Computing the $z^{-1}$ is very cheap. We can do this by Euclidean algorithm, and in logarithmic time. But you can set $z=1$ in projective coordination. This make computations easy($O(1)$ since $z^{-1}=1$), and with this representation you can see the equality of points evidently.

Also in powerful programs such as MAGMA, $z$ is equal to $1$, by default.

Computing the $z^{-1}$ is very cheap. We can do this by Euclidean algorithm, and in logarithmic time. But you can set $z=1$ in projective coordination. This make computations easy($O(1)$ since $z^{-1}=1$), and with this representation you can see the equality of points evidently.

Note that, instead of computing $y=b*z^{-3}$ you can check $y*z^3=b$. In this state, you don't need to finding multiplicative inverse.

Also in powerful programs such as MAGMA, $z$ is equal to $1$, by default.

Computing the $z^{-1}$ is very simple. We can do this by Euclidean algorithm. But you can set $z=1$ in projective coordination. This make computations easy, and with this representation you can see the equality of points evidently.

Also in powerful programs such as MAGMA, $z$ is equal to $1$, by default.

Computing the $z^{-1}$ is very cheap. We can do this by Euclidean algorithm, and in logarithmic time. But you can set $z=1$ in projective coordination. This make computations easy($O(1)$ since $z^{-1}=1$), and with this representation you can see the equality of points evidently.

Also in powerful programs such as MAGMA, $z$ is equal to $1$, by default.