General form of plane equation

[p1ayaone1]

...is Ax + By + Cz + D = 0.

The vector <A, B, C> is a normal vector of the plane. My question is: does the value of D have any geometric significance/interpretation?

I have an algorithm (that I didn't write myself) to evaluate the best-fit plane for a set of points in R3, and the value of D is coming back extremely large (10^20 or something obviously ridiculous). I wonder if D is just a mathematical artifact and I shouldn't worry, or if there is a problem with the algorithm (or my usage of it).

I don't think D should be that large based on the equation for distance between a point (x0, y0, z0) and a plane Ax + By + Cz + D, which is

D = abs(A*x0 + B*y0 + C*z0+ D) / sqrt(A^2 + B^2 + C^2)

My values of A, B, and C are all -1<value<1, but D is so big that it will completely dominate that expression.

Maybe this is a junior question and this thread should be re-categorized as such.

Thanks

 

[Anthony]

The equation of a plane with normal $$\mathbf{n}$$ going through the point $$\mathbf{a}$$ is given by $$(\mathbf{x} - \mathbf{a})\cdot \mathbf{n}=0$$. Can you see how that helps?

 

[p1ayaone1]

sure. Expanding the dot product gives the point-normal form (as opposed to the general form) of the plane equation. If a = (x0, y0, z0), x = <x, y, z> and n = <A, B, C>, we have A(x-x0) + B(y-y0) + C(z-z0) = 0

That means D = -(x0 + y0 + z0), but I still don't see what the significance of that quantity is...

 

[Anthony]

Well, using your notation we have:

$$(\mathbf{x} - \mathbf{a})\cdot \mathbf{n} =0 \quad \textrm{means} \quad Ax+By+Cz = \mathbf{n}\cdot \mathbf{a}$$

So your D is given by:

$$D = - \mathbf{n}\cdot \mathbf{a} = -|\mathbf{a}| \cos \theta = |\mathbf{a}| \cos (\theta - \pi)$$

where $$\theta$$ is the angle made between the vectors $$\mathbf{a}$$ and $$\mathbf{n}$$. So you have a geometrical interpretation of D.

 

[blue_raver22]

The value of D in the general form of a plane equation does have geometric significance. It represents the distance of the plane from the origin in the direction of the normal vector <A, B, C>. This means that if D is positive, the plane is on the side of the origin where the normal vector is pointing, and if D is negative, the plane is on the opposite side.

In terms of your algorithm, it is possible that there is a problem with either the algorithm itself or your usage of it. It is unusual for D to be such a large value, especially if your A, B, and C values are all between -1 and 1. It may be helpful to double check your input values and make sure they are correct, and also to check the algorithm for any errors. If the issue persists, it may be helpful to seek assistance from a colleague or expert in the field.