Is there a geometric way to understand the group law on an elliptic curve over a finite field?

Question

For elliptic curves over $\mathbb{R}$ or (some over)$\mathbb{C}$ or even $\mathbb{Q}$(for $\mathbb{Q}$, one can act like the points are not discrete and imitate the method for $\mathbb{R}$), one could use the chord tangent method to add two points on curve. However, for elliptic curves over finite fields, there's no apparent 'curve' like structure and it doesn't seem like the chord tangent method can be used there to add points. I can't find a pattern for how points are added. $x^3+x+1$ over $\mathbb{F}_{13}$." />