nonlinear interpolation question

Question

Hi all, here is an screenshot from book called rotation transforms for computer graphics by John Vince. I think here is an error ,if it doesn't please explain how do m+n equals 1 Thank you

Answer 1

To add to my comment.
Draw the line passing through the tip of $\mathbf v$ and parallel to $\mathbf v_2$ and find $\mathbf v$ as the sum of two vectors parallel to $\mathbf v_1$ and $\mathbf v_2$ respectively, so $$\mathbf v=a\mathbf v_1 + b\mathbf v_2$$ If $\mathbf v_2$ is separated from $\mathbf v_1$ by $\theta$ and $\mathbf v$ is separated from $\mathbf v_1$ by $\varphi$, the sine law gives $$\frac 1{\sin \theta}=\frac a{\sin(\theta-\varphi)}=\frac b{\sin \varphi}$$ and then $a$ and $b$ .

Search by spherical interpolation or slerp .