Surface joining two points in a family of concentric spheres

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Discussion Overview

The discussion revolves around the mathematical and geometric problem of finding a surface that connects two points located on concentric spheres, with a focus on optical properties and the concept of a mirror surface that can focus light at a third point. Participants explore various approaches to defining this surface in three-dimensional space.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes a mathematical description of the surface S using a general equation involving coordinates and radii of the spheres, but expresses concern that it may be too abstract.
  • Another participant seeks clarification on whether the surface is derived from the revolution of a line segment connecting the two points.
  • Some participants suggest that the surface S could be symmetrical about point P and may resemble a conic section.
  • A different approach involves rotating the graph to find the equation of line S and using surface area formulas to derive the surface.
  • One participant describes a method using Geometer's Sketchpad to visualize the relationship between the points and suggests that surface S could be a patch of a sphere.
  • Another participant emphasizes that any three points can define a new circle, which can be used to create a mirror surface that focuses light at point P.
  • Concerns are raised about the completeness of the original question, noting that an infinite number of surfaces can pass through two points.
  • A later reply clarifies that the goal is to connect three points to define a circle, which can then be extended into a 3D surface.
  • One participant challenges the notion of a focus in relation to circles and spheres, asserting that a true focus requires a parabolic mirror.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the surface S and its properties. While some agree on the potential for a spherical patch, others contest the definitions of focus and surface characteristics, indicating that the discussion remains unresolved.

Contextual Notes

There are limitations in the assumptions made regarding the definitions of surfaces and focuses, as well as the mathematical steps involved in deriving the surface S. The discussion also highlights the complexity of visualizing geometric relationships in three dimensions.

DivGradCurl
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Hi,

What's the surface joining two points in a family of concentric spheres? Shown below is the general idea; it's actually optical. Two rays meet at P from P1 and P2, respectively, where each point comes from a different sphere. How do I find surface S if I know the coordinates of P1 and P2?

Question_about_Surface.jpg


My best bet is that one can describe S as
(x-h)^2+(y-k)^2=r^2 (\phi _i), \qquad R_1 \leq r (\phi _i ) \leq R_2 \mbox{ and } \phi_2 \leq \phi_i \leq \phi_1
but that seems too abstract and 2D. I'm looking for something like an even asphere description with radius of curvature and coefficients if I know P1 and P2. How can I do that?

Thanks
 
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I want to confirm one thing, does the question ask you to find the surface by revolution of red line S?
 
I guess you can look at it that way, I'm looking for a surface in 3-space. It would be symmetrical about P. I was hoping this shape S would be a piece of another (simple) conic section.
 
I will first rotate the graph, so that the R2 is on the axis, then find the equation of the line S. use surface area formula to solve it. I think that equation is
∫2 π f(x) √(1+f'(x)2) dx
 
I'm trying a 2D approach to this where all three points P1, P2, and P belong to one circle. It seems that surface S is just the 3D version of this 2D projection. I'm doing this the quick and dirty way using Geometer's Sketchpad - basically just changing the circle size and moving it around. That would mean the shape of S as a mirror would be a patch of a sphere.
 
DivGradCurl said:
I'm trying a 2D approach to this where all three points P1, P2, and P belong to one circle. It seems that surface S is just the 3D version of this 2D projection. I'm doing this the quick and dirty way using Geometer's Sketchpad - basically just changing the circle size and moving it around. That would mean the shape of S as a mirror would be a patch of a sphere.
but you told me the surface is by revolution of line S? did I misunderstand your question?
 
I concluded any 3 points (here P1, P2, and P) can be fit into a new circle and that a mirror prescribed by that new circle shape "C", truncated by the P1 P2 arc "S", should be able to focus light at point P, also on the same circle C. It's very easy to find that new circle in Geometer's Sketchpad by stretching and translating a custom circle until all 3 points are part of the circle.

Ultimately, it is indeed a patch of a sphere, so it is a surface of revolution of the line S. The key problem really was defining S. I'm sorry if I confused you. And the 3D surface comes out easily by bumping "S" out of 2D into 3D with a polar angle.

I think I should be all set. Thanks for the conversation. It helped me think.
 
Your question
"What's the surface joining two points in a family of concentric spheres? Shown below is the general idea; it's actually optical. Two rays meet at P from P1 and P2, respectively, where each point comes from a different sphere. How do I find surface S if I know the coordinates of P1 and P2?"
doesn't seem to be complete. There exist an infinite number of surfaces that contain any finite number of points. There even exist a infinite number of planes that contain two points. Are you asking for a plane that contains two given points on the surface of two concentric spheres and the center of those spheres?
 
Hi,

I'm sorry for the confusing, perhaps incomplete question! I was really looking for a way to connect 3 points in a manner where 2 points could represent a mirror (i.e. surface S) focusing light at the 3rd point, as I represented in the drawing.

As I discovered, this is easiest done with interactive geometry software. You can draw 3 points and then insert a circle and play with its radius and position so that it overlays the 3 points. The fact that 3 points can not only define a plane (as most people immediately associate) but also define a circle is something I didn't quite appreciate before. Once you know the circle and its radius, you can turn that P1-P2 section into a 3D surface with a polar angle with respect to P, defining a solid angle. If you look for the definition of solid angle, it's similar to the resulting picture I'm taking about:

point.gif

except that in the case I'm describing, the focus P is not at the center of the circle or sphere.

I hope this makes sense. I think I'm all set. Best regards.
 
  • #10
That doesn't seem to have anything to do with your original question- and a "circle or sphere" does NOT HAVE a 'focus' although the point at, if I remember correctly, r/2 where r is the radius of the sphere, will act like a "blurry" focus. I order to have a true "focus" your mirror must be parabolic. A paraboloid can be approximated by a sphere with radius equal to twice the focal length of the sphere.
 

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