[Math] What equation can produce these curves

Question

Does an equation exist that can produce the curves shown in the attached image, by varying a single variable?

Answer 1

Consider the equilateral hyperbola $xy=1$ and map the points $(1,0)$ and $(0,1)$ to two symmetrical points on the hyperbola $(t,t^{-1})$ and $(t^{-1},t)$ by translation/scaling (as if you were zooming in).

$$(x(t-t^{-1})+t^{-1})(y(t-t^{-1})+t^{-1})=1.$$

The straight line corresponds to two infinitely close points, $t=t^{-1}=1$, which is a degenerate case of the equation.

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Previous answer:

You can think of a pencil of parabolas, $y=\frac1{\sqrt2}+\frac\lambda{\sqrt2}(2x^2-1)$, which you rotate by $45°$ right, giving

$$x+y=1+\lambda\left((x-y)^2-1\right).$$

You can solve the quadratic equation for $y$.