Balance-spring is one of the essential elements that made timepieces portable. It is attached to the center of the balance wheel to a ferrule and outside the balance wheel to a pin.
In many cases it is by changing its length that the speed of a watch is adjusted. If the balance-spring is shortened the watch will go faster and if the balance spring is lengthened the watch will go slower.
This modification is made by modifying the position of the baton which, opposite to where it is handled, has two pins that slide along the last spiral of the balance spring. Most watches have flat balance springs.
In other words, the balance-spring is attached at the same level in the center as it is on the outside. This creates a defect because the balance spring cannot expand 360 degrees. It will expand primarily away from the outer attachment point of the balance spring, which creates a balance defect because the center of gravity of the balance spring does not remain in the center. This is why the rising terminal curves, also sometimes called the Breguet curve, were created.
This rising terminal curve rises in the last 3/4 turn of the balance-spring. Thus the balance spring is completely free to expand 360 degrees because its outer attachment point is at a higher level. The rising terminal curve seen from above is sometimes called a Phillips curve, which is correct when a Phillips curve is used.
Some houses calculate their own curves and name it after the name of their house. The so-called Breguet curve describes the curve that can be seen from the side. It is the curve with two elbows.
Another curve exists which is called Geneva curve, it is also seen from the side and rises gradually thanks to two pinches made by the watchmaker at 180 degrees of angle. For some chronometry competitions the inner curve at the ferrule was also used.