What is the role of 'd' in defining a plane's normal form?

[sundar0206]

Hi guys,

I am sort of new here. So I am not pretty sure if I am to post this question in here.

I am a software programmer and I need to write a class for defining a plane. I came across the plane in its normal form nx+ny+nz+d=0

I need to feed in the values of the plane from another part of my program.

I can understand that nx,ny and nz are the normals of the plane. So where does the d come from. How exactly do you arrive at the value of d;

May sound very basic but then it would be nice if some one could help me out

 

[mathman]

The equation should read n_xx+n_yy+n_zz+d=0. d essentially defines how far the plane is from the origin of the coordinate system.

 

[sundar0206]

If I may rephrase my question:

If the orign of my co ordinate system is (0,0,0) then d is the distance between (0,0,0) and which point on the plane ? Or am I totally misunderstanding this?? Can you please explain

 

[sutupidmath]

Ok, let us first try to come up with the vector equation of the plane, and then we will switch to cartesian coordinates, and you will probbably see how the d comes into play.

A plane is generally uniqely determined by a point call it $$P_o(x_o,y_o,z_o)$$ and a vector normal on the plane $$n=<a,b,c>$$

Now, let P(x,y,z) be any other point in the plane, then its position vector would be:

$$r=<x,y,z>$$

while let

$$r_o=<x_o,y_o,z_o>$$ be the position vector to the point P_o.

Now, if you draw a picture you will se that the following relation holds:

$$(r-r_o)*n=0$$

"*" holds for the dot product. Notice that (r-ro) and n are normal vectors.

Now, switching to the coordinate representation of the above vectors we get:

$$<x-x_o,y-y_o,z-z_o>*<a,b,c>=0=>a(x-x_o)+b(y-y_o)+c(z-z_o)=0$$

After rearranging the stuff in there we get:

$$ax+by+cz-(ax_o+by_o+cz_o)=0$$

So,

$$d=-(ax_o+by_o+cz_o)$$

 

[sundar0206]

oh..
many thanks for explaining stuff to me.. I got confused after looking at many websites none of which gave me what d is .

Thanks anyways

 

[mathman]

If I may rephrase my question:

If the orign of my co ordinate system is (0,0,0) then d is the distance between (0,0,0) and which point on the plane ? Or am I totally misunderstanding this?? Can you please explain

Take a line normal to the plane starting at the origin. This line will hit the plane at a distance d from the origin. The hit point will have coordinates (-dn_x,-dn_y,-dn_z)