The green curve is the family of exact solutions. As Δ sweeps from zero to infinity, the exact solution leaves the origin along −g and bends toward the Newton step. Every trust radius picks one point on this curve.

The dogleg is that curve, approximated by two segments. Watch the red and green dots as you drag Δ. The "dogleg gives up" readout is the price, in model value, of using two straight segments instead of the true curve.

Push Δ past ‖pᴺ‖. Both dots land on the Newton step, the multiplier λ drops to zero, and the constraint goes slack. The trust region only matters when it is actually in the way.