GIS Draw a parabola flight path in kml for google earth

Question

Hi, I have to display flight paths on google earth and (still I got the exact flight plan path) want to show in a parabola way (markting side). From the taking off to the landing points. Basically I'm looking for the math formula to calculate latlng point with altitudes to display the parabola path.

I see how to do the parabolic view with the altitude parameter. But there is less easy, how to get points on a path from two cordinates (start; end).

Thank you !

Answer 1

Maybe I'm wrong about this, but if I remember my physics right, and if the distance between the two points is relatively small compared to the earth's radius, aside from altitude, the path will just follow a great circle.

If you write the coordinates as parametric equations as a function of time, you'd get:

gamma = v_horz/R * t
altitude = g * t * (T0 - t) / 2    
(where T0 = 2*v_vert/g = flight time, R = earth radius, g = earth's gravity) 
(vertical velocity = derivative of altitude = g/2*T0 - gt = v_vert - gt

where gamma = the angle along the great circle arc followed by the projectile. You know the beginning and endpoints, so you can use spherical trigonometry to figure out the arc distace G0 between start and endpoints. G0 = gamma at time T0 (when projectile lands). This tells you what omega_horz must be (= G0*R / T0).

You can then use spherical trigonometry again to figure out the lat/longitude at any point along the great circle. (use similar triangles -- it's too late in the day for my brain to work through the math properly, sorry)

assumptions:

A.

• distance between start/end points is small compared to earth's radius
• the flight in question is a ballistic trajectory (some mass M going up and down under the force of gravity alone, no thrust or lift)
• we're talking about the planet Earth so you can make certain assumptions for the radius and the force of gravity g
• ignore air resistance (good luck in real life)

OR

B.

• this is for marketing types so you just want something that looks parabolic, so just use the assumptions in A anyway

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EDIT: See also these wikipedia articles on projectile trajectories and great circle distance.