Duals: Points & Lines

A demo applet will appear below in a Java enabled browser.

Demo Instructions:

Manipulating a point:

Create: click the left mouse button.
Move: use the standard click and drag operation.
Delete: use ctrl-click, or the third mouse button.

To manipulate a line, move its dual point.

Explanation:

A point can be represented as a pair of coordinates, (x,y). A line, y=mx+b, can also be represented as a pair of coordinates (m,b). If we define the dual of a point, p: (x,y), as p^*: (x,-y), and the dual of a line, l: (m,b), as l^*: (m,-b), we maintain the following properties:

p is incident to l if and only if l^* is incident to p^*.
p lies above l if and only if l^* lies above p^*.

For example, three points on a line in primal space become three lines through a point in dual space.

References:

M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf, Computational geometry: algorithms and applications, Second Edition, Springer-Verlag, Berlin, 2000.