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Equation of planes

  1. Mar 20, 2009 #1
    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
     
  2. jcsd
  3. Mar 20, 2009 #2

    mathman

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    The equation should read nxx+nyy+nzz+d=0. d essentially defines how far the plane is from the origin of the coordinate system.
     
  4. Mar 20, 2009 #3
    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
     
  5. Mar 20, 2009 #4
    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 [tex] P_o(x_o,y_o,z_o)[/tex] and a vector normal on the plane [tex]n=<a,b,c>[/tex]

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

    [tex]r=<x,y,z>[/tex]

    while let

    [tex]r_o=<x_o,y_o,z_o>[/tex] be the position vector to the point P_o.

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

    [tex](r-r_o)*n=0[/tex]

    "*" 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:

    [tex]<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[/tex]

    After rearranging the stuff in there we get:

    [tex]ax+by+cz-(ax_o+by_o+cz_o)=0[/tex]

    So,

    [tex]d=-(ax_o+by_o+cz_o)[/tex]
     
  6. Mar 20, 2009 #5
    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
     
  7. Mar 21, 2009 #6

    mathman

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    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 (-dnx,-dny,-dnz)
     
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