Question

Relative acuity of the human eye The fovea centralis (or fovea) is responsible for the sharp central vision that humans use for reading and other detail- oriented eyesight. The relative acuity of a human eye, which measures the sharpness of vision, is modeled by the function $$R(\theta)=\frac{0.568}{0.331|\theta|+0.568}$$ ,where \(\theta\) (in degrees) is the angular deviation of the line of sight from the center of the fovea (see figure). a. Graph \(R,\) for \(-15 \leq \theta \leq 15\) b. For what value of \(\theta\) is \(R\) maximized? What does this fact indicate about our eyesight? c. For what values of \(\theta\) do we maintain at least \(90 \%\) of our maximum relative acuity? (Source: The Journal of Experimental Biology, 203, Dec 2000)

Short answer

Answer: The maximum value of R represents the highest relative acuity achievable. This occurs when θ = 0, meaning that our eyesight is the sharpest when looking straight ahead and aligned with the center of the fovea. The values of θ for which we maintain at least 90% of our maximum relative acuity are in the interval -0.1906 ≤ θ ≤ 0.1906, indicating that as the line of sight deviates from the center of the fovea, our relative acuity drops.

Step-by-step solution

Graph the function

To graph the function \(R(\theta) = \frac{0.568}{0.331|\theta| + 0.568}\), you can use graphing software like Desmos or a graphing calculator. Use the given interval of \(-15 \leq \theta \leq 15\) for the graph. Once you have drawn the graph, you will notice that the graph is symmetric with respect to the y-axis, meaning that the maximum relative acuity occurs when \(\theta = 0\). **Step 2: Find the value of \(\theta\) where \(R\) is maximized**

Maximum value of \(R\)

From the graph, we notice that the function is symmetric around the y-axis and so it is reasonable to assume that the maximum value of \(R\) occurs when \(\theta = 0\). We can confirm this by evaluating the function at \(\theta = 0\): $$R(0) = \frac{0.568}{0.331|0| + 0.568} = \frac{0.568}{0.568} = 1$$ Since \(R(0) = 1\) and the function is symmetric, the maximum value of \(R\) is 1 and occurs when \(\theta = 0\). **Step 3: Determine the values of \(\theta\) for which we maintain at least \(90 \%\) of our maximum relative acuity**

Relative acuity at least \(90 \%\)

Since the maximum relative acuity is 1, we want to find the values of \(\theta\) for which the relative acuity is at least \(0.9\). To do this, we need to solve the inequality $$R(\theta) = \frac{0.568}{0.331|\theta| +0.568} \geq 0.9$$ Firstly, we can find the value of \(|\theta|\) from the inequality. $$|\theta| \leq \frac{0.568(0.1)}{0.9(0.331)} \Rightarrow |\theta| \leq \frac{0.0568}{0.2979}$$ Now, since the function is symmetric, we can say that the values of \(\theta\) for which the relative acuity is at least \(0.9\) are in the interval $$-\frac{0.0568}{0.2979} \leq \theta \leq \frac{0.0568}{0.2979}$$ This fact indicates that our eyesight is sharpest when looking straight ahead and decreases gradually as the line of sight deviates from the center of the fovea.

Key concepts

Fovea Centralis

The fovea centralis, commonly referred to as the fovea, is a small pit located in the retina of the human eye. It plays a crucial role in our vision by facilitating sharp central vision. This sharpness is critical for activities that require detailed vision, such as reading, driving, and recognizing faces.
Here are some key points about the fovea centralis:

• It is densely packed with cone cells, which are responsible for color vision and fine detail.
• The fovea covers only about 1-2 degrees of the human visual field but crucially impacts visual acuity.
• When focusing directly on an object, the light from that object is directed onto the fovea, thus providing the clearest vision.

This optical structure essentially explains why our eyes move to align objects into the center of our field of vision for optimal clarity.

Angular Deviation

Angular deviation refers to the angle measured in degrees from the line of sight to the center of the fovea. If you imagine looking directly at a point in front of you, that point is centered on your fovea with an angular deviation of 0 degrees.
When discussing the human eye's visual acuity:

• The larger the deviation from the foveal center, the less acute the vision becomes.
• As depicted in the graph of the function \(R(\theta)\), acuity decreases symmetrically on both sides of the fovea due to increasing angular deviation.

This deviation helps us understand how peripheral vision operates and confirms why clarity is reduced when viewing objects indirectly.

Maximum Relative Acuity

Maximum Relative Acuity is achieved when the angular deviation \(\theta\) is zero, meaning the fovea is directly fixated on the object. In the function pertaining to relative acuity, \(R(\theta) = \frac{0.568}{0.331|\theta|+0.568}\), the maximum value of \(R(\theta)\) occurs precisely when \(\theta = 0\), giving \(R(0) = 1\).
The significance of this maximum:

• The value of 1 represents 100% visual acuity, the sharpest an object can be perceived.
• Our eyesight directly faces any observed object for the highest definition possible.

This understanding is essential in contexts such as vision testing and designing tasks that require precision.

Graphing Functions

Graphing functions like \(R(\theta) = \frac{0.568}{0.331|\theta|+0.568}\) visually represents relationships and dependencies in data, making it crucial for practical understanding in fields such as biology and vision science.
The importance of graphing:

• It displays symmetries and patterns found in mathematical relationships, as seen in \(R(\theta)\) being symmetrical about the y-axis.
• Graphs help identify maximum and minimum points, such as the peak of visual acuity at \(\theta = 0\).
• The area where \(R(\theta)\) remains at least 90% of its maximum value further quantifies the practical range of sharp vision.

Understanding how to graph functions is vital in interpreting data and making informed conclusions in a systematic manner.