Quadrilaterals Research Articles

Pigeon Pattern Perception: Limits on Perspective Invariance

Two experiments are reported in which the response of pigeons to perspective transformations of a pattern target was measured. In the first experiment an alphabet letter was taken as the pattern target (the 'positive') and the response to its perspective transforms was compared to the response to nontarget letters ('negatives') over the course of discrimination training. In the second experiment irregular quadrilaterals were used as positives and the responses to slight perspective deformations of the prototypes were compared to the responses to random transformations of the same magnitude under steady-state conditions. The amount of differential responding depended on the type of transformation. There was no differential response to targets rotated in the picture plane or around the horizontal axis in either experiment. There was differential response to small reductions and enlargements of the target. There was also differential response to translated targets, and this was seen irrespective of the amount of displacement. Results from targets rotated around the vertical axis were erratic, some target-plus-angle combinations elicited differential responding, but most did not. The erratic responses are attributed to symmetries in pattern elements that were abstracted as critical features. Pigeons therefore exhibit no true rotation invariance, limited size invariance, and complete shift invariance. It is argued that size invariance, but not position invariance, may depend on prior exposure to the alternatives.

Infant perception of the shapes of tilted plane forms

This paper describes three experiments based on Caron, Caron, and Carlson (1979) . In each case, 3-month-old infants were familiarized with a plane form presented at various tilts. In three subsequent test trials they saw the familiar form in a familiar (60°) orientation, the familiar form in a novel frontoparallel orientation (novel-orientation test), and a novel form in frontoparallel (novel-form test). The novel form was the frontoparallel projection of the familiar form when tilted at 60°. In Experiment 1, recovery of fixation occurred in the novel-form test when the infants were familiarized with a rectangle and the novel form was a trapezoid. No recovery occurred when both the familiar and novel forms were irregular quadrilaterals. The results for the rectangle were interpreted as evidence of shape constancy. The results for the irregular forms were attributed to a poor resolution of these forms. In Experiment 2, the infants were familiarized with a rectangle as in Experiment 1. However, the novel-orientation test stimulus was reduced in size so that its area equalled that of the trapezoid seen in the novel-form test (novel-size test). While a recovery of fixation occurred in the novel-form test, no recovery occurred to the reduction of size in the novel-size test. This was taken as evidence that the rectangle/trapezoid discrimination was based on the perception of form rather than size. In Experiment 3, the procedure of Experiment 1 was repeated but the infants were familiarized with either a parallelogram or a trapezoid. In neither case was there a recovery in the novel-form test. It was concluded that the rectangle/trapezoid discrimination was based upon a perception of orthogonality.

The proportion of quadrilaterals formed by random lines in a plane

An expression is derived for the proportion of quadrilaterals among the polygons formed by random lines in the plane. Its value is 0.381 466….

The proportion of quadrilaterals formed by random lines in a plane

An expression is derived for the proportion of quadrilaterals among the polygons formed by random lines in the plane. Its value is 0.381 466….

Minimum Conditions for Congruence Of Quadrilaterals

Minimum Conditions for Congruence Of Quadrilaterals

Activities: Midpoints and Quadrilaterals

Activities: Midpoints and Quadrilaterals

Rational Quadrilaterals Second Communication

Rational Quadrilaterals Second Communication

Rational quadrilaterals

Rational quadrilaterals

Polygonal Aspects of Cell Faces. II. Quadrilaterals as the Prevailing Type

Polygonal Aspects of Cell Faces. II. Quadrilaterals as the Prevailing Type

PRECISE TELLUROMETER TRAVERSING

35. The transmission of azimuth along a traverse is likely to be weaker than along a chain of triangulation. For example, a chain of braced quadrilaterals is loosely equivalent to four simultaneous traverses, one running up each side and two zigzagging across the middle. So we may expect the p.e. in azimuth after a line of traverse to be about twice the p.e. after one figure of a chain.

The Quadrilaterals of Pascal's Hexagram

The Quadrilaterals of Pascal's Hexagram

The Quadrilaterals of Pascal's Hexagram

The Quadrilaterals of Pascal's Hexagram

Quadrilaterals inscribed and circumscribed to a plane cubic

Quadrilaterals inscribed and circumscribed to a plane cubic

III.Placing plane quadrilaterals in perspective. Application to photography

III.<i>Placing plane quadrilaterals in perspective. Application to photography</i>

Triangles and Quadrilaterals Inscribed to a Cubic and Circumscribed to a Conic

Triangles and Quadrilaterals Inscribed to a Cubic and Circumscribed to a Conic