[?] CountourPlot of (Cos(2x))^2 ==0 ?

Questions
,
1

Hello,

I am trying to plot the countour curve of (cos(2x))^2 ==0 on the xy-plane for {x, -pi/2, pi/2} and {y,0,2pi}, but the results is an empty plot in mathemaitca. However, I obtained the correct plot for cos(2x)==0.

I used the following command lines:

ContourPlot[Cos[2x]^2 ==0, {x, -Pi/2, Pi/2}, {y, 0, 2Pi}]

How can I get the correct graph for this?

Thank you,

Bhagya

2

It’s interesting that the graph may be obtain if the right part is not exactly 0:

ContourPlot[
 Cos[2 x]^2 == 0.000000000000000001, {x, -\[Pi]/2, \[Pi]/2}, {y, 0, 
  2 \[Pi]}]

3722g.jpg
3722g.jpg387×375 13.9 KB

3

The numerical algorithms of ContourPlot assume nonzero derivative, or transversality, otherwise they may fail spectacularly:

ContourPlot[x^2 == 0, {x, -1, 1}, {y, 0, 1}]

In your case you get a correct plot by symbolic pre-processing:

ContourPlot[
 Evaluate[List @@ 
   Reduce[Cos[2*x]^2 == 0 && -Pi/2 < x < Pi/2, x, Reals]],
 {x, -Pi/2, Pi/2}, {y, 0, 2 Pi}]
4

Apropos, dear Gianluca,

How to interpret the following?

ContourPlot[x^2 == .0001, {x, -1, 1}, {y, -1, 1}]

8043g1.jpg
8043g1.jpg374×360 11.4 KB

ContourPlot[x^2 == .0001, {x, -1, 1}, {y, -1, 1}, PlotRange -> All]

g2.jpg
g2.jpg366×362 27.5 KB

5

Apropos, dear Gianluca,

How to interpret the following?

ContourPlot[x^2 == .0001, {x, -1, 1}, {y, -1, 1}]

8043g1.jpg
8043g1.jpg374×360 11.4 KB

ContourPlot[x^2 == .0001, {x, -1, 1}, {y, -1, 1}, PlotRange -> All]

g2.jpg
g2.jpg366×362 27.5 KB

6

There are curious discretization artifacts:

ContourPlot[x^2 == 1/10000, {x, -1, 1}, {y, -1, 1}, 
 PlotRange -> {{.0022, .01}, {-.3, .3}}]
ContourPlot[10000 x^2 == 1, {x, -1, 1}, {y, -1, 1}, 
 PlotRange -> {{.0022, .01}, {-.3, .3}}]

Also, in the current Mathematica version the tooltip looks like a job left incomplete.

7

Really curious… You are right!

8

And one more additional curiosity:

ContourPlot[x^2 == 1/10000, {x, -1, 1}, {y, -1, 1}, PlotRange -> {{.0022, .01}, {-.3, .3}}, PlotPoints -> 30]
ContourPlot[10000 x^2 == 1, {x, -1, 1}, {y, -1, 1}, PlotRange -> {{.0022, .01}, {-.3, .3}}, PlotPoints -> 30]

g3.jpg
g3.jpg273×542 17.7 KB

Why are not identical the results?