A curve(?) where every tangent intersects it at distance 1

Roy Wright
James Tanton asked: A curve has the property that its tangent line at any point intersects the curve at a unit distance. Must the curve be a straight line? Consider the set of points that satisfy for some positive whole number . If we can consider this set of points to be a "curve," then it has the required property. The set of points is shown in green below. The tangent line can be moved by dragging the red point.
Do you have another answer? A more elegant or otherwise better one? (Follow-up in progress: http://tube.geogebra.org/m/1735925)