Parabolas and Quadratic Bezier Curves

As a follow-up to my as3geometry update on parabolas and quadratic bezier curves, I wanted to record the mathematics used to make the translation from parabola to quadratic bezier curve. Hoping to avoid the work of thinking about this for myself, I went online to understand the relationship, but could not find the calculation explained anywhere. The answer presented itself once I started to draw this diagram in Illustrator. It is amazing how a well-drawn, careful diagram can give you the insights you need!

The parabola is defined by the directrix AB and the focus C

It was apparent from the diagram that:

• H is the midpoint of the line segment AC;
• AD is perpendicular to AB, and HD is perpendicular to AC, so D can be calculated as the intersection of these lines;
• J is the midpoint of the line segment BC;
• BE is perpendicular to AB and JE is perpendicular to BC, so E can be calculated as the intersection of these lines;
• K is the intersection of the lines JE and HD, so E can be calculated as the intersection of these lines;
• The parabolic segment defined by the line segment directrix AB and the focus C is also defined by a quadratic bezier curve with the start point D, the end point E and the control point K.

The full calculation (with some different variable names I’m afraid!) was codified in the ParabolaDrawer.as class.