Hi,

I love this paper and I have several small questions.

The equation (42) is

$$p(w_{u,v})=p(w)\left|\frac{dw}{dw_{u,v}}\right|\sin\theta_{dc}$$

Here why $\theta_{dc}$ is used instead of $\theta_{cb}$?

And from equation (27) we know the singularity originates from $\frac{1}{\sin \theta_{cb}}$, which can be cancelled just by the transformation

$$p(\theta_{cb},\phi_{cb})=\sin\theta_{cb} p(w_{cb})$$

It seems $p(\theta_{cb},\phi_{cb})$ can be tabulated directly, though it is less flat than $p(u,v)$. So I am wondering if the warp $w_{u,v}=w(\pi (1-\sqrt{u}),2\pi v)$ is necessary?

Moreover, the equation (46) reads $\left|\frac{du}{dv}\right|=\frac{1}{\sqrt{1+u^2}}$. Since $v=\operatorname{asinh}u$, I think it is $\left|\frac{du}{dv}\right|=\sqrt{1+u^2}$.

These may also apply to the case of bidirectional factorization.

It turns out that I missed something.

Thanks,

Guo