Finding x-intercepts of epicycloid

Questions
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1

I am plotting an epicycloid and trying to find the x values of where the plot intercepts the x-axis. Here’s what I’ve done:

pic.jpg
pic.jpg650×659 138 KB

When I solve (setting y = 0), I get an output that I don’t know how to interpret:
pij.jpg
pij.jpg776×415 126 KB

2

Hi Bryan,
Try FindInstance.
Best,
David

PS. If you put your code in a spikey box we can copy/paste from it.

(Remove semicolons and evaluate)

xcoordinate[a_, b_, t_] := (a + b) Cos[t] - b Cos[(a + b)/b t];

ycoordinate[a_, b_, t_] := (a + b) Sin[t] - b Sin[(a + b)/b t];

plot1 = ParametricPlot[{xcoordinate[2/5, 46/35, t], 
    ycoordinate[2/5, 46/35, t]}, {t, 0, 46 \[Pi]}];

(* Find t for up to 1000 instances *)
tValues = 
  t /. FindInstance[
     ycoordinate[2/5, 46/35, t] == 0 && 0 <= t <= 46 \[Pi], {t}, 
     Reals, 1000] // N;

(* Make points at the intercept coordinates *)
points = {Red, 
   Point[{xcoordinate[2/5, 46/35, #], ycoordinate[2/5, 46/35, #]}] & /@
     tValues};

Show[plot1, Epilog -> points];

plt.png
plt.png360×360 43.1 KB

3

You can also use NSolve, like this, with David Keith’s setup:

NSolve[ycoordinate[2/5, 46/35, t] == 0 && 0 <= t <= 46 \[Pi], {t}, Reals]
4

thank you both very much! i will spikey in the future

5

You’re welcome, Bryan.
BTW, I like Michaels’s use of NSolve better than my FindInstance. It seems more likely that Mathematica is commited to finding all the solutions.