Question

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?

Answer 1

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

Answer 2

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?

Answer 3

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.

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

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.

Answer 8

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?

Answer 9

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.

Answer 10

ahh.. Thanks you so much! :)