Paper folding method for the parabola

First we will describe how to obtain a parabola by folding a sheet of paper (Johnson, 1995). Wax paper or patty paper works best. It is easier to see the creases. On a sheet of paper that has a straight edge, mark a point F not on the edge. Fold the paper so that the edge passes through point F (figure 1a).

(a)

(b)

Figure 1. Folding a crease.

Let Q be the point on the edge that coincides with F, and let P be the point on the crease so that QP is perpendicular to the edge (figure 1b). The distance from point P to F is equal to the distance from point P to the edge, so that point P is on the parabola with F as focus and the edge as directrix. In addition, the crease is the angle bisector of angle FPQ. So this line is tangent to the parabola at P. Unfold the sheet and fold again so that a different point of the edge passes through F. Repeat this procedure several times.

Using the same principle with a dynamic geometric program.

Students can construct their own parabolas using the same method. Let C be a fixed point not on a fixed line, and let D be a point on the line. They can construct the segment CD and its midpoint. Then they construct the line perpendicular to segment CD through its midpoint (light blue line in Figure 2). The parabola is formed when this line is traced as drag point D on the line. The family of lines so obtained will have a parabola as envelope. What is the role of the fixed point C? What is the role of the fixed straight line? Notice that in this case, the line traced will play the role of the crease in the paper folding example. See interactive figure 8.

Figure 2. The parabola as paper creases.