Abstract
When a moving (real or virtual) camera images a stationary object, the use of a rotation-minimizing directed frame (RMDF) to specify the camera orientation along its path yields the least apparent rotation of the image. The construction of such motions, using curves that possess rational RMDFs, is considered herein. In particular, the construction entails interpolation of initial/final camera positions and orientations, together with an initial motion direction. To achieve this, the camera path is described by a rational space curve that has a rational RMDF and interpolates the prescribed data. Numerical experiments are used to illustrate implementation of the method, and sufficient conditions on the two end frame orientations are derived, to ensure the existence of exactly one interpolant. By specifiying a sequence of discrete camera positions/orientations and an initial motion direction, the method can be used to construct general rotation-minimizing camera motions.
Similar content being viewed by others
References
Barton, M., Jüttler, B., Wang, W.: Construction of rational curves with rational rotation-minimizing frames via Möbius transformations. In: Dæhlen, M., et al. (eds.) MMCS 2008. Lecture Notes in Computer Science 5862, pp. 15–25. Springer, Berlin (2010)
Choi, H.I., Han, C.Y.: Euler–Rodrigues frames on spatial Pythagorean–hodograph curves. Comput. Aided Geom. Des. 19, 603–620 (2002)
Choi, H.I., Lee, D.S., Moon, H.P.: Clifford algebra, spin representation, and rational parameterization of curves and surfaces. Adv. Comput. Math. 17, 5–48 (2002)
Christie, M., Machap, R., Normad, J.-M., Oliver, P., Pickering, J.: Virtual camera planning: a survey. In: Butz, A., et al. (eds.) Proceedings of Smart Graphics 2005. Lecture Notes in Computer Science 3638, pp. 40–52. Springer, Berlin (2005)
Farouki, R.T.: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer, Berlin (2008)
Farouki, R.T.: Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves. Adv. Comput. Math. 33, 331–348 (2010)
Farouki, R.T., Giannelli, C.: Spatial camera orientation control by rotation-minimizing directed frames. Comput. Anim. Virtual World 20, 457–472 (2009)
Farouki, R.T., Giannelli, C., Manni, C., Sestini, A.: Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures. Comput. Aided Geom. Des. 25, 274–297 (2008)
Farouki, R.T., Giannelli, C., Manni, C., Sestini, A.: Quintic space curves with rational rotation-minimizing frames. Comput. Aided Geom. Des. 26, 580–592 (2009)
Farouki, R.T., Giannelli, C., Manni, C., Sestini, A.: Design of rational rotation-minimizing rigid body motions by Hermite interpolation. Math. Comput. (2011). doi:10.1090/S0025-5718-2011-02519-6
Farouki, R.T., Sakkalis, T.: Rational rotation-minimizing frames on polynomial space curves of arbitrary degree. J. Symb. Comput. 45, 844–856 (2010)
Farouki, R.T., Sir, Z.: Rational Pythagorean–hodograph space curves. Comput. Aided Geom. Des. 28, 75–88 (2011)
Han, C.Y.: Nonexistence of rational rotation-minimizing frames on cubic curves. Comput. Aided Geom. Des. 25, 298–304 (2008)
Nieuwenhuisen, D., Overmars, M.H.: Motion planning for camera movements. In: Proceedings, IEEE International Conference on Robotics & Automation, New Orleans, LA, pp. 3870–3876 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Helmut Pottmann.
Rights and permissions
About this article
Cite this article
Farouki, R.T., Giannelli, C. & Sestini, A. An interpolation scheme for designing rational rotation-minimizing camera motions. Adv Comput Math 38, 63–82 (2013). https://doi.org/10.1007/s10444-011-9226-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-011-9226-z