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S Joseph, V Lakshminarayanan; A New Analytical Expression To Represent The Horizontal Horopter . Invest. Ophthalmol. Vis. Sci. 2002;43(13):4673.
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© ARVO (1962-2015); The Authors (2016-present)
Purpose: Ogle's equation (Researches in Binocular Vision, Hafner, NY, 1972) to represent the Horizontal horopter ((tan alpha2/tan alpha1)=R0+H tan alpha2)is not accurate mathematically, because at higher angles, the right hand side dramatically increases whereas the left hand side is observed to remain close to one. The above equation is not symmetric with respect to alpha2 and alpha1 as one might expect. If we consider the right hand side of the above equation as a Taylor's expansion of the left hand side on the independent variable alpha2, we get a second order curve. But, if we choose to expand the LHS as a function of tan(alpha1) and choose to retain only the first two terms, we get a third order curve. Is there a way to get rid of these problems and get an accurate equation to represent the horizontal horopter? Methods: We expand the LHS of the above equation as a function of tan(alpha2-alpha1). We choose this particular Taylor's expansion based on physical reasoning of how a person perceives the magnification of various curves. We keep only the first two terms of the expansion since tan(alpha2-alpha1)is always very small irrespective of the values of alpha2 and alpha1 as long as alpha2 is very close to alpha1, which usually is the case. We fit the empirical data with the new equation and compare it with the Ogle's equation and find that ours is a better equation. We also derive the equations to determine the spatial form of the curve, which is a fourth degree curve. The horopter data given by Ogle was fitted to both the original and the newly derived formulas. Results:1. We got a fourth order curve as opposed to the second order curve obtained from Ogle's formula. 2. We have expressed the equation of the curve in a Cartesian coordinate system where we will have the least number of independent terms. 3. We got a better R2 value when using the new formulation as compared to Ogle's formula. Conclusion: The newly derived formula gives a better analytic representation of the horizontal horopter.
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