# Plot intersection (elliptic curve) of two quadrics

Consider a $$3d$$ system with variables $$x,\,y,\,z$$ and two surfaces defined by the functions

$$\qquad f(x,y)=0$$ and $$g(x,y,z)=0$$.

I am interested in the intersection of these two surfaces giving an elliptic curve. That is, I would like to plot $$f(x,y)=0$$ under the condition that $$0=g(x,y,z)$$ (or equivalently the other way around). I've tried with ContourPlot3d the following

ContourPlot3D[
{f(x, y) == 0, 0 == g(x, y, z)}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1},
AxesLabel -> {"x", "y", "z"},
Contours -> {0},
ContourStyle -> Opacity[0.0],
Mesh -> None,
BoundaryStyle -> {1 -> None, 2 -> None, {1, 2} -> {{Thick, Green}}},
PerformanceGoal -> "Quality"]


However, I wonder if there is a better way and also if one could go around the BoundaryStyle-option by some '&&' enforcing the simultaneous plotting/solution of both equations.

All remarks are welcome!

### Edit

After a remark the second equation was correctly written down.*

The functions are given as

$$\qquad f(x,y)=4 x y - \left(\frac{660}{4x+4y-45}+15-2x\right)^2=0$$

and

$$\qquad g(x,y,z)=\left(\frac{660}{2x +z-15}+45- 4x -4y\right)=0$$.

It would also be great to know, how to extract the algebraic form of the curve, perhaps also in parameterized form.

• Could you define $f(x,y)$ and $g(x,y)$? Jan 8, 2021 at 21:03
• Surely you can't have used {f(x,y)==0,z==g(x,y)} exactly, since that is not correct Mathematica syntax. So what was your expression, and the definitions of f and g? Jan 8, 2021 at 22:02
• If the answers to the linked question do not suit your needs,then edit your question to explain why. Jan 8, 2021 at 22:26
• Thanks, the question was edited. @J. M., I've edited the question. Apart from plotting a nice result it would be great to know how to extract the algebraic form of the curve itself. Jan 8, 2021 at 22:38
• I am not certain what you are expecting as the algebraic form, since the two algebraic equations taken together already (implicitly) represent your curve. If it's a parametrization, that can be nontrivial, but one of the answers in the linked question talks about how to use an ODE formulation to numerically compute one. Jan 8, 2021 at 22:40

You may get the intersections parametrized by e.g. x:

res = Solve[{f[x, y] == 0, g[x, y, z] == 0}, {y, z},Reals];


This gives 3 lengthy solutions. E.g. the first one:  Found Daniel's solution interesting and wanted to extend it just a bit to plot the result. Only the first 2 solutions I found were needed. Notice I had to draw two plots and hard-code the limit 0<x<0.12949. Might be a challenge to extract the limits directly and express the solution as a single, perhaps conditional function, for a single parameter say

$$h(t)=\{t,y(t),z(t)\}; \quad 0<=t<=1$$

Ran into problems doing this though so left as is.

f[x_, y_] := 4 x y - (660/(4 x + 4 y - 45) + 15 - 2 x)^2;
g[x_, y_, z_] := (660/(2 x + z - 15) + 45 - 4 x - 4 y);

theSols = NSolve[{f[x, y] == 0, g[x, y, z] == 0}, {y, z}, Reals];

myFun1[x_, y_, z_] = Flatten[{x, {y, z} /. theSols[]}];
pp1 = ParametricPlot3D[myFun1[x, y, z], {x, 0, 0.12949},
PlotStyle -> Blue];
myFun2[x_, y_, z_] = Flatten[{x, ({y, z} /. theSols[])}];
pp2 = ParametricPlot3D[myFun2[x, y, z], {x, 0, 0.12949},
PlotStyle -> Red];
Show[{pp1, pp2}, BoxRatios -> {1, 1, 1}, AxesLabel -> {"x", "y", "z"}] Edit

Region work in this simple situation.

f[x_, y_] = 4 x*y - (660/(4 x + 4 y - 45) + 15 - 2 x)^2;
g[x_, y_, z_] = 660/(2 x + z - 15) + 45 - 4 x - 4 y;
Region[ImplicitRegion[{f[x, y] == 0 ,
g[x, y, z] == 0}, {{x, -1, 1}, {y, -1, 1}, {z, -1, 1}}],
Axes -> True, Boxed -> True, BaseStyle -> {Thick, Green}] Original

BoundaryStyle -> {1 -> None, 2 -> None, {1, 2} -> {{Thick, Green}}} to find intersection of two surfaces is too slow.

Here we just draw one of the implicit surface and add Mesh according to the second condition.

f[x_, y_] = 4 x*y - (660/(4 x + 4 y - 45) + 15 - 2 x)^2;
g[x_, y_, z_] = 660/(2 x + z - 15) + 45 - 4 x - 4 y;
ContourPlot3D[{g[x, y, z] == 0}, {x, -1, 1}, {y, -1, 1}, {z, -1,
1}, AxesLabel -> {"x", "y", "z"}, Mesh -> {{0}},
MeshFunctions -> Function[{x, y, z}, f[x, y]],
MeshStyle -> {Thick, Green}, ContourStyle -> Opacity[0.02],
BoundaryStyle -> None] Or

f[x_, y_] = 4 x*y - (660/(4 x + 4 y - 45) + 15 - 2 x)^2;
g[x_, y_, z_] = 660/(2 x + z - 15) + 45 - 4 x - 4 y;
ContourPlot3D[{f[x, y] == 0}, {x, -1, 1}, {y, -1, 1}, {z, -1,
1}, AxesLabel -> {"x", "y", "z"}, Mesh -> {{0}},
MeshFunctions -> Function[{x, y, z}, g[x, y, z]], MeshStyle -> {Thick, Green}, ContourStyle -> Opacity[0.02], BoundaryStyle -> None]