# Tag Info

## Hot answers tagged computational-geometry

1

The curve $z=\log_b(xR+1)$ has $z(0)=0$ and $z(C)=k$ provided $R$ is defined as $$R=\frac{b^k-1}{C}.\tag{1}$$ From a comment it appears you want $B$ to denote the value of $x$ at which $z(x)=1$, and since $z$ is defined as a log base $b$ this means that at $x=B$ the input of the log should be $b$ [since $\log_b(b)=1$]. This gives $BR+1=b$ or ...

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There's a linear-algebraic solution to this problem too. You can take $P$ and $Q$ as vectors, and then construct the matrix $$A=\begin{pmatrix} p_x & q_x \\ p_y & q_y \end{pmatrix},$$ Take the singular value decomposition of $A$, $U\Sigma V^T$, and you'll get the axes as the columns of $U\Sigma$. The reason this works is that $A$ transforms ...

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