The mathematical framework for finding the minimum distance between two line segments. It defines the squared distance function, handles degenerate cases, distinguishes between parallel and non-parallel segments, and identifies potential critical points. It concludes by mentioning that a more efficient algorithm is needed but doesn't provide the details.
1. Parametric Representation of Line Segments:
The two line segments are defined parametrically:
$P(s) = (1-s)P0 + sP1$, where s ranges from 0 to 1.$Q(t) = (1-t)Q0 + tQ1$, where t ranges from 0 to 1.$P_0$ and $P_1$ are the endpoints of the first segment, and $Q_0$ and $Q_1$ are the endpoints of the second segment.2. Squared Distance Function:
The squared distance between any two points on the segments, P(s) and Q(t), is given by the quadratic function:
$R(s, t) = |P(s) - Q(t)|^2 = as^2 - 2bst + ct^2 + 2ds - 2et + f$
where the coefficients a, b, c, d, e, and f are defined using dot products of the segment endpoints' differences.
3. Degenerate Cases:
The algorithm handles degenerate cases where one or both segments have zero length (becoming points). This simplifies to a point-segment or point-point distance calculation.
4. Parallel and Non-Parallel Segments:
$Δ = ac - b^2$ determines whether the segments are parallel.$Δ > 0$, the segments are not parallel. $R(s, t)$ represents a paraboloid.$Δ = 0$, the segments are parallel. $R(s, t)$ represents a parabolic cylinder.5. Minimization Strategy:
The goal is to find the minimum of $R(s, t)$ within the unit square [0, 1] x [0, 1] (representing the valid range of s and t). The minimum can occur either:
R is zero.