# EXPLICIT SOLUTION FOR THE MINIMUM DISTANCE BETWEEN TWO SOLID SEMI-INFINITE CIRCULAR CONES

### Baruch E. Karlin

#### Abstract

Multi-body kinematics and object rendering often involve minimum distance calculations. Explicit solutions exist for the distance between spheres, cylinders and other simple objects. Deriving the minimum distance between cones requires numerical minimization or geometrical approximations combined with analytical solutions for the simpler objects. This paper describes an explicit solution for the minimum distance between two solid semi-infinite circular cones. The method combines geometrical reasoning with analytical derivation. The solution also includes the location of the intersection points. Solution regions are identified and discussed. A numerical method based on minimizing the distance between two cone generators was used as part of the verification process. The exact solution was compared to results of approximation by regular polytopes. The explicit solution is robust, independent of coordinate system and invariant under rigid translation and rotation of the setup.

#### References

- Gilbert, E. G., Johnson, D.W. and Keerthi, S.S., 1988, “A Fast Procedure for Computing the Distance Between Complex Objects in Three-Dimensional Space,” IEEE Journal of Robotics and Automation, Vol. 4, No. 2, pp. 193-203, April 1988.
- Jovanoski, D., 2008, “The Gilbert-Johnson-Keerthi (GJK) Algorithm,” Department of Computer Science University of Salzburg, February 2008.
- Manchem, S. and Mukund, R., 2009, “The GilbertJohnson-Keerthi Algorithm,” Indian Institute of Technology Guwahati, October 31, 2009.
- Chung, K. and Wang, W., 1996, “Quick Collision Detection of Polytopes in Virtual Environments,” ACM Symposium on Virtual Reality Software and Technology 1996, 1-4, July, 96, University of Hong Kong, Hong Kong.

#### Paper Citation

#### in Harvard Style

E. Karlin B. (2010). **EXPLICIT SOLUTION FOR THE MINIMUM DISTANCE BETWEEN TWO SOLID SEMI-INFINITE CIRCULAR CONES** . In *Proceedings of the International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2010)* ISBN 978-989-674-026-9, pages 154-159. DOI: 10.5220/0002849901540159

#### in Bibtex Style

@conference{grapp10,

author={Baruch E. Karlin},

title={EXPLICIT SOLUTION FOR THE MINIMUM DISTANCE BETWEEN TWO SOLID SEMI-INFINITE CIRCULAR CONES},

booktitle={Proceedings of the International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2010)},

year={2010},

pages={154-159},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0002849901540159},

isbn={978-989-674-026-9},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2010)

TI - EXPLICIT SOLUTION FOR THE MINIMUM DISTANCE BETWEEN TWO SOLID SEMI-INFINITE CIRCULAR CONES

SN - 978-989-674-026-9

AU - E. Karlin B.

PY - 2010

SP - 154

EP - 159

DO - 10.5220/0002849901540159