Solving Geometry Problem: Disparity in Terms of a, D, d, e, & f

[SoggyBottoms]

Homework Statement

Disparity is defined as $\delta = \alpha - \beta$. Find $\delta$ in terms of interocular distance a, viewing distance D and d, e and f.

http://img220.imageshack.us/img220/7576/43519392.jpg

The Attempt at a Solution

I'm not getting anywhere. Any tips to get me started?

 

[bbbeard]

I would start by writing down the tangent relations. For example, α is the difference between the two angles that Q and P make from the vertical (forward?) direction. So

α = atan(f/(D+d)) - atan(e/D)

and so forth.

BBB

 

[SoggyBottoms]

Thanks. Then:

$\beta = \arctan(\frac{a - e}{D}) - \arctan(\frac{a - f}{D + d})$

Correct? Then I already have my answer it seems.

 

[bbbeard]

Yes. It seems like you're about there. Don't forget the problem asks for δ=α-β.

 

[asteriatic]

I would first clarify the definitions of the variables involved in this problem. Interocular distance, a, refers to the distance between the two eyes. Viewing distance, D, refers to the distance between the observer and the object being viewed. The variables d, e, and f are not defined in the problem statement, so it is unclear what they represent.

To solve this problem, we need to understand the geometric relationships between these variables. From the image provided, we can see that \delta is the difference between the angles \alpha and \beta. These angles can be calculated using trigonometric functions, specifically tangent and inverse tangent.

Using the definition of tangent, we can write \tan{\alpha} = \frac{d}{a} and \tan{\beta} = \frac{e}{D}. Solving for \alpha and \beta, we get \alpha = \tan^{-1}{\frac{d}{a}} and \beta = \tan^{-1}{\frac{e}{D}}.

Substituting these values into the definition of disparity, \delta = \alpha - \beta, we get \delta = \tan^{-1}{\frac{d}{a}} - \tan^{-1}{\frac{e}{D}}.

Therefore, the disparity, \delta, can be expressed in terms of interocular distance, a, viewing distance, D, and the variables d and e. However, without a clear definition of d and e, it is not possible to simplify this expression further.

In conclusion, to solve this geometry problem, we need to use trigonometric functions and understand the geometric relationships between the variables involved. It is important to have a clear understanding of the definitions of the variables and to carefully follow the given instructions.