As someone with basic algebra knowledge, I am having trouble understanding Paul Bourke's explanation on "Intersection of two circles" on this page.
The specific part that I don't understand is where the last three equations come from:
1) $P_2 = P_0 + a \cdot \dfrac{ P_1 - P_0 } {d}$
2) $x_3 = x_2 \pm h \cdot\dfrac{ y_1 - y_0 }{d}$
3) $y_3 = y_2 \mp h\cdot \dfrac{ x_1 - x_0} {d}$
Can anyone help shed some light on how they are derived? Or at the very least point me in the right direction?
On
In eq. 1) the second term is a vector distance (P1-P0) which is then factorized by a/d so that P2 lies in the line connecting P0 and P1 at a distance a/d starting at P0. This equation is not used for calculations anyway.
To better understand equation 2 you could draw the axis x and y in anyway you want and you'll find out that (y1-y0)/d corresponds to sin(θ) where θ is the angle between the line P0P1 and the x axis.
h⋅(y1−y0)/d = h.sin(θ)
The angle θ happens to be also the same angle between the line P2P3 and the y axis which you use to derive the x component of P3 by computing h.sin(θ).
Eq. 3 can be explained in the same way as eq. 2, but in this case (x1−x0)/d corresponds to cos(θ).
It would be clearer by looking at a drawing but hopefully you'll be able to draw it yourself after this explanation.
All segment lengths are considered as positive. You have to add in the negative signs somewhere when necessary.
$x_3 = x_2 + k$
$= x_2 + h \sin \theta$ ... [from the smallest triangle]
$= x_2 + h \cdot \dfrac {y_1 – y_0}{d}$ ... [from the largest triangle]
It is part of the statement #2. Other statements can be proved in a similar fashion.