Does anyone know an equation that calculates the z value of a point within a triangle with known xyz values?

Question

I'd like to find an equation that I can use in Excel to calculate an estimated ground water elevation at any point within a triangle of three wells with known GW elevations. If it would make it any easier, doing it with a fixed point with a known x and y coordinate would be fine too. I'd like to keep it simple if possible, Kriging is a bit more than I can handle!

Answer 1

Doug's answer is correct. You can certainly use the plane equation for your problem. Generally speaking, you are trying to perform interpolation. Another method that is used in computer graphics may be a little simpler for your problem.

First, consider the x coordinate of your interior point, and choose the two triangle sides that span this x value. Use linear interpolation to find a depth value on each triangle side for this x value. That is, for each side, evaluate

d = d0 + (d1-d0) * (x - x0)/(x1-x0)

where x0,d0 and x1,d1 are the x coordinate and depth values found at the two vertices of an edge. x is the x coordinate of your interior point.

So, now you have two interpolated depth values that lie on two edges of the triangle at the same x value as your interior point. If you draw a picture, you'll see that the locations of the interpolated points and your interior point all fall along a horizontal line. All that remains is to reapply the linear interpolation formula along that line to get the interpolated depth value for the interior point.

One major pitfall of this algorithm is that you have to choose the edges very carefully. With a pencil and paper you can draw out lots of different triangles and see that there are several different cases. This makes it much harder to code a general solution in Excel. However, if you are just solving a single problem, it is easy to see which edges to choose from a picture.

A third approach also works well, and is computationally cheaper if you want to interpolate more than one attribute. For example, imagine if you had to handle water salinity as well as depth. With the plane method you'd have to start all over again, because you'd be working with a new plane for salinity. The third method doesn't have this problem. Rather than explain it, I'll point you to a short pdf on the topic, which you can find at the URI below.

Answer 2

If you assume that the GW elevation levels are all on a plane defined by the three sets of (x,y,z) coordinates found from the three wells it's not too bad. I they're not, all bets are off

The general form of the equation of a plane is

Ax + By + Cz +D = 0

which probably doesn't help you much. But, if you assume D = 0 and let z be the depth, then you can re-write it as

Ax + By + Cz = 0

Take your samples (the x,y,z values) at the three points
and set it up as three simultaneous equations. Plug in your (x,y,z) values and solve it for the A,B,and C coefficients.

Now, to get z at any point (x,y) use

z = -(Ax + By)/C

I don't know if Excel has any built in 'equation solvers' for simultaneous equations or not. If not you'll have to 'roll your own' using Gaussian reduction or determinants.

You can get details on those from any linear algebra book.


Doug

Answer 3

Sorry, if I understand your question (and I may not), can't be done. What I think you're asking is for a point anywhere within that triangle defined at each point by the x,y,z values of each well.

First, you are looking for GW elevation, not a well elevation; so the well elevations (z?) would only approximate GW elevation.

Second, knowing the locatiion of the three points in no way defines a fourth location within a triangle defined by those three points.

What I think you are asking is like asking "Given the location of the four corners of my living room, tell me where my sofa is." Can't be done...sorry. If I've misunderstood your question, please recast it in different words for clarification. Thanks.