Log-Aesthetic Curves for Shape Completion Problem

Algorithm 1

REMARK: Endpoints , inflection point and are user defined. The tangent angle at both endpoints can be

obtained either from three point circular arc approximation around the endpoints or the user defines it. Let represent the

coordinate of a point.

INPUT: , , , .

OUTPUT: , , , LAC segment.

Begin

Step  1. Set , where is the center of circle, is radius and represents angle.

Step  2. Identify unit tangent vector, and set .

Step  3. Calculate directional angle, .

Step  4. Solve simultaneous equations , where to get position of second control point .

Step  5. If , then ,

            else ; .

            If , then ; ,

              else ; .

Step  6. Translate , rotate triangle such that for ; for .

   If or , then reflect the triangle through -axis.

Step  7. Compute .

Step  8. If , then ,

          else .

Step  9. Identify bound of . If , then ; ,

               else if , then ; ,

                  else ; .

Step  10. Set tolerance ;

      iteration number .

Step  11. Calculate using Bisection method.

Step  12. Determine scaling factor, .

Step  13. Scale to the curve and transform inversely the triangle to its original position.

Step  14. Construct LAC segment using (2).

Step  15. OUTPUT (, , , LAC segment).

End