On the Geometries of the Rational Unfoldings of ±Xk


W. Kuyk (in real life, Willem Kuijk - here are a few other of his many works) and Lieven Smits

Bibliographical Reference

Acta Applicandae Mathematicae, volume 19 (1990), pages 77-86.


With a view to applications to self-regulating dynamical processes in biology, we determine the geometric structure of what we call 'isotangent curves', i.e., curves parametrized by the slopes of their points. They come up naturally as bifurcation curves of rational unfoldings of Xk and we classify them according to degree and number of cusps. They, as well as their isotangent involute curves, turn up in simulations of these processes.

AMS Subject Classifications (1985)

52A10, 28A75, 26B15.


There are a few reprints left. If your library does not have this particular journal issue, ask for a reprint by emailing me your postal address. Remove "unwanted" from the address below.


  1. Bröcker, Th. and Lander, L., "Differential Germs and Catastrophes," Cambridge University Press, Cambridge 1975.
  2. Guggenheimer, H.W., "Differential Geometry," McGraw-Hill, New York 1973.
  3. Janssens, J. and Kuyk, W., A Set of Integral-differential Equations for the Ventricular Cardiac Suction Pump, in B. Schmidt (ed.), "Proc. 3rd European Simulation Congress," Erlangen 1990, pp. 567-572.
  4. Kuyk, Willem, The Human Thermostat: a Catastrophe Type Controlling Biological Processes, pp. 713-719 in Vansteenkiste, G.C., Kerckhoffs, E.J.H., Dekker, L. and Zuidervaart, J.C. (eds.), "ESC 86: Proceedings of the Second European Simulation Congress," Antwerp 1986.
  5. Kuyk, Willem, Servocontrol of the Heart Rate in Homeothermic Animals, Bull. Math. Biol. 48 (1986), 107-124.
  6. Kuyk, Willem and Smits, Lieven, Rational Unfoldings of ±Xk and Their Relation with Geometric Equilibrium Theory, Acta Applic. Math. 19 (1990), 87-95.
  7. Poston, T. and Stewart, Ian, "Catastrophe Theory and its Applications," Pitman, London 1978.
  8. Zeeman, E.C., "Catastrophe Theory: Selected Papers 1972-1977," Addison-Wesley, Reading, Mass. 1977.

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