Simplify project() Jacobians. - #2

Plane: $\mathbf{n}^\top \mathbf{X} = d$
Camera equations: $\lambda \mathbf{x} = \mathbf{R} \mathbf{X} + \mathbf{t} \iff \mathbf{X} = \lambda \mathbf{R}^\top \mathbf{x} - \mathbf{R}^\top \mathbf{t}$
Solve for $\lambda$:
$\mathbf{n}^\top \left( \lambda \mathbf{R}^\top \mathbf{x} - \mathbf{R}^\top \mathbf{t} \right) = d \iff \lambda = \frac{d + \mathbf{n}^\top \mathbf{R}^\top \mathbf{t}}{ \mathbf{n}^\top \mathbf{R}^\top \mathbf{x}}$

Partial derivatives:

$\frac{\partial \lambda}{\partial d} = \frac{1}{\mathbf{n}^\top \mathbf{R}^\top \mathbf{x}}$

$\frac{\partial \lambda}{\partial \mathbf{n}} = \frac{\left( \mathbf{R}^\top \mathbf{t} \right)^\top \left( \mathbf{n}^\top \mathbf{R}^\top \mathbf{x} \right) - \left( \mathbf{R}^\top \mathbf{x} \right)^\top \left( d + \mathbf{n}^\top \mathbf{R}^\top \mathbf{t} \right)}{\left( \mathbf{n}^\top \mathbf{R}^\top \mathbf{x} \right)^2} = \frac{\mathbf{t}^\top \mathbf{R} - \mathbf{x}^\top \mathbf{R} \lambda}{\mathbf{n}^\top \mathbf{R}^\top \mathbf{x}} = \frac{-\mathbf{X}^\top}{\mathbf{n}^\top \mathbf{R}^\top \mathbf{x}}$