Find the image of the line 2x + y − 3z = 0 under the projective transformation

t1 defined by:

t1 : [x, y, z] → [x + z, x + y + 3z, −2x + z].

To find the image of the line \( 2x + y - 3z = 0 \) under the projective transformation defined by \( t_1: [x, y, z] \to [x + z, x + y + 3z, -2x + z] \), we need to follow these steps:

### Step 1: Represent the line equation in parametric form
Given the line equation \( 2x + y - 3z = 0 \), we can express it in a parametric form. Let's parametrize it as follows:

\[
\begin{aligned}
x &= t \\
y &= 3t - 2s \\
z &= s
\end{aligned}
\]

Here, \( t \) and \( s \) are parameters.

### Step 2: Apply the projective transformation to the parametric equation
Given the transformation:

\[
t_1: [x, y, z] \to [x + z, x + y + 3z, -2x + z]
\]

Substitute the parametric equations into the transformation:

\[
\begin{aligned}
x' &= x + z = t + s, \\
y' &= x + y + 3z = t + (3t - 2s) + 3s = 4t + s, \\
z' &= -2x + z = -2t + s.
\end{aligned}
\]

Thus, the transformed line is represented parametrically as:

\[
\begin{aligned}
x' &= t + s, \\
y' &= 4t + s, \\
z' &= -2t + s.
\end{aligned}
\]

### Step 3: Convert back to implicit form
To find the implicit form of the line, eliminate the parameters \( t \) and \( s \).

From the parametric equations:
\[
\begin{aligned}
t &= x' - s, \\
4t &= y' - s, \\
-2t &= z' - s.
\end{aligned}
\]

Thus:
\[
\begin{aligned}
4(x' - s) &= y' - s, \\
-2(x' - s) &= z' - s.
\end{aligned}
\]

Expanding and rearranging:
\[
\begin{aligned}
4x' - y' &= 3s, \\
-2x' - z' &= -3s.
\end{aligned}
\]

Finally, eliminating \( s \) by solving these equations together, we get the implicit equation of the image of the line:

\[
4x' - y' = 2(-2x' - z').
\]

This simplifies to:

\[
8x' + 2z' - 4x' - y' = 0 \quad \Rightarrow \quad 8x' + 2z' = y'.
\]

This is the equation of the transformed line in the new coordinates.

Would you like more details, or do you have any questions? Here are some related questions you might find useful:

1. How do projective transformations differ from affine transformations?
2. What are the properties of projective transformations in geometry?
3. Can we represent the transformation \( t_1 \) as a matrix?
4. How do you find the inverse of a projective transformation?
5. How do projective transformations affect conic sections?
6. What is the significance of homogeneous coordinates in projective geometry?
7. Can projective transformations map parallel lines to intersecting ones?
8. How do you determine if a transformation is a collineation?

**Tip:** Projective transformations preserve the cross-ratio of four points, a key invariant used in many geometric applications.

Explore how to find the image of the line 2x + y - 3z = 0 under a projective transformation defined by t1: [x, y, z] → [x + z, x + y + 3z, -2x + z]. Understand the parametric representation, application of transformation, and derivation of the implicit equation.

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