Consider a line l in the image, given by parameters (a, b, c), in the image coordinate system.
We know that the corresponding 3-D line casting this image lies in a plane. Derive the equation of this plane in the camera coordinate system. You may assume that the intrinsic calibration matrix, K, is given

To derive the equation of the plane in the camera coordinate system, we follow these steps:

### 1. **Image Line in 2D**:
The equation of the line \( l \) in the image plane is given by:

\[
a x + b y + c = 0
\]
where \( (x, y) \) are the pixel coordinates in the image.

### 2. **Projection of 3D Points to the Image Plane**:
In 3D, let a point \( P = (X_c, Y_c, Z_c)^T \) be a point in the camera coordinate system, and its corresponding projection onto the image plane is \( p = (u, v)^T \), where \( u \) and \( v \) are the pixel coordinates.

The relationship between the 3D point and its projection is given by the camera projection equation:

\[
\begin{bmatrix} u \\ v \\ 1 \end{bmatrix} = \frac{1}{Z_c} \mathbf{K} \begin{bmatrix} X_c \\ Y_c \\ Z_c \end{bmatrix}
\]
where \( \mathbf{K} \) is the intrinsic calibration matrix, typically written as:

\[
\mathbf{K} = \begin{bmatrix} 
f_x & 0 & c_x \\
0 & f_y & c_y \\
0 & 0 & 1
\end{bmatrix}
\]
with \( f_x \) and \( f_y \) representing the focal lengths, and \( (c_x, c_y) \) being the principal point coordinates.

Thus, the pixel coordinates \( (u, v) \) are related to the 3D coordinates \( (X_c, Y_c, Z_c) \) by:

\[
u = \frac{f_x X_c}{Z_c} + c_x
\]
\[
v = \frac{f_y Y_c}{Z_c} + c_y
\]

### 3. **The Line in Homogeneous Coordinates**:
We represent the line in homogeneous coordinates as \( \mathbf{l} = (a, b, c)^T \), which represents the image line in the form:

\[
a u + b v + c = 0
\]

Substituting the expressions for \( u \) and \( v \) from the camera projection equation, we get:

\[
a \left( \frac{f_x X_c}{Z_c} + c_x \right) + b \left( \frac{f_y Y_c}{Z_c} + c_y \right) + c = 0
\]

Simplifying, we obtain:

\[
\frac{a f_x X_c}{Z_c} + \frac{b f_y Y_c}{Z_c} + a c_x + b c_y + c = 0
\]

Multiplying by \( Z_c \) to eliminate the denominator:

\[
a f_x X_c + b f_y Y_c + (a c_x + b c_y + c) Z_c = 0
\]

### 4. **Equation of the Plane in the Camera Coordinate System**:
The equation derived above is linear in \( X_c \), \( Y_c \), and \( Z_c \). Therefore, it represents the equation of a plane in the camera coordinate system. We can express the plane equation as:

\[
a f_x X_c + b f_y Y_c + (a c_x + b c_y + c) Z_c = 0
\]

or more compactly:

\[
A X_c + B Y_c + C Z_c = 0
\]
where:

- \( A = a f_x \)
- \( B = b f_y \)
- \( C = a c_x + b c_y + c \)

This is the equation of the plane in the camera coordinate system, where the 3D line lies.

---

### Do you want more details or have any questions?

### Related questions:
1. How is the intrinsic calibration matrix \( \mathbf{K} \) typically obtained?
2. How does distortion in the camera affect the projection of 3D points?
3. What changes if the line in the image is a horizon line?
4. How does the focal length affect the shape of the plane in 3D?
5. What are the implications if the camera is rotated with respect to the plane?

### Tip:
When working with camera projections, always ensure that the intrinsic matrix \( K \) is accurately calibrated to avoid significant errors in 3D reconstructions.

Consider a line l in the image, given by parameters (a, b, c), in the image coordinate system. We know that the corresponding 3-D line casting this image lies in a plane. Derive the equation of this plane in the camera coordinate system. You may assume that the intrinsic calibration matrix, K, is given.

This solution explores how to derive the equation of a plane in the camera coordinate system from a 2D line given in image coordinates. Using concepts like projective geometry and the camera projection theorem, the solution walks through the projection of 3D points and camera calibration. Ideal for students in engineering and computer vision fields.

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