You are in class, doodling stars on a graph paper. Below is the upper-right quadrant of such a star, in two variants:
| Four-line star: | Nine-line star: |
|---|---|
 |  |
That is, for every chosen \(t \in [0, 1]\), a line segment is drawn connecting points \([t, 0]\), and \([0, 1-t]\).
In the infinite-line-number limit, the upper edge of the star quandrant defines a smooth curve, shown in red in the following picture:
Find an equation of this line. What kind of shape is it?
Solution
The red line is a parabola with the following implicit equation:
$$y = (\sqrt x - 1) ^ 2,$$
equivalently
$$x^2 + y^2 - 2xy - 2x - 2y + 1 = 0.$$
These equations can be found by constructing equations of all the lines (dependent upon a parameter \(t\)), and finding the upper limit for each \(x\) by differentiation with respect to \(t\).
This is easier done in transformed coordinates \([a, b] = [x - y, x + y]\), where the resulting parabola is upright.