ODE modeling a curved reflective surface, describe shape of curve

In summary, the conversation discusses a problem involving a curve C and a surface of revolution created when C is revolved around the x-axis. It is known that all light rays parallel to the x-axis will be reflected to a single point O on the surface. The goal is to determine a differential equation that describes the shape of C using the fact that the angle of incidence is equal to the angle of reflection. The conversation also mentions using an appropriate trigonometric identity to determine the shape of C, and suggests that it may be parabolic. However, without a diagram it is difficult to follow the discussion. Possible starting points for solving the problem include labeling the point of reflection on the curve and using the slope of the curve and the angle between
  • #1
Breedlove
27
0

Homework Statement


Assume that when the curve C shown below is revolved about the x-axis, it generates a surface of revolution with the property that all light rays L parallel to the x-axis striking the surface are reflected to a single point O (the origin). Use the fact that the angle of incidence is equal to the angle of reflection to determine a differential equation that describes the shape of the curve C.

(I drew up the diagram in MSpaint, but can't figure out how to upload it. HELP!)

Homework Equations



It's a model problem, so I assume that there is going to be some sort of proportionality constant, but as for equations I'm not sure at this point.

The Attempt at a Solution


Because I can't figure out how to upload the figure, I will attempt to describe what's happening and how far I was able to go. There is a curve C and a tangent line to C at the point where light hits C and reflects. The angle of incident, the angle at which the light hits C, is also the angle at which the light leaves C (theta), as measured from the tangent line, NOT C. The obtuse angle that the light makes with the x-axis is phi, which equals 2(theta) because of alternate interior angles.
Okay. There is little likelihood that I would be able to follow all this without a diagram, but I hope it's not too lost. I think I should just get this question out there so someone can enlighten me so I can better describe the problem. There was a hjint that once you figured out that phi=2(theta) you could use "an appropriate tironometric identity". What kind of identitiy could give me an idea about the shape of C? I'm pretty sure it's parabolic, although my picture doesn't really look like that, but it's paint. Okay. Thanks!

okay I attached it, I hope it works!

Okay, in the attached picture, the thetas are supposed to be the angle between the tangent line and the path of the light ray, not the angle between the curve C and the light ray, sorry about that.
 

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  • #2
I haven't worked it all out but here's where I might start. Label the point where the light hits on the curve (x,y). You know the slope of the curve:

[tex] y' = \tan(\theta)[/tex]

and from the picture

[tex] \frac y x = \tan(\phi) [/tex]

and you know

[tex] \phi = 2\theta[/tex]

Take that and run with it.
 

1. What is an ODE model?

An ODE (ordinary differential equation) model is a mathematical representation of a system or process that involves the rate of change of one or more variables. It is commonly used in scientific and engineering fields to describe and predict the behavior of complex systems.

2. How does ODE modeling relate to curved reflective surfaces?

ODE modeling can be used to describe the shape of a curved reflective surface by representing the surface as a function of time and using equations to calculate the rate of change of the surface's curvature.

3. What factors influence the shape of a curved reflective surface?

The shape of a curved reflective surface is influenced by several factors, including the material of the surface, the angle of incidence of the light, and the curvature of the surface itself. Additionally, external forces such as gravity can also affect the shape of the surface.

4. Can ODE modeling accurately predict the shape of a curved reflective surface?

ODE modeling can provide a good approximation of the shape of a curved reflective surface, but it is important to note that it is only a model and not a perfect representation of the real world. Factors such as imperfections in the surface or external disturbances can affect the accuracy of the model.

5. How can ODE modeling be used in practical applications involving curved reflective surfaces?

ODE modeling has many practical applications in fields such as optics, materials science, and engineering. It can be used to design and optimize the shape of curved mirrors or lenses, to predict the behavior of light in optical systems, and to study the effects of external forces on reflective surfaces.

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