A smooth manifold M comes equipped with a tangent space at each point on the manifold. If M is embedded in Euclidean space, then the tangent space to M at a point x ∈ M may be identified with (a translate of) the affine subspace of the ambient Euclidean space best approximating M at x.

Of course, there are more formal definitions of the tangent space. For example, if M is described near x as the locus where a sequence of smooth functions f₁, …, f_n-m of n variables vanish (where m = dim(M)), then the tangent space to M at x may be identified with the kernel of the Jacobian matrix (∂ f_i/∂ X_j).

For Zariski geometries, the notion of the tangent space itself is not available, but one can make sense of tangency. Roughly speaking, given a family { C_b }_{b ∈ B} of plane curves on strongly minimal Zariski geometry X all passing through a sufficiently generic point (x,y), we say that C_b and C_b' are tangent if C_b ∩ C_b' is smaller than expected.

From the family of lines { Γ_λ | λ ∈ K } (where Γ_&lambda = { (x, λ x) | x ∈ K}) on the affine plane over the field K, one can recover multiplication on K via composition of the curves in this family.

In general, the composition of two curves in some family of curves will not produce another curve in that family. However, if the family is sufficiently large and well-prepared (for instance, one might want to look at curves of the form C_b • C_b'^-1 to ensure that they pass through a point of the form (x,x)), the compositions will be tangent to curves in the orginal family. In this way, one may recover the algebraic operations on the field from families of curves.