Why Inverse Kinematics Can Have Zero, One, or Two Solutions
Forward kinematics (see the Forward Kinematics 2R Arm Simulator) answers "given these joint angles, where does the arm end up" — a straightforward, always-solvable calculation. Inverse kinematics asks the opposite, harder question: "given a target position, what joint angles get the arm there" — and for a simple 2-joint planar arm, this can have zero, one, or two valid solutions depending on the target's distance from the base.
If the target is farther away than the arm's total reach (L1 + L2) or closer than the difference between the two link lengths, no solution exists at all — the position is physically unreachable regardless of joint angles. Within the reachable range, there are typically exactly two valid configurations, commonly called "elbow up" and "elbow down" — both place the end-effector at the identical target coordinate, just via a mirrored joint-angle configuration, and a real robot arm needs an explicit rule (or planning logic) for choosing between them, since the inverse kinematics math alone doesn't prefer one over the other.
This ambiguity (multiple valid solutions, or none at all) is exactly why inverse kinematics is a genuinely harder problem than forward kinematics, and why more complex arms with additional joints (redundant manipulators) can have infinitely many valid solutions for a single target position, requiring an optimization approach rather than a closed-form formula.