Draw a parabola using a pen attached to a robot

Theory

The robot follows the blue line.

A parabola is a function (or mapping) $y=ax^{2}$ (without the constant). The inverse function of the positive root is $x={\sqrt {y/a}}$. We drive the robot in $N$ steps to the vertex, see the image.

The angle $\alpha _{n}$ of $n$th slope with respect to the resetting can calculated using $tan$ function. The distance travelled by the robot can be obtained using trigonometric functions or Pythagorean theorem.

Note that all parabolas are similar, see is only One True Parabola by Matt Parker

Exercises

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