I need to parametrize the curve of intersection of 2 surfaces, where the surfaces are:
$$x^2+y^2 = z$$ $$z^3 = 5x+y$$
I follow the standard step of substitution as below $$(x^2+y^2)^3 = 5x+y$$ However, this will give me a mass after I open the cube on the left side. Could anyone tell me what is the right approach for this problem?
Best Answer
substitute $x = r\cos(\theta) \ and \ y = r\sin(\theta) $ which yeilds $$ r^3 = 5\cos(\theta)+\sin(\theta) $$ the parametric equation of the curve is $$ \Big(\cos(\theta)\big(5\cos(\theta)+\sin(\theta)\big)^\frac{1}{3},\sin(\theta)\big(5\cos(\theta)+\sin(\theta)\big)^{\frac{1}{3}},\big(5\cos(\theta)+\sin(\theta)\big)^\frac{2}{3}\Big) $$