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EXPLICIT SOLUTION FOR THE MINIMUM DISTANCE
BETWEEN TWO SOLID SEMI-INFINITE CIRCULAR CONES
Baruch E. Karlin
Rafael, Aerodynamics Department, POBox 2250, Haifa, Israel
Keywords: Minimum Distance, Semi-infinite Cone, Geometric Reasoning, Explicit Solution.
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.
1 INTRODUCTION
Multi-body kinematics and object rendering often
involve minimum distance calculations. Explicit
solutions exist for simple objects, including points,
lines, flat patches, spheres and cylinders. The
minimum distance between two circular cones can
be derived by numerical minimization or by
polyhedral approximation, combined with explicit
solutions for the simpler objects (The GJK
Algorithm: Gilbert 1988, Jovanoski 2008, Manchem
2009. Polytopes: Chung 1996).
The geometrical approximations have inherent
geometric inaccuracies and require iterative
refinements. Numerical procedures based on exact
parametric modeling require a good initial guess and
some number crunching. Nearly tangent cones
generators, steep slopes and discontinuities may
cause convergence difficulties. The computational
time of an iterative procedure may vary significantly
depending on the parameters of the problem.
This paper describes an explicit solution for the
minimum distance between two solid semi-infinite
circular cones. The method is based on geometrical
reasoning and vector algebra derivation. The
solution also includes the location of the intersection
points. Rendering and examples were implemented
with Matlab®.
2 METHOD
2.1 Scope
The cones dealt with in this paper are solid, circular,
semi-infinite, with positive generator angles smaller
than π/2. Axes referred to in this paper are the axes
of the cones. Each cone extends from apex to
infinity in the positive direction of its axis.
2.2 Nomenclature
a,b - cones’ axes
a,b - distances to intersection points on a and b
d - minimum distance vector between the cones
d - minimum distance between the cones
f,g - apex shift along a and b
p - minimum distance vector between axes
p - distance between cones' axes
r - extended minimum distance vector
r - length of extended minimum distance vector
α,β - generator angles of the cones
γ - positioning angle (between cones' axes)
γ
crit
- critical positioning angle
ω - rotational positioning angle of polytope
2.3 Statement of the Problem
Given the positioning of two known cones, the pro-
154
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, pages 154-159
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