|
|
|
|
|
Welcome to Fermat's Pond. Like a real pond, the Java applet below is somewhat complicated. When you look toward the water of a real pond you often see two images -- a reflected image from above the water's surface and a refracted image from below the water's surface.
The applet below has three panels arranged side-by-side. The middle panel shows a picture of a fisherman who is standing in water that is approximately three feet deep. The fisherman wouldn't look like this, however, because the water would distort his appearance. Exactly how the water would distort his appearance would depend on where you were as you looked at him. The diagram to the left shows how this happens. In this diagram the fisherman is standing at the black line separating the diagram from the picture of the fisherman. Your eye and the location from which you are looking is represented by a black dot toward the left side of the diagram. There are several lines emanating from your eye toward the fisherman. These lines represent light rays arriving at your eye from the right side of the diagram -- the side occupied by the fisherman. Notice that when you look toward the water you see two light rays -- a red light ray that is reflected by the surface of the water and a blue light ray that is bent as it passes from the water to the air. Your brain is used to light rays that travel in straight lines with no bends. Your brain thinks both light rays traveled along the black line. Thus, you see the fisherman as shown in the panel to the right. To see how your location changes the appearance of the fisherman click on the diagram at a new location for your eye. Notice if you click below the water's surface you won't see much at all. This is because you are not wearing a diver's mask. Your eyes are designed to work in air, not in water.
You are not the only visitor today. Just below the Fermat's Pond applet below are five other visitors. You can click on any of them to see what they are thinking.
|
|
|
|
|
|
We are interested in why and how light rays bend as they travel across the water's surface. Willebrord Snell and Pierre de Fermat each offered their own explanations. You can click on each of their pictures above to see their explanations. Next you should answer the questions below.
Question 1: Use the diagram (applet) above to explain why the fisherman's appearance is distorted by the water.
Question 2: Suppose that two fishermen are standing in water that is three feet deep and they are six feet apart. Suppose one fisherman whose eye is three feet above the surface of the water looks at the shoe of the other fisherman. The shoe is three feet below the surface of the water.
Question 3: Write a Mathematica notebook that solves problems like the one above. Check your notebook by comparing the answer it gives for the previous question with the one you found.
There are a number of questions you might have about how objects underwater appear as viewed from a viewpoint above the water's surface. You can use the Mathematica notebook that you created above to work out some examples investigating these questions.
Question 4: As the viewer approaches the viewee how does the viewee's apparent image change?
Question 5: As the viewer crouches down so that his eye is closer to the surface of the water how does the viewee's apparent image change?
Question 6: In the examples above the portion of the viewee that is below the water appears to be shortened. Is it shortened in a uniform way? For example, is the portion of the viewee's body that is located from the surface of the water to a point one foot below the surface of the water shortened by the same factor as the portion of the viewee's body that is located from a point one foot below the surface of the water to a point two feet below the surface of the water?
Question 7: Do the problem below from Calculus, Concepts and Contexts, by James Stewart.
Question 8: How does a fisherman appear to a fish?
After you have answered the last question reflect on your answer. Is it surprising? Do you believe it? After you have consided your answer click here to open a new window with the same applet as the one above but this time you have a diver's mask and can go below the surface of the water to check your conclusions about how a fisherman appears to a fish.