# 31 October 2020

## GeoGebra Discovery: Symbolic support for points attached to an algebraic curve

A long awaited improvement has arrived in yesterday's release. It supports attaching points to algebraic curves, and then the computation can be continued with symbolic means. An example is shown below.

Here the algebraic curve $y=\frac{1}{x^2+1}$ is shown (in black), also known as Agnesi's curve (or Agnesi's witch). To emphasize that this is an algebraic curve and GeoGebra should consider it like that, the user needs to enter $c:y\cdot(x^2+1)=1.$ In the next steps a point $$M$$ on the curve $$c$$ and the segment $$f=AB$$ are created, and a circle $$d$$ with center $$M$$ and radius $$f$$.

At this stage the center of the circle can be moved on the curve. We can be interested of what type of another curve will be touched by the set of circles. Therefore we issue the command Envelope(d,M) to learn what happens if $$M$$ is moving along, and we consider the set of circles $$d$$ meanwhile.

The obtained result is a curve of degree 14. It is also called a parallel curve (or offset curve). We can continue our experiments by changing the length $$f$$ by dragging the point $$A$$ or $$B$$—however, the degree of the curve remains 14.

Hint: Enable the trace for circle $$d$$ (by using the right mouse click on the circle). Then you will see that the envelope $$e$$ is indeed touched by the circles. Also, you can try to change the algebraic formula to get different outputs!