# Visualize manifold specified by equalities [duplicate]

This question already has an answer here:

Suppose I have two nonlinear equalities $$x^3 = y^2, y = z^3$$. How can I visualize the manifold in $$\mathbb{R}^3$$ that is generated by simultaneously satisfying the two equalities? I think ContourPlot3D is the one to use but I couldn't get it to work show the set of points in $$\mathbb{R}^3$$ that satisfy the two equalities. The best I can do is make it show the intersection of the surfaces:

How can I plot the curve defined by the intersection in 3D?

## marked as duplicate by J. M. is away♦Feb 27 at 0:29

You can use the option BoundaryStyle to mark the intersection of the two contour surfaces as follows:

ContourPlot3D[{x^3 == y^2, y == z^3}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},
Mesh -> None, ContourStyle -> Opacity[.3],
BoundaryStyle -> {1 -> None, 2 -> None, {1, 2} -> Directive[Thick, Red]}]


Also

SliceContourPlot3D[y - z^3,  x^3 == y^2, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},
Contours -> {{0}}, BoundaryStyle -> None, ContourShading -> None,
ContourStyle -> Directive[Red, Thick]]


• That's exactly what I needed. I follow most of it: Since a list was passed as first argument the '{1 -> None, 2-> None ... }' but how Mathematica knew to handle {1,2} -> is just magic! – ITA Feb 26 at 23:44
r = 1;
R = ImplicitRegion[{x^3 == y^2, y == z^3}, {{x, -r, r}, {y, -r, r}, {z, -r, r}}];
Region[R]