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I need to convert a plane's equation from Parametric form to Cartesian form. For example: 

(1, 2, -1) + s(1, -2, 3) + t(1, 2, 3)

to:

ax+yb+cz+d=0

So basically, my question is: how do I find the a, b, c and d, and what's the logic behind the conversion.

DMEM
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1 Answers1

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Calculate normal vector to this plane :
N = s x t (vector product of two vectors belonging to plane)
Now you have coefficients a, b, c:

N = (a, b, c)

enter image description here

then substitute base point (in general - any point in the plane)
(1, 2, -1) to equation ax+yb+cz+d=0 

a+2b-c+d=0

and find d

MBo
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  • What is the intuition behind the coefficients of the Cartesian form being equal to the numbers of the normal vector? – Thomas Wagenaar Sep 10 '17 at 12:47
  • @ThomasW the only useful vector in determining the orientation of a plane is the vector ortogonal to it (if you use vectors parallel to it, you need 2) – Caridorc Jan 20 '18 at 18:12