A Toy Geometric Model of the "Blurred Point": From Circle Tangency to the One-Point Compactification of the Plane

We present an elementary geometric construction illustrating the idea that the Planck length lp sets a lower bound on the distinguishability of geometric objects. The starting point is the classical problem of two tangent circles of radius R: if points separated by a gap smaller than lp are considered physically indistinguishable, the ideal tangency point turns into a finite “fusion” segment. We trace the consequences of this construction: the limiting transition to a straight line as R→∞ (without ever reaching actual infinity), the construction of a Cartesian coordinate system from four such circles with a “blurred” origin, and finally the connection to the classical onepoint compactification of the plane and of space. The model does not claim the status of a physical theory and is not derived from the dynamical equations of any candidate quantum-gravity theory; its value lies in reproducing, by elementary means, a qualitative structure familiar from existing “minimal length”.