Consider the space of all plane curves of degree $d$ in $\mathbb{P}^2$. This is a space of dimension $\frac{(d+1)(d+2)}{2} -1$. The subspace of all those curves which pass through a given point in $\mathbb{P}^2$ has dimension one less. That is, the condition that the curve has to pass through a given point, cuts down the dimension by one.
I am interested in the space of curves of degree $d$ which are tangential to another given curve say $C$ at a point. What will be the dimension of such curves. Will this condition cut down the dimension by two?