Dictionary for Duality Translations
General Note: This dictionary is for parabola duality, where a point p with coordinates (a, b) has a line dual, D(p), with the equation y = 2ax - b. Conversely, the line l with the above equation has a point dual D(l) with coordinates (a,b).
Incidence: Given a point p and a line l, for which p is on l, then the point D(l) is is on the line D(p).
Above/Below Properties: Given a point p and a line l, where p is above l, the point D(l) is above the line D(p).
Slope: The slope of a line l is the x-coordinate of is dual D(l).
Intersection: The intersection point p of two lines, m and n, has a dual D(p) which is the line formed by the two points D(m) and D(n).
Using these properties, the QucikHull algorithm for finding the convex hull of a set of points can be converted into an algorithm for finding the upper and lower envelopes of an arrangement of lines.