Nowadays, most of the progress in mathematical achievements still relies on the difficult problems encountered in reality, so people turn to seek help from mathematics.

The second and most important point is that the development of mathematics cannot obtain feedback from the world.

Even though Reiner proposed the polar coordinate system, the feedback from the world almost did not exist. Thales Anakshi proposed the Anakshi theorem of triangles a thousand eight hundred years ago. This major discovery did not receive feedback from the world at all, which made him think he had made a mistake.

The calculus founded by His Excellency Alberton did not provide any help in his construction of spell models and gaining students' resentment. Therefore, until now, there is no school that specializes in mathematics in the mage faction, let alone mathematicians. Most researchers are distributed in law systems and element systems, focusing on optimizing magic arrays and spell models with mathematical knowledge, and tend to apply mathematics.

The reason why the academic system in this world is flourishing and people are eager for the truth is that they can obtain feedback and gain power through exploration of the world's reality, and mathematics that seems to be "useless" will naturally no one cares about it.

“This is amazing.”

Dana whispered that if she had used the formula derived by Reiner, even she could quickly obtain the trajectory equation of the magic channel, she had never realized that mathematics had such a wonderful power before today.

Claire fell into deep thought. She thought for a while before she raised her hand and asked.

"But this can only explain the trajectory of the parabola. There are more and more complex curves in the spell model, such as ellipses and hyperbolas. What should I do?"

“That’s the problem.”

Reiner smiled slightly, then drew an ellipse on the blackboard, established polar coordinates, and began to deduce.

"The definition of an ellipse is the set of points whose distance from the plane to two fixed points is equal to a constant and is greater than the distance between two fixed points. There is also a line and a focus. The definition can be converted into a set of points whose ratio of the distance from the plane to the fixed point and the distance from the line to the line is constant, and it is brought in in a similar way to the parabola..."

Reiner's blackboard writing is very regular and simple, and Dana can quickly understand it.

Finally, after introducing polar coordinates, the ellipse obtains a formula r=E/(1-e*cosθ), E=b^2/a, e=c/a, a is the general of the major axis of the ellipse, b is half of the minor axis, and c is the distance between the two focal points.

"These two formulas are very similar."

Dana realized some problems, but couldn't draw a conclusion.

Without waiting for them to think carefully, Reiner began to deduce the polar coordinate equation of the hyperbola.

Hyperbola is the set of points whose absolute value of the difference between the distances to two fixed points is equal to a constant and whose distances are smaller than the two points. Reiner has derived the polar coordinate equations of parabola and ellipse, so he quickly obtained the polar coordinate equations of hyperbola.

r=E/(1-e*cosθ).

The three equations are surprisingly consistent in their form, which makes Claire and Dana so surprised that they can't speak.

"In fact, we can assume that there is an e, but the value of this e is 1, and the length of the focal point and the length of the long and short axes can also be unified. In this way, ellipses, hyperbolas, and parabolas can actually be represented by the same polar coordinate equation, and the one that determines them is this e, which I define as the eccentricity."

Looking at three completely different curves and a large string of derivation formulas on the blackboard, Reiner said.

"When the centrifugal rate is less than 1, then it is a hyperbola, when the centrifugal rate is greater than 1, it is an ellipse, and when the centrifugal rate is equal to 1, it is a parabola, and when the centrifugal rate is equal to 0, then this is a perfect circle."

His conclusions seem unacceptable, but the step-by-step derivation process is so clear that Claire and Dana can't find any fault.

"From this, we can prove that these curves are actually the changes of the same curve under different circumstances, and at the same time, we give these curves a more streamlined and unified definition: on the plane, the set of points whose ratio of the distance from a fixed point to a fixed line is constant, and this constant is the eccentricity e!"

Putting down the chalk, Reiner whispered.
Chapter completed!