Pablo Baldivieso
Assignment 6

Problem 1

The parametric graph x = a cos(t), y = a sin(t). We set a parameter n which ranges from 0 to 20 with increments equal to 1 and the parameter s which ranges from 0 to 1 with increments 0.1.  The curve is created using command <curve>. The curve can be evaluated at a particular point, let's say c(i). Then a sequence of  segments connecting points on the curve, c(i) to c(2*i).




If we include the polar curve r(t) = 1+cos(t) we can see that the envelopes follow the curve r(t).




Now we change the envelopes lines to be lines from c(i) to c(k*i) for k= 1,2,.. 10. and the corresponding r(t)=a/2 + a/2cos((k-1)t). The envelopes and the graphs have the same shapes.









Problem 2

For the ellipse case, if the free point is located either inside the circle, or outside the cirlce the shape of the envelopes curves are different. For example when the freepoint is located outsised circle we have Figure 1 and figure 4, which are known as Limacon. however when the free point is inside the circle we have Figure 2 and Figure 3.



                                                  Figure 1

                                                              Figure 2

                                                                                      Figure 3





                                                                            Figure 4


For the hyperbola the special case is when the free point is located at equidistance from the foci the envelopes leave two holes creating a symbol similar to the infinity symbol, see Figure 6.



                                                Figure 5





                                                    Figure 6