From what I understand, projective plane curves $F$ of degree $d$ which pass through a point $P\in\mathbb{P}_K^{2}$ with multiplicity $\geq m$ form a linear system of projective curves in $\mathbb{P}_K^{D}$, where $D=\frac{1}{2}d(d+3)$ and it has co-dimension $\frac{1}{2}m(m+1)$. That is curves of degree $d$ passing through $P$ form a hyperplane in $\mathbb{P}_K^{D}$.
I would like to however come up with a concrete example of such a hyperplane that I can produce in MATLAB. I understand the intuition but I think such an example would help me visualise how such linear systems form hyperplanes.
Note that a plane curve of degree $d$ is given by the zeros of a degree $d$ polynomial $$ F(x,y,z) = \sum_{i+j+k=d} a_{ijk}x^iy^jz^k $$ which gives a vector $v = (a_{ijk})\in K^{\frac{(d+2)(d+1)}{2}}$. Since F is defined up to a scalar multiple $v$ is better suited as a point in $\mathbb{P}_K^{\frac{(d+2)(d+1)}{2}-1}= \mathbb{P}_K^{\frac{d(d+3)}{2}}$. Now note that, for a fixed $P$, $F(P)$ is a linear combination of the $a_{ijk}$ with coefficients in $K$. This linear expression defines a hyperplane $H$ in $\mathbb{P}_K^{\frac{d(d+3)}{2}}$ with the property that $F(P)=0$ if and only if $v\in H$.
To compute it explicitly just write a polynomial $F$ with arbitrary coefficients, choose a point $P$ and compute $F(P)$.
If we choose $P=(0:0:1)$ and write $$F(x,y,z) = A(x,y)_d + A_{d-1}(x,y)z + \cdots+ A_1(x,y)z^{d-1}+A_0z^d$$ we can see how the other hyperplanes appear as the multiplicity grows.