The image you uploaded displays the least squares method in the context of solving for a circle. The equation provided is in the form:

$\mathcal{L}_{\text{least squares}} = \sum_i w_i \|d_i\|^2$

Where:

• $w_i$ represents weights,
• $d_i$ is the residual, and
• $\|d_i\|^2$ is the squared magnitude of the residual vector.

For solving for a circle using the given points $(2, 0, 1), (0, 1, 1), (-1, 0, 1), (0, -1, 1)$, we need to apply the method of least squares to find the best-fitting circle to these points.

Step-by-step Approach to Solve Numerically:

1. General Equation of a Circle in 3D:

The general equation for a circle in 3D space, given by:

$(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$

Where:

• $(h, k, l)$ is the center of the circle,
• $r$ is the radius.

2. Residual Vector:

For each point $(x_i, y_i, z_i)$, the residual $d_i$ is the difference between the actual distance from the point to the center $(h, k, l)$ and the desired radius $r$.

$d_i = \sqrt{(x_i - h)^2 + (y_i - k)^2 + (z_i - l)^2} - r$

3. Weighting:

Each residual $d_i$ can be weighted by a factor $w_i$ based on the importance of each point.

The least squares function becomes:

$\mathcal{L} = \sum_i w_i \left( \sqrt{(x_i - h)^2 + (y_i - k)^2 + (z_i - l)^2} - r \right)^2$

4. Minimization:

To find the optimal values for $h$, $k$, $l$, and $r$, we would minimize the least squares error function with respect to these parameters. This involves setting the partial derivatives of $\mathcal{L}$ with respect to each parameter equal to zero and solving the system of equations.

Numerical Setup:

For this set of points $(2, 0, 1), (0, 1, 1), (-1, 0, 1), (0, -1, 1)$, we will assume that the weights $w_i$ are equal (if not specified otherwise), and the function can be minimized using calculus.

1. Compute the distance from the center to each point.
2. Square the difference between the distance and the radius for each point.
3. Sum up the squared residuals weighted by the weights $w_i$.
4. Apply minimization techniques (often via gradient descent or solving the system of equations) to find the optimal values of $h$, $k$, $l$, and $r$.

Conclusion:

This is a conceptual overview of how the least squares method is applied to fit a circle in 3D space using the given points. To find the specific circle parameters numerically, you would typically need to solve the resulting system of equations through techniques like Newton's method or other optimization algorithms.

Would you like more detail on any specific part of the solution or any further steps in the process?