The late High Mage of the Law System Andrew. Loire's definition of a parabola is the trajectory of a point on a plane at a distance equal to the distance to a fixed line of no more than this point, which is the focal point of the parabola, and that fixed line is the alignment of the parabola.

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

Reiner wrote smoothly on the blackboard, he had already deduced it himself, so it was nothing more than a retelling now.

"Then, the distance from the point A on this parabola to the alignment is r*cosθ+p, and the distance to the focal point is r, according to the definition, the two should be the same, that is, r=r*cosθ+p, a little simplification, with θ as the independent variable, you can get an expression r=p/(1-cosθ)."

Calculations are constantly written on the blackboard, like mysterious spells, guiding a wonderful world.

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

Obviously, the equation as a function of polar coordinates is so simple that even Dana can quickly figure out the value of it.

When Reiner consulted the mathematical materials of this world, he found that the development of mathematics here is surprisingly much behind the development of other aspects, although the development of various curvilinear equations and trigonometric functions has been very fast, and most of the mathematical concepts have been determined, but the knowledge related to calculus and number theory is rarely discussed, and the field of imaginary numbers does not yet exist.

The legendary mage of the Law System, Isaris. His Excellency Alberton was the founder of calculus, but he started out as a description of his three laws of motion, and he did not think of carrying them forward.

The popularization of calculus was still a few years later, when the school where the newly advanced mage Alberton was facing a financial crisis, he thought of making calculus a compulsory course for law students, and the school's retake fee income increased by more than 500% that year, successfully surviving the crisis, and calculus began to become a reference for middle and advanced mages when building spell models.

There are two reasons for this, according to Reiner.

First of all, this is a magical world after all, ancient mages have developed a splendid civilization without any mathematical theory, and for the vast majority of mages, empirical intuition is far more convenient than calculation, and the more advanced the mage, the more obvious this is reflected.

A simple example of this is measuring the volume of an irregular bucket, which one can choose to break down and integrate to get the final answer, or simply fill it with magic and get the answer, which is obviously much simpler and crude.

High-level mages are like machines with powerful computing power, and can complete the calculations of most spell models even with simple exhaustive methods.

In the final analysis, mathematics in this world is just a shortcut, and the strong do not need shortcuts, and the knowledge of the weak is not enough to find new shortcuts, so the development of this discipline has not been promoted.

Nowadays, most of the progress of mathematical achievements still depends on the fact that people encounter difficult problems in reality, and people turn to seek help from mathematics.

The second, and most important, point is that the development of mathematics does not receive feedback from the world.

Even though Reiner proposed the polar coordinate system, the world's feedback was almost non-existent, and 1,800 years ago Thales. Anakhy proposed the Anakhy theorem for triangles, but this major discovery was completely unfed back by the world, and he thought he was mistaken.

The calculus founded by Lord Alberton did not help him in building spell models and harvesting the resentment of his students, and because of this, until now, there is no school in the mage's faction that specializes in mathematics, let alone mathematicians, and most of the researchers are distributed in the law system and the element system, focusing on using mathematical knowledge to optimize the magic circle and spell model, and are more inclined to apply mathematics.

A large part of the reason why the world's academic system is thriving, and why people are thirsty for truth, is that the exploration of the truth of the world can get feedback and gain power, and mathematics, which seems to be "useless", is naturally ignored.

"It's amazing."

Dana whispered that if with Reiner's formula, even she could quickly get the trajectory equation of the magic channel, she had never realized that mathematics had such wonderful power before today.

Claire was lost in thought, she thought for a moment before she raised her hand and asked.

"But this only explains the trajectory of the parabola, and there are more complex curves in the spell model, such as ellipses and hyperbolas, what about these?"

"That's the problem."

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

"The definition of an ellipse is the set of points on a plane whose distance to two fixed points is equal to a constant and greater than the distance between two fixed points, and there are also alignments and focal points, and the definition can be transformed into a set of points whose ratio of the distance to the fixed point on the plane to the distance to the alignment is constant, and is brought into the ...... in a similar way to a parabola."

Reiner's board book is neat, simple and straightforward, and Dana can understand it quickly.

Finally, the ellipse is given a formula r=E/(1-e*cosθ), E=b^2/a, e=c/a, where 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 foci.

"These two formulas, very similar."

Dana was aware of something, but couldn't draw conclusions.

Without waiting for them to think carefully, Reiner began to derive the polar equations of the hyperbola.

A hyperbola is a set of points whose absolute value is equal to a constant and less than the distance between two points, and Reiner has derived the polar coordinate equations for parabolas and ellipses, so the polar coordinate equations for hyperbolas are quickly obtained.

r=E/（1-e*cosθ）。

The three equations are strikingly consistent in form, leaving Claire and Dana speechless with surprise.

"Actually, we can assume that there is also an e in the parabola, but the value of this e is 1, and the focal point and the length of the major and minor axes can also be unified, so that the ellipse, hyperbola, and parabola can actually be represented by the same polar equation, and it is this e that determines the difference between them, which I define as eccentricity."

Looking at three very different curves and a long list of derivative formulas on the blackboard, Reiner said.

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

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

From this, we can prove that these curves are actually variations of the same kind of curve in different situations, and at the same time give these curves a more concise and unified definition: on the plane, the ratio of the distance to a fixed point to the distance of a fixed line is the set of points with a constant, and this constant is the eccentricity e!"

Putting down the chalk, Reiner whispered.

"Proof done."