As we have already observed, there is a whole family of curves called limaçons. A limaçon is best describe by a polar equation of the form:
This places the curve in the position shown at right. We can simplify this equation by scaling the coordinates suitably, to get
and we shall use this form here. When It is not helpful to try to find the Cartesian equation of the general limaçon, but we can find the Cartesian coordinates of a point The members of the limaçon family have one particular point of interest on the curve. In the case of the cardioid we call it the Click on the figure at right to open the applet. You can operate the applet by clicking the ‘Animate‘ button, or by manually dragging the point Consider the limaçon shapes for different positions of the sliding point Notice the values of Experiment with the applet linked at right. We know that when the slider point There seems to be no interesting new tangent behaviour, apart from the fact that near the ends of the
http://homepage.mac.com/paulscott.info/trisectrix/
The little exercise above gives us some insight into how mathematical research is done. Often a new result is obtained by varying the conditions of an existing problem. Here I was looking for some alternative behaviour of the three tangents, and just happened to notice this (supposedly) new result. From the applet the result appears to be true for all positions of the generating point
Play with the applet linked at right. Investigate the length of the chord through the cusp / cross-over point, and also the locus of the midpoint The general equation of the limaçon is Length = ( Again, for the cardioid, the locus of the midpoint is the circle centre (1/2, 0) and radius 1/2. I was really quite surprised to find that this appears to remain true for all limaçons in the family. Let’s see if we can prove it. In fact, the proof is quite easy. In the diagram, let the blue circle meet the positive So we have proved the
Check out the applet at right. You can show or hide the locus of When the curve is a cardioid, the tangent at the cusp is horizontal and meets the other vertical tangent at (2, 0). In all other cases the tangents at these two points are both vertical and meet ’at infinity’. It is interesting to investigate the different loci of
New trisectrix result: http://homepage.mac.com/paulscott.info/trisectrix/ Wikipedia : http://en.wikipedia.org/wiki/Limaçon Wolfram MathWorld: http://mathworld.wolfram.com/Limacon.html |