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.