xEnOnn 4 years ago Is it possible to convert from an explicit equation back to its implicit equation? Suppose I have the following explicit equation: $\left \{ (\frac{1}{2}-\frac{3}{2}t, -\frac{1}{2}+\frac{1}{2}t,t)|t \in \mathbb{R}\right \}$ Possible to change this back to its implicit form?

1. JamesJ

Do you, mean for example, as z = t, then x = 1/2 ( 1 - 3z) y = 1/2 ( z - 1) ?

2. xEnOnn

oh..I meant the above explicit form of the equation was derived from the system of equations (implicit form): $x+y+z=0$$x-y+2z=1$ So after getting the explicit form, can I change the explicit form back to this implicit form?

3. xEnOnn

But how do I find the equations of the planes? Is there a systematic way to do this? Probably a linear algebra way? I tried to solve for its nullspace and all of that but none of the ways lead me back to the implicit form of the 2 equations.

4. JamesJ

Oh wait. No. Think about this for a minute. The number of planes that contain one line in 3-D space is infinite.

5. JamesJ

Therefore there are an infinite number of planes that can intersect to give that plane.

6. JamesJ

Hence once given such a line, there is no unique solution to the question "what are the planes that intersect at that line."

7. JamesJ

So coming back to your question: yes, you can find any number of "implicit forms". But you may not find the original pair of planes/pair of equations.

8. xEnOnn

So it's kind of like a one way thing where I could change from an implicit form to an explicit form. After which, from the explicit form, I cannot change back to the original implicit form any more?

9. JamesJ

You can change back to A implicit form. But it many not be THE implicit form with which you started. So if you like, the operation from implicit forms to explicit forms is not a one-to-one operation.

10. xEnOnn

ahh.. Thanks you so much! :)