Elliptic Curves (not to be confused with ellipses!) are defined by equations of the type
where the coefficients a and b determine the exact form. As you can easily see from the equation, such curves are symmetric with respect to the x-axis. Depending on the values of a and b you get either a connected curve or two separated parts. Which of these two cases applies can be seen from the sign of the discriminant
Elliptic curves are used by a well-known encryption method (ECC, Elliptic Curve Cryptography). They also play an important role in some areas of modern mathematics, for example in the proof of Fermat's Last Theorem by Andrew Wiles (1994).
On elliptic curves an addition is defined which assigns for two given points P and Q (blue) of the curve a further point of the curve which is designated as "sum" P+Q. You find the result by drawing the secant to the given points (for equal points the tangent). The drawn line (grey) generally intersects the curve at a further point. By reflecting this point through the x-axis you get the point P+Q (red).
In this app the coefficients of an elliptic curve are set using sliders. The given points can be moved with the mouse respectively finger.