Intersecting lines · scipython.com

E10.3Intersecting lines

Consider the two lines given by the equations:

\begin{align*} y &= x\\ y &= mx + c \end{align*}

These lines intersect at $(x_\star, y_\star) = (c/(1-m), c/(1-m))$ . Finding the intersection point is an ill-conditioned problem when $m \approx 1$ (lines nearly parallel).

For example, the lines $$y=x$$ and $$y = (1.01)x + 2$$ intersect at $(x_\star, y_\star) = (-200, -200)$ . If we perturb $$m$$ slightly by $\delta m = 0.001$ , to $m' = m + \delta m = 1.011$ , the intersection point becomes $(x_\star', y_\star') = (-181.8182, -181.8182)$ . That is, a relative error of $\delta m / m \approx 0.001$ in $$m$$ has created a relative error of $|(x_\star' - x_\star)/x_\star| \approx 0.091$ , almost 100 times larger.

Conversely, if the lines have very different gradients, the problem is well-conditioned. Take, for example, $$m = -1$$ (perpendicular lines): the intersection $$(1,1)$$ becomes (1.0005, 1.0005) under the same perturbation to $m' = m + \delta m = -0.999$ , leading to a relative error of 0.0005 which is actually smaller than the relative error in $$m$$ .