1. Visual Estimation of Three- and Four-body Center of Mass Jay Friedenberg Psychology Department Manhattan College Jay Friedenberg Psychology Department Manhattan College Bruce Liby Physics Department Manhattan College Bruce Liby Physics Department Manhattan College Acknowledgement: Damien Germino Contact: Jay.Friedenberg@Manhattan.Edu

2. Introduction A center of mass (COM) is the point about which the bulk of an object or objects are evenly distributed. A center of mass (COM) is the point about which the bulk of an object or objects are evenly distributed. COM is useful in estimating location and for grasping and manipulating objects (Lukos, Ansuini, & Santello, 2007). It is also utilized during multiple-object tracking (Fehd & Seiffert, 2008). COM is useful in estimating location and for grasping and manipulating objects (Lukos, Ansuini, & Santello, 2007). It is also utilized during multiple-object tracking (Fehd & Seiffert, 2008). Image source: http://graphics.cs.cmu.edu/nsp/projects/hands/hands. html

3. Motivation Do holistic configural properties like elongation and symmetry influence center estimation in multiple object displays? Do holistic configural properties like elongation and symmetry influence center estimation in multiple object displays? COM accuracy decreases with an increase in the length of an elongation axis relative to the length of a reflective symmetry axis in 3-dot isosceles triangle patterns (Friedenberg & Liby, 2008). COM accuracy decreases with an increase in the length of an elongation axis relative to the length of a reflective symmetry axis in 3-dot isosceles triangle patterns (Friedenberg & Liby, 2008). Experiment 1: What role does elongation play by itself, in the absence of symmetry in 3-dot patterns? Experiment 1: What role does elongation play by itself, in the absence of symmetry in 3-dot patterns? Experiment 2: How do symmetry and elongation interact when both are present together in 4-dot patterns? Experiment 2: How do symmetry and elongation interact when both are present together in 4-dot patterns?

4. Experiment 1 Stimuli In this experiment we studied visual estimation of COM in 3-dot displays organized to form virtual right triangles varying in elongation. In this experiment we studied visual estimation of COM in 3-dot displays organized to form virtual right triangles varying in elongation. The equal-sized, black-filled dots were viewed against a white background on a computer screen. The equal-sized, black-filled dots were viewed against a white background on a computer screen. Only virtual lines connect the dots to form a geometric triangle. No contours other than those of the dots themselves are visible in the stimulus. Only virtual lines connect the dots to form a geometric triangle. No contours other than those of the dots themselves are visible in the stimulus.

5. Right Triangle Axis Ratio The shortest inter-dot distance (A) was fixed at 30 mm. The shortest inter-dot distance (A) was fixed at 30 mm. Distance B was varied proportional to A in whole number increments from one to five (1 - 5). Distance B was varied proportional to A in whole number increments from one to five (1 - 5). Axis ratio was the length of side B relative to side A. Axis ratio was the length of side B relative to side A.

6. Right Triangle Orientation Triangles were horizontally and vertically oriented based on the alignment of side B. Triangles were horizontally and vertically oriented based on the alignment of side B. For each orientation the hypotenuse (side C) could face one of the four screen quadrants, either Upper Right, Lower Right, Upper Left, or Lower Left. For each orientation the hypotenuse (side C) could face one of the four screen quadrants, either Upper Right, Lower Right, Upper Left, or Lower Left.

7. Procedure 19 participants were first instructed on what a COM is and given common examples. 19 participants were first instructed on what a COM is and given common examples. They used a mouse pad to move a small blip to what they perceived as the center. They used a mouse pad to move a small blip to what they perceived as the center. When satisfied with their estimate, they pushed the space bar to begin the next trial. When satisfied with their estimate, they pushed the space bar to begin the next trial.

8. Dependent Measures There were two dependent measures, each analyzed separately. There were two dependent measures, each analyzed separately. Error is the Euclidean distance between the estimate and the true COM and is an indication of overall task difficulty. Error is the Euclidean distance between the estimate and the true COM and is an indication of overall task difficulty. Response orientation is the angular deviation of the estimate from the the true COM measured in degrees clockwise from the upright. It is a measure of orientation bias. Response orientation is the angular deviation of the estimate from the the true COM measured in degrees clockwise from the upright. It is a measure of orientation bias.

9. Triangle Error Results Triangle Error Results COM estimation error increased linearly with axis ratio. COM estimation error increased linearly with axis ratio. There was no interaction with or main effect for orientation and hypotenuse direction. There was no interaction with or main effect for orientation and hypotenuse direction.

10. Triangle Orientation Results Responses point generally toward the gravitational down (180°). Responses point generally toward the gravitational down (180°). They are also “pulled” in the direction of the laterally-oriented dot, CCW ( 180º) for left-facing triangles. They are also “pulled” in the direction of the laterally-oriented dot, CCW ( 180º) for left-facing triangles.

11. Experiment 2 We next employed 4-dot patterns made to form squares and rectangles. We next employed 4-dot patterns made to form squares and rectangles. These patterns have reflective and rotational symmetries that can be used to localize the COM. These patterns have reflective and rotational symmetries that can be used to localize the COM. Accuracy should be better than in the right triangle case where there are no symmetries present. Accuracy should be better than in the right triangle case where there are no symmetries present.

12. Symmetry Squares have four axes of reflective symmetry and four-fold rotational symmetry. Squares have four axes of reflective symmetry and four-fold rotational symmetry. The intersection of the reflective axes and the center of rotation mark the COM. The intersection of the reflective axes and the center of rotation mark the COM. Rectangles have a long and a short reflective axis and two-fold rotational symmetry. Rectangles have a long and a short reflective axis and two-fold rotational symmetry. If observers use these symmetries to locate a center then estimation for squares should better. If observers use these symmetries to locate a center then estimation for squares should better.

13. Quadrilateral Axis Ratio Distance A was fixed. Distance B was varied so that it was one to five times as long as distance A, as in the first experiment. Distance A was fixed. Distance B was varied so that it was one to five times as long as distance A, as in the first experiment.

14. Square Orientations Squares were presented at two orientations (VH) and (LR). Squares were presented at two orientations (VH) and (LR). Diagonals were at ±45º. Diagonals were at ±45º.

15. Rectangle Orientations Rectangles were presented at four orientations (V, H, L, R). Rectangles were presented at four orientations (V, H, L, R).

16. Quadrilateral Error Results

17. Summary of Error Data Errors increase linearly with elongation replicating this result from Experiment 1. Errors increase linearly with elongation replicating this result from Experiment 1. Accuracy was better for squares than for rectangles suggesting the use of symmetry to estimate the center. Accuracy was better for squares than for rectangles suggesting the use of symmetry to estimate the center. For rectangles, performance for vertical and horizontal orientations is better than for diagonals as is the case in symmetry detection. For rectangles, performance for vertical and horizontal orientations is better than for diagonals as is the case in symmetry detection.

18. Differential Reinforcement For horizontal rectangles, gravity reinforces the vertical symmetry axis while elongation reinforces the horizontal axis. Accuracy is greatest in this case. For horizontal rectangles, gravity reinforces the vertical symmetry axis while elongation reinforces the horizontal axis. Accuracy is greatest in this case. For vertical rectangles gravity and elongation both reinforce the vertical axis. The horizontal axis receives no support. Accuracy is intermediate here. For vertical rectangles gravity and elongation both reinforce the vertical axis. The horizontal axis receives no support. Accuracy is intermediate here. In diagonal orientations, none of the axes are reinforced. Accuracy in these conditions is the lowest. In diagonal orientations, none of the axes are reinforced. Accuracy in these conditions is the lowest.

19. Quadrilateral Orientation Data There is again a predominant gravity effect. There is again a predominant gravity effect. Deviations from gravity are biased by nearby symmetry axes and by diagonals Deviations from gravity are biased by nearby symmetry axes and by diagonals

20. Conclusions In 3- and 4-dot patterns, center estimation accuracy decreases with elongation. In 3- and 4-dot patterns, center estimation accuracy decreases with elongation. In 4-dot patterns, accuracy improves with an increase in symmetry and follows the pattern of results seen in the symmetry detection literature. In 4-dot patterns, accuracy improves with an increase in symmetry and follows the pattern of results seen in the symmetry detection literature. This implies that observers are utilizing the symmetry in the pattern to infer the location of a center. This implies that observers are utilizing the symmetry in the pattern to infer the location of a center.

21. A Process Model Connect dots with virtual lines. Connect dots with virtual lines. Determine medians and symmetry axes. Determine medians and symmetry axes. Center is intersection of medians or axes. Center is intersection of medians or axes. Our study does not test this model. Our study does not test this model. There are other ways of determining a center that may be stimulus-specific. The visual system could adopt multiple strategies. There are other ways of determining a center that may be stimulus-specific. The visual system could adopt multiple strategies.

22. Holistic Pattern Perception Both elongation and symmetry are global holistic shape properties. They emerge from the relationship between parts. Both elongation and symmetry are global holistic shape properties. They emerge from the relationship between parts. Their influence strongly suggests that participants perceived these patterns as coherent single shapes and not simply as aggregates of individual dots. Their influence strongly suggests that participants perceived these patterns as coherent single shapes and not simply as aggregates of individual dots. This is probably a function of their geometric regularity and familiarity. This is probably a function of their geometric regularity and familiarity.

23. COM Formula The COM of a system of three objects can be determined by the following equations: Where x and y are the x- and y- coordinates of the masses from an arbitrary reference point and m are the masses of the individual bodies.