circles with different radius and center

  • With the green point M=O we get a semicircle.
  • If we drag M rightwards on the segment AB, we get a pointed arch.
But if we drag M anywhere in the blue rectangle we can make a construction that combines circles with a different radius and center in one arch:
  • With P as a point on the segment AB construct a circle with radius AP and center P.
  • Draw the line MP and create the intersection point N with the circle.
  • Now create the circular arc AN with center P and the circular arc NO' with center M.
Mirroring the arcs the result is an arch that combines in its shape two circles with different radius and center. Experiment with dragging P and M and examine their influence on the shape of th arch.
The mathematical logic of this construction with different curvatures is simple. 3 points have to be collinear: the transition point of the two circular arcs and their two centers. Because if so:
  • The radii of the two cricular arcs may be different, but their direction is coinciding.
  • The tangent of a circle in a point on that circle is perpendicular to the radius.
  • conclusion: the tangents in point N of the two parts are the same and thus their transition is smooth.