Computing cone equation using data points

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Discussion Overview

The discussion revolves around a mathematical problem involving the computation of cone equations based on three points in 3D space. Participants explore how to determine the apex of right circular cones that pass through these points and the implications of noise in the data points on the solution.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant describes a problem involving the propagation of waves in 2D space and the relationship between three points in 3D space and the cones that can be formed around them.
  • Another participant suggests a geometric relationship involving the apex of the cone and the points, hinting at an equation that relates the lines from the apex to the points and the axis of the cone.
  • There is a mention of the need to consider noise or inaccuracies in the data points and a request for methods to address this issue.
  • Clarification is sought regarding the term "vertical" in relation to the cone, with a distinction made between right circular cones and oblique cones.

Areas of Agreement / Disagreement

Participants express different interpretations of the term "vertical" in the context of cones, indicating a lack of consensus on this aspect. The discussion remains unresolved regarding the exact mathematical formulation and methods to handle inaccuracies in data points.

Contextual Notes

Participants have not fully defined the assumptions regarding the geometry of the cones or the specific mathematical relationships involved, leaving some aspects of the problem open to interpretation.

Saeid
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Dear All
I have a problem that can be represented in two different forms.
Problem is related to propagation of waves in 2D space with respect of time.
I have three random points in the 3D Space.
How many right circular, infinite cones with specific predetermined angle between conical surface and axis we can find that passes these three points.
I need the 3D coordinates of the apex point.
Three points are in one side of cone. I.e., if we draw a plane perpendicular to axis of cone that meets the axis in apex point, all three points will fall in one side of the plane.
Another representation of the problem is, if I draw three right circular, infinite cones with specific predetermined angle between conical surface and axis which their apex points are these three points in the 3D space, how many points I can find that belongs to surface of these three cones. Or, in how many points these three cones coincide with each other.
What will be the result if I use four different points in the space?
If my points (data points) have noise (inaccuracy) what solutions or method exists or you suggest for solving it?


Thank you
 
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Welcome to PF!

Hi Saeid! Welcome to PF! :smile:

Hint: if the points are A B and C, and the axis of the cone is the line L, and the apex is at the point X on L, and the half-angle of the cone is θ, then the equation relating the lines XA XB XC and L is … ? :smile:
If my points (data points) have noise (inaccuracy) what solutions or method exists or you suggest for solving it?

No idea! It might be better to ask that question in the computing part of the forum. :smile:
 
tiny-tim said:
Hi Saeid! Welcome to PF! :smile:

Hint: if the points are A B and C, and the axis of the cone is the line L, and the apex is at the point X on L, and the half-angle of the cone is θ, then the equation relating the lines XA XB XC and L is … ? :smile:


No idea! It might be better to ask that question in the computing part of the forum. :smile:
Thanks tiny-tim
Another hint, required cone is vertical.
 
right circular cones

Saeid said:
Thanks tiny-tim
Another hint, required cone is vertical.

Hi Saeid! :smile:

What do you mean by "vertical"? :confused:

The question only says "right circular" … in other words, an "ordinary" cone.

See http://en.wikipedia.org/wiki/Cone_(geometry) :
In common usage in elementary geometry, however, cones are assumed to be right circular, where right means that the axis passes through the centre of the base (suitably defined) at right angles to its plane, and circular means that the base is a circle. Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the centre of the base.
 

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