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I am attempting to determine the TCAS-protected volume by utilizing the tau values listed in the TCAS table enter image description here

(Reference: Page 23 FAA TCAS-II guilines)

The TCAS table provides distinct tau values for the Traffic advisory (TA) and resolution advisory (RA) zones. Based on these values, one would expect the protected volume to take the shape of a circle when viewed from the top. However, upon further examination of figures provided by the FAA FAA-TCAS-II or on Wikipedia, it appears that the TA and RA zones are actually represented as ellipses. TCAS figure Wiki Wikipdia TCAS

Can anyone provide insight on the reason for the TA and RA zones being represented as ellipses instead of circles and the methodology used to determine these ellipses?

Edit: Based on the discourse in the comments section, it has been determined that the zones in question should adopt an elongated shape oriented in the direction of heading, as opposed to a circular shape. Despite this conclusion, the issue of determining the precise shape remains unresolved. It is imperative to consider multiple parameters in order to accurately determine the shape, and the utilization of specific methods or equations may prove beneficial in this regard.

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  • $\begingroup$ Perhaps they are ellipses because if two airplanes are on track to collide, one or both will enter the end of the others' ellipse before it would enter a sphere? $\endgroup$
    – Someone
    Commented Jan 18, 2023 at 22:58
  • $\begingroup$ @Someone, But in some figures, those two aircrafts are not aligned on the major axis of the ellipse. [link] (tinyurl.com/ep4bu4hw) $\endgroup$ Commented Jan 18, 2023 at 23:49
  • $\begingroup$ I think my explanation still applies to that diagram? A plane behind you isn't as much of a problem as one in front of you, right? $\endgroup$
    – Someone
    Commented Jan 19, 2023 at 1:00
  • $\begingroup$ @Someone, Thank you, that makes sense. If my understanding is correct, the TA and RA volumes are determined solely by the characteristics of the interrogated aircraft and are not affected by the presence or characteristics of any intruding aircraft. Additionally, it is my understanding that these volumes should take the form of an elongated shape, such as an ellipse when viewed from above, as collisions are more likely to occur in the aircraft's heading direction. My question is, what methods or equations can be used to determine the locus of that shape? $\endgroup$ Commented Jan 19, 2023 at 5:11

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Figure 14 suggests that the shape is not an ellipse. Figure 14

The reason they take such a shape is that the range tau calculations is based off of closing speed, as defined on page 23. This means that the TCAS protection volume with respect to an aircraft directly ahead should not include anything aft of the subject aircraft.

The locus of the projection of this shape onto a 2d plane can be found by calculating the range tau for a given target and comparing to Table 2.

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  • $\begingroup$ Agreed, it may not be an ellipse. Can you please elaborate more on how to calculate the locus ? Thank you, $\endgroup$ Commented Jan 19, 2023 at 18:06

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