# Descriptive Models for Perception of Optical Illusions: Part I (UMAP)

### Author: David A. Smith

In this unit, the historical background of the study of stable illusions of angle is sketched to show how a 19th century conecpt leads to formalization of a first order differential equation model for the apparent curves in such illusions. Some difficulties with the model are identified, and these lead to a refined formulation based on parametrization by arc length and polar coordinate transformations to solve the refined equations. ** Table of Contents:1. BACKGROUND OF THE PROBLEM2. THE HOFFMAN MODEL** 2.1 Vector Interpretation of the Bretano Hypothesis

2.2 Parametrization of Curves

2.3 The Differential Equation Model

2.4 Another Example: The Hering Illusion

2.5 Some Difficulties with the Hoffman Model

**3. THE NORMALIZED MODEL**

3.1 Removing Dependency on Position in the Plane

3.2 The Orbison Illusion

3.3 The Hering Illusion

3.4 Polar Coordinate Transformations

**4. FURTHER EXERCISES**

5. PREVIEW OF UNIT 535

6. ANSWERS TO EXERCISES

7. REFERENCES

APPENDIX A: SPECIAL ASSISTANCE SUPPLEMENT

5. PREVIEW OF UNIT 535

6. ANSWERS TO EXERCISES

7. REFERENCES

APPENDIX A: SPECIAL ASSISTANCE SUPPLEMENT

#### Mathematics Topics:

#### Application Areas:

#### Prerequisites:

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