Parabola: Vertex and (Focus or Point on arc)
- Ryan Hirst
I take V as the origin of local coordinates . F must then lie on the line through x2. P is a free point in space. One point controls one property: Translation (Vertex V), Rotation (A), Scale/Shape* (F or P). If F is moved, P is not uniquely determined. I choose to hold k constant, and vary h. In words, F changes the width of the parabola. Now, F and P have the same job. This is a good time to introduce a control structure:
The control diagram relates GGB objects V, A, F, P, p, h, k. Now, it is possible to determine the parabola locus using only two points. Say, V and F. Suppose I had done this. There are two problems:
- Unwieldy. F change both the shape and rotation of the curve. This does not help me construct measured figures, although sometimes it is pretty.
- Not enough information to orient the curve. For example:Does the parabola open up or down? What is the direction of increasing t? Direction in each dimension is a free choice. There is no answer. We must give the direction of increasing t (here, x1). The UP direction (x2).