Greetings, Mister Principal

The late high-ranking mage Andre Loire defined a parabola as the distance to a fixed point on the plane was equal to the trajectory of a point with the same distance to a fixed straight line that did not exceed this point. And that fixed point was the focal point of the parabola, and the fixed straight line was the guideline of the parabola.

"The directrix equation of this parabola is y=-p/2, and the focal point is (0, p/2). By introducing polar coordinates, we can get x=r*sinθ, y=r*cosθ+p/2."

Reiner wrote fluently on the blackboard. He had already derived it himself before, so now he was just repeating.

"Then, the distance from point a on the parabola to the directrix is r*cosθ+p, and the distance to the focal point is r. By definition, the two should be the same, that is, r=r*cosθ+p. To simplify it, take θ as the independent variable, and you can get an expression r=p/(1-cosθ)."

Calculation formulas were constantly being written on the blackboard, just like mysterious spells, guiding a wonderful world.

"Putting it into the original functional equation, it is easy to see that the two are equivalent, but they are just different mathematical expressions of the same parabola in different coordinate systems."

Obviously, the functional equation of polar coordinates was very concise, even Dana could quickly calculate the value.

When Reiner looked up the mathematics of this world, he found that the development of mathematics here was far behind other developments. Although the development of various curve equations and trigonometric functions had been rapid, and most of the mathematical concepts had been determined, the knowledge related to calculus and number theory was rarely discussed, and the field of imaginary numbers did not yet exist.

His Excellency Isaris Alberton, the legendary mage of the law system, was the founder of calculus, but he only used it to describe his three laws of motion at first, and he did not seem to want to promote this theory.

The popularization of calculus happened several years later. The academy of His Excellency Alberton, who had just become a high-level mage, was facing a funding crisis, so he thought of making calculus a compulsory course for students of the law department. The academy's income immediately increased more than 500% that year, and it helped the academy to get through the crisis smoothly. Since then calculus became a reference for middle and high-level mages when they built spell models.

In Reiner's opinion, there were two main points.

The first point was that this was a magical world after all. Ancient mages had developed brilliant civilizations without any mathematical theory. For most mages, experience and intuition were far more convenient than calculations. Regarding this reason, the higher level a mage was, the more obvious it was.

Using a simple example to illustrate this point was to measure the volume of an irregular barrel. People could either choose to measure it and then calculate it to get the final answer, or they could choose to fill it with magic power to get the answer, and the latter was obviously much simpler.

High-level mages were like machines with powerful computing power, even with a simple method of exhaustion, they could complete most of the calculations of spell models.

In the final analysis, mathematics was only a shortcut in this world. The strong did not need shortcuts, and the knowledge of the weak was not enough to find new shortcuts. Therefore, the development of this discipline had not been promoted.

Nowadays, most of the progress of mathematics results depended on the difficult problems encountered in reality, and people would turn their heads to seek help from mathematics.

The second and most important point was that the development of mathematics would not obtain feedback from the world.

Even if Reiner proposed the polar coordinate system, the feedback from the world hardly existed. One thousand and eight hundred years ago, Thales Anakshi proposed the Anakshi theorem of triangles. However, this major discovery didn't get any feedback from the world. It made him think that he had made a mistake.

Other than helping him to build a model of spells and reap the frustration of students, the calculus created by His Excellency Alberton did not have much use. For this reason, until now, there were no academies that specialized in mathematics. Most of the researchers in the law system and element system that were focusing on optimizing mathematics and spell models with mathematical knowledge were more inclined to applied mathematics.

The reason why the academic system of this world was flourishing and people were thirsty for truth was that a large part of the reason was that the real exploration of the world could get feedback and gain power, so mathematics that seemed "good for nothing" naturally would not arouse people's interests.

"This is amazing."

Dana exclaimed in a low voice. If using Reiner's formula, even she could quickly obtain the trajectory equation of the magic channel. Before today, she had never realized that mathematics had such a wonderful power.

Claire was lost in thought, she thought for a while, then raised her hand and asked.

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

"That's the problem."

Reiner smiled slightly, then drew an ellipse on the blackboard, established the polar coordinates system, and started the deduction.

"The definition of an ellipse is a collection of points whose distance to two fixed points on the plane is equal to a constant and greater than the distance between the two fixed points. There are also guidelines and focal points. The definition can be transformed into a collection of points where the ratio of the distance to the fixed point to the distance to the directrix on the plane is constant. Bring it in in a similar way to the parabola..."

Reiner's blackboard writing was very neat, simple, and clear, and Dana could understand it quickly.

Finally, after the introduction of polar coordinates, the ellipse got a formula r=e/ (1-e*cosθ), e=b^2/a, e=c/a, a was half of the long axis of the ellipse, b was half of the short axis, and c was the distance between the two focal points.

"These two formulas are very similar."

Dana realized some problems, but couldn't draw conclusions.

Without waiting for them to think carefully, Reiner began to derive the hyperbolic polar coordinate equation.

A hyperbola was a collection of points whose absolute value of the difference between the distance to two fixed points was equal to a constant and less than the distance between the two points. Reiner had derived the polar coordinate equations of parabola and ellipse, so he quickly obtained the polar coordinate equations of the hyperbola.

r = e / （1-e * cosθ）。

The forms of these three equations were surprisingly consistent, leaving Claire and Dana speechless in surprise.

"In fact, we can assume that there is also an e for the parabola, but the value of this e is 1, and the length of the focal point and the long and short axis can also be unified. From this point of view, ellipse, hyperbola, and parabola can actually be expressed by the same polar coordinate equation, and the difference between them is this e, which I define as eccentricity."

Looking at three very different curves and a large series of deduction formulas on the blackboard, Reiner said.

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

His conclusion may seem unacceptable, but the step-by-step derivation process was so clear that Claire and Dana couldn't find any fault in it.

"From this, we can prove that these kinds of curves are actually the changes of the same kind of curve in different situations. At the same time, it also gives these types of curves a more streamlined and unified definition: On the plane, a collection of points whose ratio of the distance from a fixed point to the distance from a fixed straight line is constant. This constant is the eccentricity e!"

Putting down the chalk, Reiner said softly.

"The proof is complete."