# Differentiating between Affine and Perspective-Based Models for the Geometry of Visual Space Based on Judgments of the Interior Angles of Squares

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Visual Space as an Affine Transformation

_{1}, y

_{1}) and (x

_{2}, y

_{2}):

## 3. Linear Perspective-Based Models for Visual Space

_{Φ}–BH

_{Φ}represent the location of the walls as experienced in visual space, and the dotted lines C

_{Φ}–D

_{Φ}and so on represent perpindicular cross-sections. The walls appear to converge in depth (and the floor to raise, if we take the diagram to reprent a saggital or median plane). Visual direction is preserved. The phenomenal space is contracted with respect to physical space. This contraction is greater in the in-depth dimension than it is for planes situated perpendicular to the line of sight—that is, the phenomenal counterpart to segment EG is more compressed than is the counterpart to segment GH. The model is related to linear perspective in that physical locations are projected along lines of sight. It is a three-dimensional to (contracted) three-dimensional projection. A two-dimensional perspective projection is obtained by considering the scene as projected onto a plane that cuts the lines of sight EP, FP, etc. perpendicularly with respect to the central line of sight, PJ. In the perspective projection onto the plane, the sides of the hallway would converge, but this convergence is contained in the plane of projection and so does not recede in depth.

_{Φ}G

_{Φ}would appear at the location where physical segment EG is found. Accordingly, there would be no phenomenal contraction of the hallway amd its sides would not seem to converge (and similarly for the railway tracks). Hatfield observed that, phenomenally, contraction is a regular and pervasive feature of human visual experience.

## 4. The Present Experiments

## 5. Experiment 1

#### 5.1. Method

**Participants.**Thirty undergraduate students (28 females and 2 males ranging from 18 to 22 years old) from Wagner College participated. Participation was voluntary. The College’s Human Experimental Review Board approved the experiment. All participants gave their informed consent. They were allowed to withdraw at any time, but none did.

**Materials and experimental layout.**Five squares were laid out on the floor of a corridor using 2.54 cm wide silver duct tape. The sides of each square were 0.5 m in length (external dimensions). Participants stood just behind a line on the floor. The squares were at various distances from the participant one behind the other, with their centers aligned with the participant’s line of sight (which was perpendicular to the line on the floor). They were oriented with two sides parallel to the participant’s frontal plane and the remaining two sides parallel to the participant’s median plane (oriented in-depth relative to the observer). The center of the near edge of the first square was 0.5 m from the participant, the second was 1.5 m away, the third was 2.5 m away, the fourth was 4 m away, and the fifth was 8 m away.

**Procedure.**Each of the 30 participants performed the experiment individually. They stood just behind the line facing the five squares on the floor. Each was given the compass and told that he or she needed to adjust it in order to match the apparent size of all 20 interior angles of the squares. Half began with the nearest square, half with the most distant square. Participants made estimates working clockwise around the interior angles of each square and then went on to estimate the angles of the next square. They held the device in front of them with the wedge specifying the angle in their frontal plane. Sometimes they moved the compass slightly to the left or right of the median plane, however.

You see five squares in front of you at different distances away. Although we all know that the interior angles for each corner of the square are physically 90 degrees, they might or might not appear or “look” that size subjectively. Please adjust this compass until it matches the apparent size of each of the interior angles of the squares. Start with the bottom right angle of the nearest [or most distant] square and go clockwise around the square, and then proceed to the next square. In between each adjustment, I will take the compass from you and record your answer. Please don’t turn the compass over, since we want you to rely on your subjective impressions, not the numbers on the back. Remember, we don’t want you to report the actual physical size of the angle, but we want you to tell us how large each angle looks or appears.

#### 5.2. Results

_{p}

^{2}= 0.102. In addition, the left vs. right location variable did not interact significantly with either distance to squares or near vs. far location within a square and need not be considered further.

_{p}

^{2}= 0.13. In addition, mean angle estimates for angles near to the observer (M = 83.71) were significantly smaller on average than those more distant from the observer (M = 87.43) within the same square, F(1,29) = 10.00, p = 0.004, η

_{p}

^{2}= 0.26. On the other hand, the interaction effect between distance and near vs. far location on angle estimates was not significant, F(4,116) = 0.34, p > 0.05.

_{p}

^{2}= 0.13.

#### 5.3. Discussion

## 6. Experiment 2

#### 6.1. Method

**Participants.**The thirty participants ranged from 18 to 30 years old. Participation was voluntary, and all participants gave their informed consent.

**Materials and experimental layout.**As in Experiment 1, five squares were laid out on the floor of a corridor using 2.54 cm wide silver duct tape. The sides of each square were 0.5 m in length (external dimensions). Participants stood just behind a line on the floor. The squares were at various distances from the participant one behind the other, with their centers aligned with the participant‘s line of sight. They were oriented with two sides parallel to the participant’s frontal plane and the remaining two sides parallel to the participant’s median plane (oriented in-depth relative to the observer). The center of the near edge of the first square was 0.5 m from the participant, the second was 1.5 m away, the third was 2.5 m away, the fourth was 4 m away, and the fifth was 8 m away.

**Procedure.**Each of the 30 participants performed the experiment individually. They stood just behind the line facing the five squares on the floor. Each was given the compass and told that he or she needed to adjust it in order to match the apparent size of all 20 interior angles of the squares. Half began their estimation with the nearest square, half with the most distant square. They made estimates working clockwise around the interior angles of each square and then went on to estimate the angles of the next square. The participants held the device in front of them with the wedge specifying the angle in their frontal plane. Sometimes they moved the compass slightly to the left or right of the median plane, however.

You see five squares in front of you at different distances away. Although we all know that the interior angles for each corner of the square are physically 90 degrees, they might or might not appear or “look” that angular size subjectively. Please adjust this compass until it matches the apparent number of degrees of each of the interior angles of the squares. For each angle we will have you do two adjustments, one where the compass angle is initially zero and must be adjusted outward and one where the compass angle is initially 180° and must be adjusted inward. In between each adjustment, I will take the compass from you and record your estimate. Please don’t turn the compass over to see the numbers, since we want you to rely on your subjective impressions, not the numbers on the back. Start with the bottom right angle of the nearest [or farthest] square and go clockwise around the square, and then proceed to the next square. Remember, we don’t want you to report the actual physical number of degrees of each angle nor the area occupied by the angle, but we want you to tell us how many degrees each angle looks or appears to span.

#### 6.2. Results

_{p}

^{2}= 0.33. In addition, ascending trials produced significantly smaller mean angle estimates (M = 85.22, std. error = 0.80) than descending trials (M = 87.12, std. error = 0.70), F(1,29) = 127.05, p < 0.001, η

_{p}

^{2}= 0.81. Thus, the data show a strong error of expectation effect. In addition, there were several significant higher-order interaction effects. For example, there were significant three-way interactions between distance, left vs. right, and trial, F(4,116) = 4.04, p = 0.004, η

_{p}

^{2}= 0.12, and between near vs. far, left vs. right, and trial, F(1,29) = 7.08, p = 0.01, η

_{p}

^{2}= 0.19. These interactions do not bear on this study’s main hypotheses and so will not be discussed further.

_{p}

^{2}= 0.38. Sidak post hoc tests showed that all angle estimates differ significantly from each other as a function of distance except that angle estimates for the second square (1.5 m from the observer) do not differ significantly from those of the first (0.5 m) or third (2.5 m) squares. In addition, within the same square, mean angle estimates for angles near to the observer (M = 84.15) were significantly smaller on average than those more distant from the observer (M = 88.18), F(1,29) = 74.27, p < 0.001, η

_{p}

^{2}= 0.72. Note that this result has very large F statistics and effect sizes. Unlike the first experiment, the interaction effect between distance and near vs. far location on angle estimates was significant (using the Greenhouse-Geisser correction for a significant Mauchly Test of Sphericity), F(2.68,77.63) = 6.51, p = 0.001, η

_{p}

^{2}= 0.18. This small interaction effect reveals that the difference between mean near and far estimates grows somewhat with increasing distance from the observer.

_{p}

^{2}= 0.38. This result is consistent with a slight positive curvature to visual space very near to the observer and increasingly negative curvature with increasing distance. This pattern of change in curvature is strikingly similar to a number of previous works [76,77,78,79].

#### 6.3. Discussion

## 7. General Discussion

## Author Contributions

## Conflicts of Interest

## Appendix A

**Figure A1.**Diagram illustrating parallel alleys transformed according to Hatfield’s Perspective-Based model.

_{1}, y’

_{1}, z’

_{1}) and (x’

_{2}, y’

_{2}, z’

_{2}) that correspond to two points in physical space (x

_{1}, y

_{1}, z

_{1}) and (x

_{2}, y

_{2}, z

_{2}) would be

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**Figure 1.**Diagram illustrating the spatial contraction proposed in Hatfield’s Perspective-Based model.

**Figure 2.**Mean angle estimate (in degrees) as a function of distance from the observer to each square (in meters) and location of angle within a square for Experiment 1. Diamonds (bottom line) indicate mean angle estimates for near angles and squares (top line) indicate mean angle estimates for far angles within a square.

**Figure 3.**Mean sum of the angle estimates (in degrees) for each square as a function of distance (in meters) to the square from the observer to the square in Experiment 1.

**Figure 4.**Mean angle estimate (in degrees) as a function of distance from the observer to each square (in meters) and location of angle within a square in Experiment 2. Diamonds (bottom line) indicate mean angle estimates for near angles and squares (top line) indicate mean angle estimates for far angles within a square.

**Figure 5.**Mean sum of the angle estimates (in degrees) for each square as a function of distance (in meters) to the square from the observer to the square in Experiment 2.

**Table 1.**Means and standard errors (in parentheses) for angles estimates as a function of distance both overall and separately for near vs. far location.

Distance to Square | Angles Estimated | ||
---|---|---|---|

Overall Estimate | Near Estimates | Far Estimates | |

0.5 m | 91.27 (0.57) | 90.27 (0.64) | 92.27 (0.60) |

1.5 m | 88.66 (1.13) | 87.50 (1.23) | 89.83 (1.12) |

2.5 m | 87.41 (1.09) | 85.57 (1.28) | 89.26 (1.04) |

4.0 m | 84.47 (1.25) | 82.46 (1.69) | 86.46 (1.04) |

8.0 m | 79.03 (1.86) | 74.96 (2.42) | 83.10 (1.42) |

**Table 2.**Means and standard errors (in parentheses) for angles estimates as a function of distance both overall and separately for near vs. far location.

Distance to Square | Angles Estimated | ||
---|---|---|---|

Overall Estimate | Near Estimates | Far Estimates | |

0.5 m | 87.28 (0.85) | 85.88 (1.03) | 88.68 (1.08) |

1.5 m | 87.79 0.70) | 84.62 (0.85) | 88.97 (0.99) |

2.5 m | 87.78 (0.88) | 85.25 (1.50) | 88.32 (1.05) |

4.0 m | 84.40 (1.34) | 82.17 (1.53) | 86.63 (1.63) |

8.0 m | 82.73 (1.89) | 80.65 (2.25) | 84.82 (2.13) |

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Wagner, M.; Hatfield, G.; Cassese, K.; Makwinski, A.N.
Differentiating between Affine and Perspective-Based Models for the Geometry of Visual Space Based on Judgments of the Interior Angles of Squares. *Vision* **2018**, *2*, 22.
https://doi.org/10.3390/vision2020022

**AMA Style**

Wagner M, Hatfield G, Cassese K, Makwinski AN.
Differentiating between Affine and Perspective-Based Models for the Geometry of Visual Space Based on Judgments of the Interior Angles of Squares. *Vision*. 2018; 2(2):22.
https://doi.org/10.3390/vision2020022

**Chicago/Turabian Style**

Wagner, Mark, Gary Hatfield, Kelly Cassese, and Alexis N. Makwinski.
2018. "Differentiating between Affine and Perspective-Based Models for the Geometry of Visual Space Based on Judgments of the Interior Angles of Squares" *Vision* 2, no. 2: 22.
https://doi.org/10.3390/vision2020022