map correspondence.
• Derivation of linear, circular, and parabolic vertex trajectories.
• Comparative analysis using six cost measures: travel-distance, distortion, stretch, local acceleration, average squared mean curvature, and maximum squared mean curvature.

Main Results:

• The proposed ball morphs are symmetric and meet input curves at a right angle (circular and parabolic).
• The linear ball-morph demonstrates the shortest travel-distance across tested scenarios.
• The circular ball-morph exhibits the least distortion among the evaluated methods.
• Performance comparison highlights the dependence of results on specific input curve characteristics.

Conclusions:

• The proposed ball morphing techniques offer a symmetric and controllable approach to curve interpolation.
• The linear and circular ball-morphs provide distinct advantages in minimizing travel-distance and distortion, respectively.
• The choice of morphing technique should consider the specific application requirements and input curve properties.