Animals as well as humans adjust their gait patterns in order to minimize energy required for their locomotion. A particularly important factor is the constant force of earth’s gravity. In many dynamic systems, gravity defines a relation between temporal and spatial parameters. The stride frequency of an animal that moves efficiently in terms of energy consumption depends on its size. In two psychophysical experiments, we investigated whether human observers can employ this relation in order to retrieve size information from point-light displays of dogs moving with varying stride frequencies across the screen. In Experiment 1, observers had to adjust the apparent size of a walking point-light dog by placing it at different depths in a three-dimensional depiction of a complex landscape. In Experiment 2, the size of the dog could be adjusted directly. Results show that displays with high stride frequencies are perceived to be smaller than displays with low stride frequencies and that this correlation perfectly reflects the predicted inverse quadratic relation between stride frequency and size. We conclude that biological motion can serve as a cue to retrieve the size of an animal and, therefore, to scale the visual environment.

*k*, however, is not as easily obtainable.

*l*and the period

*T*of an ideal pendulum is with

*g*being gravitational acceleration. In order to obey this relation, smaller animals have to move with a higher stride frequency

*f*=

*1/T*than larger animals.

*mv*

^{2}/

*L*, where

*m*is body mass,

*L*is leg length, and

*v*is forward speed (Kram, Domingo, & Ferris, 1997). The ratio between the centripetal force and the gravitational force (

*mv*

^{2}/

*L*)/

*mg*=

*v*

^{2}/

*gL*is the dimensionless Froude number (Alexander, 1989). Therefore, if animals travel with equal Froude number, their speeds

*v*are proportional to the square root of the leg length

*L*. If they move in dynamically similar fashion (Alexander & Jayes, 1983), the stride length

*l*is proportional to the leg length and hence the stride frequency

*f*=

*v/l*is inversely proportional to the square root of the leg length. Pennycuick (1975) measured the stride frequencies of African mammals moving spontaneously in their natural habitat and found that they are in fact inversely proportional to the square root of the stride length to a very good approximation. Thus, the findings show that the relation between spatial and temporal scales expressed in Equation 1 is also reflected in the locomotion patterns of animals.

*f*of an animal and its estimated size

*s*

_{dyn}is where

*c*

_{1}is a constant factor quantifying the spatio-temporal scaling relation. The absolute value of

*c*

_{1}depends on gravitational acceleration and on the gait pattern (e.g., trotting, cantering, etc.).

*s*

_{dyn}and

*s*

_{stat}exist simultaneously and both may contribute to a size estimate. Here, we assume linear integration, and we introduce a factor

*λ*accounting for the relative weight of the two terms: .

^{2}and was displayed in a bright green coloring. An additional set of 20 black dots represented the shadows of the dots depicting the dog’s body. Adding a shadow ensures that observers perceive the animal’s legs to have contact to the ground. The point-light display had a size of 4 cm on the screen corresponding to 4 deg of visual angle at the viewing distance of 58 cm. This distance was fixed by using a wooden chinrest. The image sizes of the point-light displays were held constant across all trials.

*p*< .001). On average across all participants, animated dogs moving with high stride frequency were perceived to be smaller than dogs moving with low stride frequency (Figure 2). This outcome confirms the hypothesis that observers retrieve size information from the stride frequency that animals use for locomotion. Recall that the instructions did not explicitly draw observers’ attention to the stride frequency of the animated animals. According to the instructions, observers were requested to adjust the position so that the scene looked as natural as possible. Therefore, observers seem to use implicit knowledge to make their size judgments.

*k*

_{1}= 141 and

*k*

_{2}= 35. With these values, the Equation 6 correlates with

*r*

^{2}= 0.96 to the means of estimated sizes across all observers. Only 4% of the variance of the data remains unexplained. A linear fit, on the other hand, correlates to the empirical data with

*r*

^{2}= 0.88, therefore leaving 12% of the variance unexplained.

*k*

_{1}(Figure 3).

*k*

_{1}.

Participant | k_{1} | k_{2} | r^{2} |
---|---|---|---|

F.N. | 308 | 27 | .38** |

K.S. | 358 | 23 | .62** |

L.J. | 47 | 56 | .00 |

C.O. | 324 | 14 | .75** |

Z.K. | 155 | 23 | .45** |

M.H. | 286 | 39 | .17** |

I.L. | 8 | 59 | .00 |

L.M. | 341 | 16 | .66** |

H.B. | −26 | 48 | .05 |

T.R. | 93 | 37 | .05 |

T.B. | −111 | 60 | .12** |

M.V. | 76 | 37 | .08 |

J.B. | 104 | 36 | .11* |

R.R. | −11 | 31 | .01 |

A.S. | 215 | 17 | .61** |

S.B. | 93 | 44 | .04 |

Parameters of the theoretical model (Equation 6) fitted to the data of individual participants. *r*^{2} = coefficient of determination. **p* < .05; ***p* < .01.

*λ*providing information about the individual weights of both sources of information (static vs. dynamic) nor the constants

*c*

_{1}and

*c*

_{2}can be calculated directly, because

*λ*is confounded with the constant scaling factors

*c*

_{1}and

*c*

_{2}(Equation 5). The constant

*k*

_{1}combining

*λ*and

*c*

_{1}only weakly reflects the tendency to what extent the temporal scaling relation is considered.

*c*

_{2}in Equation 5. In combination with measurements

*k*

_{1}and

*k*

_{1}obtained from the first part of Experiment 2, this was used to derive values for

*λ*and

*c*

_{1}. By this procedure, we are able to separate size information from static and dynamic sources and to calculate how the sources of information are integrated.

*p*< .001).

*r*

^{2}= 0.98. A linear fit correlates to the model with

*r*

^{2}= 0.94. Comparing the proposed model fit with a linear fit, the proposed model leaves only 2% of the variance unexplained, whereas the linear fit leaves 6% of the variance unexplained.

*c*

_{2}, representing size information independent of any temporal scaling cue. On average across all observers,

*c*

_{2}assumes a value of 61.47 cm. The standard deviation of 13.90 cm is relatively small, indicating a generally uniform behavior in this subtask. Individual measures for

*c*

_{2}were used to determine the weight factor

*λ*= 1 −

*k*

_{2}/

*c*

_{2}and the spatio-temporal scaling factor

*c*

_{1}=

*k*

_{1}*

*c*

_{2}/(

*c*

_{2}−

*k*

_{2}) for each observer, according to Equation 5 (Table 2).

Participant | k_{1} | k_{2} | c_{1} | c_{2} | λ | r^{2} |
---|---|---|---|---|---|---|

A.C. | 491 | 20 | 732.84 | 60.00 | .67 | .71** |

J.A. | 186 | 30 | 413.33 | 55.00 | .45 | .25** |

H.O. | 219 | 38 | 521.43 | 65.43 | .42 | .29** |

U.A. | 204 | 34 | 340.02 | 84.82 | .60 | .24** |

A.A. | −13 | 69 | 86.67 | 60.00 | −.15 | .00 |

S.I. | 68 | 53 | 566.67 | 60.00 | .12 | .02 |

J.N. | 143 | 49 | 572.01 | 65.43 | .25 | .08 |

N.K. | 427 | 12 | 514.46 | 71.34 | .83 | .81** |

C.N. | 184 | 33 | 408.89 | 60.00 | .45 | .52** |

D.M. | −5 | 70 | −20.83 | 92.50 | .24 | .00 |

P.P. | 97 | 33 | 440.91 | 42.41 | .22 | .18** |

M.H. | −27 | 43 | 128.57 | 35.67 | −.21 | .06 |

A.G. | 205 | 34 | 427.03 | 65.43 | .48 | .26** |

C.K. | 180 | 29 | 382.98 | 55.00 | .47 | .36** |

M.K. | 283 | 21 | 435.38 | 60.00 | .65 | .45** |

J.C. | 373 | 33 | 1065.71 | 50.43 | .35 | .18** |

Characteristics of the theoretical model (Equation 5) fitted to the data of individual participants. Note: *k*_{1} = *λ c*_{1}; *k*_{2} = (1−λ)*c*_{2}. *c*_{2} was derived from the median of the size estimations per observer given in the static stick-figure trials. *r*^{2} = coefficient of determination. **p* < .05; ***p* < .01.

*p*< .01), there was no correlation at all for the remaining 5 observers (

*p*> .05). Showing very flat curves, these observers did not seem to pay any attention to the different stride frequencies. Their response patterns seemed to be completely ignorant with respect to the independent variable (i.e., the stride frequency). Two observers (J.N. and D.M.) also showed very large variances across similar stimulus repetitions, which indicates that they responded in a disoriented manner. Observers from this group also gave the largest and smallest values for the size of the statically displayed dog. Consequently, for some of them, very low (and in two cases even negative) values for

*λ*are obtained.

*k*

_{1}when compared to Experiment 1, in which we had attempted to provide a method for transforming observers’ size impression into a corresponding response while maintaining a constant retinal size of the stimulus.

*c*

_{1}in Equation 3. Summarizing the results of Experiment 2, we compute

*c*

_{1}as the median of the 11 observers that did respond in a consistent manner. The resulting value amounts to 435 cm s

^{−2}.

*c*

_{1}to amount to 410 cm s

^{−2}for cantering animals. This value is very close to the one obtained from our data.

*c*

_{1}as derived from our experiments. For instance, the perceived height of the reference objects in the scenery may deviate from their “real” height. The posts were intended to have a height of 1 m and the cactuses a height of 2 m. Those numbers were given to the observers in their introduction to the experiment. However, the reference objects may still have been perceived to be larger or smaller, changing the reference frame used to indicate the dog’s size. Another critical point is the determination of the constant

*c*

_{2}in Equation 5. In the second subtask of Experiment 2, we tried to measure the perceived size as given by cues that are independent from stride frequency. We did that by asking the observers to estimate the size of a static stick-figure display. However, this procedure may not be sufficient to accurately derive the desired information. It is still possible that a moving dog does provide cues about its size, which are not available in the static display but which are still not depending on the stride frequency. A last factor that adds uncertainty is the fact that living animals, even if they try to minimize energy consumption during locomotion, are still different from inanimate dynamic systems. In a swinging pendulum or a bouncing ball, the relation between temporal and spatial parameters is exactly defined by gravity, because no other forces affect these motions. In contrast, in dynamic animate systems, muscular forces controlled by intentional behavior play an important role. They are not used only to simply compensate for damping effects in the articulated pendulum system of the body; they can also be used to significantly alter the motion pattern to cover a wider range of stride frequencies within a given gait pattern.

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