Final Answer:
Fitting a conic through points P₁ (x₁, y₁),…,Pₘ(xₘ, yₘ) amounts to finding the kernel of an m×6 matrix A.
Step-by-step explanation:
When fitting a conic through points P₁ to Pₘ, each point represents an equation of the form Ax² + Bxy + Cy² + Dx + Ey + F = 0, where (x, y) satisfy the conic equation. This translates to a system of linear equations for each point, forming a matrix A with m rows (one for each point) and 6 columns (coefficients A, B, C, D, E, F).
The i-th row of matrix A corresponds to the coefficients of the conic equation for the i-th point, constructed as [xᵢ², xᵢyᵢ, yᵢ², xᵢ, yᵢ, 1]. To fit a conic, we solve the system of equations represented by this matrix, and finding its kernel provides a one-dimensional subspace that defines a unique conic. The kernel represents the set of coefficients for the equation of the conic that satisfies all given points simultaneously.
This process transforms the problem of fitting a conic through multiple points into a linear algebra problem, where the kernel of the matrix A encapsulates the solutions for the coefficients of the conic equation. By finding the null space (kernel) of the matrix A, we obtain the coefficients that define the conic that passes through the given set of points. This approach allows for a systematic and mathematical way to determine the equation of a conic that best fits a given dataset of points, providing a unique solution within a one-dimensional subspace of the kernel.