Infinitesimal, Snell's law, and ray tracing

In summary, the author is seeking a relationship between d(alpha1) and d(v1), in which d represents differentiation. They say that v1 is always constant, and that alpha1 and alpha2 change when v1 changes. They also say that direct differentiation of Snell's law can give them the result.
  • #1
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With FIXED SOURCE AND RECEIVER, I have a light incident from fluid 1 with velocity v1 into fluid 2 with velocity v2. Obviously, according to Snell's law, v1/v2=sin(alpha1)/sin(alpha2), where alpha1 and alpha2 are the angles with regard to the vertical line.

My question is: how to calculate d(alpha1)/d(v1)?

Obviously, I can obtain the result from numerical calculation, but I am seeking analytical method. I guess maybe I need some approximation to calculate that?

Please note v2 is always constant, please also note when v1 changes, alpha1 and alpha2 change. Please note direct differentiation of Snell's law, I can obtain d(alpha1)/d(alpha2). I am clueless and have already tried several methods but obtain no result, please help!
 
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  • #2
Have you drawn a diagram? If you do so, you ought to be able to see a relationship between ##\alpha_1## and ##\alpha_2##.
 
  • #3
Thank you very much for your kind answer. Yes, I have drawn it. I am NOT seeking a relationship between alpha1 and alpha2. I am seeking relationship between d(alpha1) and d(v1), in which d represents differentiation.
 
  • #4
And you can get that by using a relationship between ##\alpha_1## and ##\alpha_2## to eliminate ##\alpha_2## from your Snell's Law expression.
 
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  • #5
Ibix said:
And you can get that by using a relationship between ##\alpha_1## and ##\alpha_2## to eliminate ##\alpha_2## from your Snell's Law expression.

Sorry you answer is NOT clear. I have tried everything that I can think of for several days, can you write down specific equations? Thank you very much!
 
  • #6
I can help you find the answer, but I can't give it to you. The aim of this site is to help you learn, not to do things for you.

You say the source and receiver are fixed. Where are they? Write down the coordinates. Then, work out where the ray traveling from source to receiver crosses from one medium into the other (this is where a diagram will help you). That will give you a second expression relating ##\alpha_1## and ##\alpha_2##.
 
  • #7
ytht100 said:
Thank you very much for your kind answer. Yes, I have drawn it.

show us your diag. so we can confirm you are on the right path
 
  • #8
Thank you all for your kind answer. I have figured it out by using the diagram. Kind regards!
 

1. What is an infinitesimal in mathematics?

An infinitesimal is a mathematical concept that refers to a quantity that is extremely small and approaches zero. It is often represented by the symbol "dx" and is used in calculus to describe the change in a function as the input approaches a specific value.

2. What is Snell's law and how is it used in optics?

Snell's law, also known as the law of refraction, describes the relationship between the angle of incidence and the angle of refraction when light passes through two different mediums. It is used in optics to predict how light will bend as it travels from one medium to another, such as from air to water.

3. How is ray tracing used in computer graphics?

Ray tracing is a rendering technique used in computer graphics to create realistic images by simulating the behavior of light as it interacts with objects in a virtual scene. It works by tracing the path of light rays as they bounce off objects in the scene, taking into account factors such as reflection, refraction, and shadows.

4. Can Snell's law be applied to other types of waves besides light?

Yes, Snell's law can be applied to any type of wave that travels through different mediums, such as sound waves or water waves. The only difference is that the speed of the wave will change depending on the properties of the medium it is traveling through.

5. What are some real-world applications of ray tracing?

Ray tracing is used in a variety of industries, including film animation, video game development, architecture, and product design. It is also used in scientific simulations to model the behavior of light in different environments, such as in medical imaging or meteorological studies.

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