Plane curves of minimal energy
ACM Transactions on Mathematical Software (TOMS)
On minimal energy trajectories
Computer Vision, Graphics, and Image Processing
IBM Journal of Research and Development
Elastica and minimal-energy splines
Curves and surfaces
The conformal map z→z2 of the hodograph plane
Computer Aided Geometric Design
Rational curves and surfaces with rational offsets
Computer Aided Geometric Design
On the Generation of Trajectories for Multiple UAVs in Environments with Obstacles
Journal of Intelligent and Robotic Systems
On the generation of feasible paths for aerial robots with limited climb angle
ICRA'09 Proceedings of the 2009 IEEE international conference on Robotics and Automation
A path planning algorithm for UAVs with limited climb angle
IROS'09 Proceedings of the 2009 IEEE/RSJ international conference on Intelligent robots and systems
On the generation of feasible paths for aerial robots in environments with obstacles
IROS'09 Proceedings of the 2009 IEEE/RSJ international conference on Intelligent robots and systems
Feasible UAV path planning using genetic algorithms and Bézier curves
SBIA'10 Proceedings of the 20th Brazilian conference on Advances in artificial intelligence
Rational Pythagorean-hodograph space curves
Computer Aided Geometric Design
C1 Hermite interpolation with PH curves by boundary data modification
Journal of Computational and Applied Mathematics
Geometric constraints on quadratic Bézier curves using minimal length and energy
Journal of Computational and Applied Mathematics
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Pythagorean-hodograph (PH) curves admit closed-form expressions for the integral of the square of the curvature with respect to arc length (the ''energy'' integral) involving only rational functions, arctangents, and natural logarithms. In particular, the complex formulation of PH curves greatly facilitates the derivation of these expressions, yielding compact and efficient implementations in any high-level language that provides complex arithmetic. Explicit formulae are presented for the case of Tschirnhausen's cubic and the regular PH quintics, and in the latter case the use of the energy integral in optimizing the ''fairness'' of geometric Hermite interpolants is discussed. Compelling empirical evidence indicates that, for ''reasonable'' derivative data, first-order PH quintic Hermite interpolants are systematically of lower energy than their ordinary cubic counterparts.