Finding Angles of Triangle Given Two Points on Earth and One in Space

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Discussion Overview

The discussion revolves around determining the angles of a triangle formed by two points on Earth and one point in space, using the known distance and direction between the Earth points, as well as the right ascension (RA) and declination (DEC) of the celestial object. The scope includes geometric interpretation, astronomical coordinates, and potential mathematical approaches.

Discussion Character

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

Main Points Raised

  • One participant, Tom, seeks assistance in calculating the angles of a triangle defined by two points on Earth and a celestial object, providing details on the known parameters.
  • Another participant challenges Tom's understanding, suggesting he has provided a direction rather than a specific point.
  • A different participant expresses uncertainty about the interpretation of the problem but attempts to clarify the known parameters, including the altitude and azimuth of the object as seen from the Earth points.
  • One participant mentions the law of sines and cosines, indicating that these cannot be applied without knowing at least one angle or all three sides of the triangle.
  • Another participant introduces a hypothetical scenario involving a helicopter and a star, emphasizing the difference in distances despite similar RA and DEC values.
  • One participant suggests that the angles between the apparent position of the object and the direction to the other observer can be used to find the third angle, proposing the use of three-dimensional vectors for calculations.
  • A request for a diagram is made to aid understanding of the problem.
  • One participant acknowledges the usefulness of external resources for converting DEC/RA to vectors, indicating a potential method for addressing the problem.

Areas of Agreement / Disagreement

Participants express differing views on how to approach the problem, with no consensus on a specific method or solution. Some participants challenge the initial understanding of the geometry involved, while others propose various mathematical approaches without agreement on their applicability.

Contextual Notes

Participants note limitations in the information provided, such as the need for specific angles or sides to apply certain laws of geometry. There is also uncertainty regarding the conversion of RA and DEC into usable angles for the triangle.

Boxturtle
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Hello Folks,

I have two points on Earth at each end of a great circle path, for which I know the length in km and direction in degrees True. Also I have the RA and DEC of an object in space as seen from one of the previous points. The RA and DEC come from the setting circles of a telescope having the object centered in its view. The three points, two on Earth and one in space, determine a plane triangle. I would like to determine the angles of that triangle in the plane, in degrees. Can you help?

Thanks, Tom
 
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Boxturtle said:
I have two points on Earth at each end of a great circle path, for which I know the length in km and direction in degrees True. Also I have the RA and DEC of an object in space as seen from one of the previous points. The RA and DEC come from the setting circles of a telescope having the object centered in its view. The three points, two on Earth and one in space, determine a plane triangle. I would like to determine the angles of that triangle in the plane, in degrees. Can you help?
No. You need to review your geometry; you've given a direction, not a point.
 
I'm not sure how to interpret this, but let me attempt to clarify. I have two points on Earth. I know exactly where they are. I know the direction from one to the other, and I know the distance between them. I also have a third point in space. I do not know the location, but I do know the direction from the points at the ends of the transect on Earth. I need to convert RA and DEC to angles of a triangle, with one Earthly point at each of two vertices, and the stellar object at the third vertex. And yes, I do need to review geometry, in that I do not know how to convert RA and DEC into direction in this particular application.

I should add, I also know the altitude and azimuth of the object in space, as seen from each of the two known points on Earth.
 
Boxturtle said:
the altitude and azimuth of the object in space, as seen from each of the two known points on Earth.
Boxturtle said:
attempt to clarify.
Stellar distances, or, "near earth?" For "near earth" see "law of sines." For stellar distances see "parsec."
 
Neither the law of sines nor the law of cosines can be used unless at least one angle (or all three sides) is known. How to determine that angle is my question.
 
Last edited:
If a helicopter were hovering nearby, positioned directly in front of a star as seen from my yard, they would both have the same RA and Dec, but their positions would be light years apart.
 
Would the helicopter appear to be directly in front of the same star as seen from my yard?
 
All you need are the two angles between the apparent position of the object in the sky and the direction towards the other observer on Earth. The third angle (at the object in the sky) will follow from that as the interior angles add to 180 degrees. If the object is close (low Earth orbit) and the baseline is long enough you can even determine its distance.

There are multiple ways to approach this, but I would use three dimensional vectors in cartesian coordinates. Various websites will tell you how to convert DEC/RA to a vector. Its length is irrelevant here, it just has to point in the right direction. This process is the same for both points on Earth.
While you can use the great arc, it is much more convenient to find longitude and latitude of the points and then find their location in 3D space, the process is nearly the same as for DEC/RA. Take the difference between the two points and you get the vector corresponding to their separation.
Once you have the vectors, calculate their scalar product and divide by the product of their magnitudes to get the sine of the angle. Convert it to an angle, repeat for the other place, and you are done.
 
A good diagram would help a lot to understand this problem .
 
  • #10
"Various websites will tell you how to convert DEC/RA to a vector."

This is the first reply with a reference to the question I asked. Thank you.
 

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