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Points of intersection between implicit f(x,y,z)=0 and explicit z=z(x,y) surfaces

f(x,y,z): cos(x)+cos(y)+cos(z)=0 and z(x,y)=x^2-y^2

intersection between implicit cos(x)+cos(y)+cos(z)=0 and explicit z(x,y)=x^2-y^2 surfaces

intersection between implicit cos(x)+cos(y)+cos(z)=0 and explicit z(x,y)=x^2-y^2 surfaces