# How To Calculate Height of Point B from Point A with X, Y, Height, Angle (Dip), Azimuth, and Depth (Distance)?

I have a survey point data which has coordinates X, Y, Height, Angle (Dip), Azimuth, and Depth (Distance).

For Example, point A:

Easting: 290694

Northing: 715927

Elevation: 1060

Angle: 65°

Azimuth:45°

Distance:150

Can you let me know how I can calculate the end point (End of trace) Height, please?

The angle, azimuth, and distance are a form of spherical coordinates for the displacement from the initial position:

• The horizontal displacement is a multiple of the distance. That multiple depends on the dip. The multiple--by definition--is called the cosine of the dip.

• The vertical displacement is also a multiple of the distance. That multiple also depends only on the dip. Once again, by definition, the multiple is called the sine of the dip. But we must be careful to consider the dip a negative angle.

• The northing and easting are, in a similar fashion, multiples of the horizontal displacement. Assuming the azimuth is measured east of north, the multiple to use for the northern displacement is the cosine of the azimuth and the multiple to use for the eastern displacement is the sine of the azimuth.

In the example given,

``````cos(dip) = cos(-65 degrees) = cos(-65/180 * pi radians) =  0.4226183;
sin(dip) = sin(-65/180 * pi radians) =  -0.9063078.
``````

Therefore the vertical displacement equals `150 * -0.9063078 = -135.9462`. For the sake of continuing the calculation, I will assume the distance and the elevation are measured in the same units. (If not, then of course the units have to be made commensurable before adding them.) Whence the elevation of at the end of the displacement is `1060 + -135.9462 = 924.0538`.

Similarly the horizontal displacement equals `150 * 0.4226183 = 63.39274`. In order to continue the calculation, I will assume the distance and the (easting, northing) coordinates are measured in the same units. (If not, then the units have to be made commensurable before proceeding.) We compute

``````cos(azimuth) = cos(45 degrees) = cos(45/180 * pi radians) = 0.7071068;
sin(azimuth) = sin(45/180 * pi radians) = 0.7071068.
``````

Therefore the northern displacement is `63.39274 * 0.7071068 = 44.82544` and the eastern displacement works out to the same (because the sine and cosine of this particular azimuth happen to be equal). Whence the ground coordinates of the end of the displacement are

``````(290694, 715927) + (44.82544, 44.82544) = (290738.82544, 715971.82544).
``````

As a double-check let's compute the distance between the start and end:

``````distance((290694, 715927, 1060), (290738.82544, 715971.82544, 924.0538))
= sqrt((290694 - 290738.82544)^2 + (715927 - 715971.82544)^2 + (1060 - 924.0538)^2)
= 150.0000,
``````

as intended. We can also plot the two points and visually check that the angles look about right: that's a good idea when testing out these calculations on an unfamiliar computing platform, because it detects the common mistakes of (1) confusing radians and degrees, (2) mixing up sines and cosines, (3) not using the correct signs for the angles, and (4) not performing units conversions where needed.

• Wow, Thanks a lot! Honestly I need to re-read this several times but to completely understand what is happening :-).So by this method you calculate both Height and Position of the point B? am I right? – user1106951 Jul 28 '12 at 17:04
• Yes, that is correct. If all you want is the height, then you can omit the calculations of position altogether. (Notice that the height depends only on the original elevation and the dip but not on the easting, northing, or azimuth.) Because variations of this question have occurred, and will likely occur again, I provided a full answer for future reference. – whuber Jul 28 '12 at 18:07