I take V as the origin of local coordinates [math] {\rm x_1 ⊥ x_2}[/math]. F must then lie on the line through x2. P is a free point in space. One point controls one property: [math]\;\;\;[/math]Translation (Vertex V), [math]\;\;\;[/math]Rotation (A), [math]\;\;\;[/math]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, [i]F changes the width of the parabola[/i]. 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: [list=1] [*]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:[i]Does the parabola open up or down? What is the direction of increasing t?[/i] Direction in each dimension is a free choice. There is no answer. We must [i]give[/i] [math] \;\;\;[/math] the direction of increasing t (here, x1). [math] \;\;\;[/math] The UP direction (x2). [/list] *On a parabola, shape and scale may be indistinguishable. I leave it this way for now. I will separate the two when it describes the problem at hand.