Norberg and Rayner (1987) considered an inverse relationship between maneuverability and agility, whereby an animal can turn tightly at a low speed or make a wide turn at higher speeds. Maneuverability is the capability to turn in a confined space and is measured as the length-specific radius of the turn trajectory ( R/ L, where R is the radius of the turn and L is total body length) agility is the rate of turn measured as the angular velocity (ω) ( Norberg and Rayner, 1987 Webb, 1994 Walker, 2000). Turning performance is assessed by measurements of maneuverability and agility. Such turning maneuvers fall within the range of performance exhibited by swimmers with rigid bodies. Data for 30 sequences of rays performing slow, steady turns showed the highest 20% of values for the turning rate and smallest 20% of turn radii were 42.65☑6.66 deg s −1 and 2.05☑.26 m, respectively. These are flexible models that minimize a cost function with a parameter controlling the balance between data fidelity and regularity of the derivative. Smoothing splines were applied to these turns. Principal component analysis was used to project the three-dimensional trajectory onto the two-dimensional turn. A mathematical analysis was performed on the coordinate data to calculate the turning rate and curvature (1/turning radius) as a function of time by numerically estimating the derivative of manta trajectories through three-dimensional space. Movements of the cephalic lobes, eye and tail base were tracked to obtain three-dimensional coordinates. To measure turning performance in a three-dimensional space for the manta ray ( Mobula birostris), a large open-water swimmer, scaled stereo video recordings were collected. For aquatic animals, turning maneuvers represent a locomotor activity that may not be confined to a single coordinate plane, making analysis difficult, particularly in the field.
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