.: Conics : Parabola :.

Parabola:
Given a point F and a line d, called respectively focus and directrix, the parabola defined by the pair (F,d) is the geometric place of the points P that the distance between P and F equals the distance between P and d.
Parabola = {P : ||P-F|| = ||P-d||    (||P-d|| = minQ em d{ ||Q-P|| })

Dettails Construction

Construction's details
  1. Build the line directrix d. Build the point F, outside the line d, that will be the focus of the parabola.

  2. Build a "free" point P over the line d and build the segment FP. Then build the midpoint M of this segment.

  3. Build the line s, perpendicular to the segment FP, passing through the point M (this line is the bisector of FP). Now, build the line r, perpendicular to the directrix d, passing through P.

  4. Set the point Q, intersection between s and r.

  5. Trace the point Q: move P over the line d and observe the "locus" defined by Q.

Interactive construction

Below we present the construction proposed above, directly on iGeom, however we used the axes x and y. The point P of the construction above, below is the point Pd, which is the ortogonal projection of the abscissa X over the directrix line d.
Besides, in the presented construction, intermediate steps are hidden. To see them, "click" the button , which is inside the primary button .
If you wish a brief description, in the form of an algorithm, put the pointer over the iGeom's drawing area, then "click" the mouse's middle button (if you don't have three buttons, try clicking both simultaneously - eventualy this emulates the middle button).

Move one of the points that define the parabola (A, B, C or E) to verify what happens with the parabola. One can also move the point D over the line r, which is the point that generates the "locus" (from the point I, in this case it will be possible to observe that this point "walks through" the parabola).

To move a point, the button "move" must be selected .