(k * cos(theta) * a.x + k * -sin(theta) * a.y + (k * cos(theta) * v.x + k * -sin(theta) * v.y)) = c.x
(k * cos(theta) * b.x + k * -sin(theta) * b.y + (k * cos(theta) * v.x + k * -sin(theta) * v.y)) = d.x
(k * sin(theta) * a.x + k * cos(theta) * a.y + (k * sin(theta) * v.x + k * cos(theta) * v.y)) = c.y
(k * sin(theta) * b.x + k * cos(theta) * b.y + (k * sin(theta) * v.x + k * cos(theta) * v.y)) = d.y

//divide by k
cos(theta) * a.x + -sin(theta) * a.y + cos(theta) * v.x + -sin(theta) * v.y = c.x / k
cos(theta) * b.x + -sin(theta) * b.y + cos(theta) * v.x + -sin(theta) * v.y = d.x / k
sin(theta) * a.x + cos(theta) * a.y + sin(theta) * v.x + cos(theta) * v.y = c.y / k
sin(theta) * b.x + cos(theta) * b.y + sin(theta) * v.x + cos(theta) * v.y = d.y / k

//use C and S for cos and sin
C * a.x + -S * a.y + C * v.x + -S * v.y = c.x / k
C * b.x + -S * b.y + C * v.x + -S * v.y = d.x / k
S * a.x + C * a.y + S * v.x + C * v.y = c.y / k
S * b.x + C * b.y + S * v.x + C * v.y = d.y / k

//rearrange
C * a.x + C * v.x + -S * v.y + -S * a.y = c.x / k
C * b.x + C * v.x + -S * v.y + -S * b.y = d.x / k
S * a.x + S * v.x + C * v.y + C * a.y = c.y / k
S * b.x + S * v.x + C * v.y + C * b.y = d.y / k

//factor
C * (a.x + v.x) + -S * (v.y + a.y) = c.x / k
C * (b.x + v.x) + -S * (v.y + b.y) = d.x / k
S * (a.x + v.x) + C * (v.y + a.y) = c.y / k
S * (b.x + v.x) + C * (v.y + b.y) = d.y / k