Intersection points of Taylor polynomial and trigonometric function

Question

I seem to be unable to find intersection points between following functions, and plot both functions and points on the same plot:

a[x_, y_] := Cos[x^2 + y^2];

d[x_, y_] := Evaluate[Series[Cos[x^2 + y^2], {x, 0, 4}, {y, 0, 4}] // Normal];

This is what I've tried so far

ContourPlot3D[{a == 0, d == 0}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, MeshFunctions -> {Function[{x, y, z, f}, a - d]}, MeshStyle -> {{Thick, Black}}, Mesh -> {{0}}, 
ContourStyle -> Directive[Orange, Opacity[0.5], Specularity[White, 30]]]

Plotting them with Plot3D:

Plot3D[{a[x, y], d[x, y]}, {x, -Pi, Pi}, {y, -Pi, Pi}]

seems to give rather different results.

OK, but should that be a[x, y] == 0, etc., in the contour plot, then? And should b and g be changed to a and d? — Michael E2, May 23 '17 at 15:25
@J. M. Thanks for link. I've looked at it, but it it deals with 2D, while I am looking for 3D implementation. @Michael E2 My mistake, changed to a and d. — Cauchy, May 23 '17 at 16:44
Your description of the problem sounds like you're solving a[x, y] == d[x, y] (not like J.M.'s link), but the ContourPlot3D code suggests you might be solving the system {a[x, y] == 0, d[x, y] == 0} (exactly like J.M.'s link). Which is it? — Michael E2, May 23 '17 at 16:48
So, you want the curves of intersection and not points? Have you seen this? — J. M.'s missing motivation, May 23 '17 at 16:49
@J.M. You keep beating me to the question I want to link! :) — Michael E2, May 23 '17 at 16:52
@Michael, it helps a lot when they're problems I've tangled with before. ;) — J. M.'s missing motivation, May 23 '17 at 16:57
@J. M. Checked both provided link, but still cannot get it. Using ContourPlot3D I get nothing even close to intersection of those two plots. Used simple Plot3D to check how it looks like, and my ContourPlot3D is far from this. — Cauchy, May 23 '17 at 19:54

score 3 · Accepted Answer · answered May 25 '17 at 00:08

Here is a brute force approach.

a[x_, y_] := Cos[x^2 + y^2]
d[x_, y_] = Series[Cos[x^2 + y^2], {x, 0, 4}, {y, 0, 4}] // Normal;

Plot3D[{a[x, y], d[x, y]}, {x, -Pi, Pi}, {y, -Pi, Pi},
  PlotRange -> 2,
  ClippingStyle -> None]

mesh = Catenate[CoordinateBoundingBoxArray[{{-π, -π}, {π, π}}, .0025]];
xyPts = Select[mesh, Norm[#] > .75 && Abs[a @@ # - d @@ #] < .01 &];
xyzPts = {Sequence @@ #, a @@ #} & /@ xyPts;
ListPointPlot3D[xyzPts, PlotRange -> All]

Notes

The evaluation of the expression for xyPts is rather slow.
The excluded the points having a norm less than .75 to speed up the plotting. The empty circle in the center of the plot would otherwise be filled.

Intersection points of Taylor polynomial and trigonometric function

1 Answers1