<s>[INST] homogeneous coordinates I am to use homogeneous coordinates to calculate a standard matrix for a projection onto the line $4x-2y=6$ from the point $(3,10)$.
I'm not sure what homogeneous coordinates are and neither how to use them to calculate my problem? Please believe me, I've been looking for information on various sources!
 [/INST] In 2D points in homogeneous coordinates have the form $P = (x,y,1)$ and lines $L=(a,b,c)$ such that the equation for the line can be found by
$$ L \cdot P = 0 \} a x + b y + c = 0 $$
So in your case, the homogeneous coordinates for the point is $P=(3,10,1)$ and the line $L=(4,-2,-6)$.
The trick with homogeneous coordinates is that if you multiply them with a scalar value, it does not change the underlying geometry. This is powerful be because the coordinates of a point $P=(a,b,c)$ are $(x,y) = (\frac{a}{c},\frac{b}{c})$. To test if a point belongs to a line simply do the dot product $P\cdot L=0$ and check for zero. The coordinates of a line joining two points $P$, $Q$ are
$$\begin{pmatrix} p_x\\p_y\\1 \end{pmatrix} \times \begin{pmatrix} q_x\\q_y\\1 \end{pmatrix} = \begin{pmatrix} p_y-q_y \\ q_x-p_x \\ p_x q_y - p_y q_x \end{pmatrix} $$
and similarly the point where two lines $L=(a,b,c)$ and $M=(u,v,w)$ meet is
$$\begin{pmatrix} a\\b\\c \end{pmatrix} \times \begin{pmatrix} u\\v\\w \end{pmatrix} = \begin{pmatrix} b w - c v \\ c u-a w \\ a v - b u \end{pmatrix} $$
The above has coordinates $(x,y) = \left( \frac{b w -c v}{a v - b u} , \frac{c u-a w}{a v - b u} \right)$ which if you do the calculations with vectors you will end up with the same value.
Futhermore, the minimum distance of the line $L=(a,b,c)$ to the origin is $d = -\frac{c}{\sqrt{a^2+b^2}}$ and the point closest to the origin has coordinates $P=(-a c,-b c, a^2+b^2) \equiv \left(-\frac{a c}{a^2+b^2},-\frac{b c}{a^2+b^2}\right)$
In addition, the minimum distance of the line $L=(a,b,c)$ to the point $P=(u,v,w)$ is
$$ {\rm dist}(L,P) = \frac{a u + b v + c w}{w \sqrt{a^2+b^2}} $$
and the projected point on the line has coordinates
$$ {\rm proj}(L,P) = \left( b( b u-a v)-a c, -a ( b u - a v)- b c, w (a^2+b^2) \right) $$
An affine 3×3 transfromation can be defined for translations and rotations.
$$  \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix} \begin{pmatrix} p_x\\p_y\\1 \end{pmatrix} =  \begin{pmatrix} p_x+t_x\\p_y+t_y\\1 \end{pmatrix}$$
$$  \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{pmatrix} p_x\\p_y\\1 \end{pmatrix} =  \begin{pmatrix} p_x \cos\theta-p_y \sin \theta\\p_x \sin\theta + p_y \cos\theta\\1 \end{pmatrix}$$
With these basics you can built what you need from here.
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