Chapter 74

Andre Loire, the late high-order mage of the law system, defined a parabola as the locus of a point on the plane whose distance to a fixed point is equal to the distance to a fixed line of this point. That fixed point is the focus of the parabola, and that fixed line is the directrix of the parabola.

"The Quasilinear equation of this parabola is y = - P2, and the focus is (0, P2). By introducing polar coordinates, we can get x = R * sin θ, y = R * cos θ + P2."

Lena wrote fluently on the blackboard. He had deduced it himself before, so now it's just a retelling.

"Then, the distance from point a on the parabola to the Quasilinear is R * cos θ + P, and the distance to the focus is R. according to the definition, the two should be the same, that is, r = R * cos θ + P. a little simplification, taking θ as the independent variable, we can get an expression r = P (1-cos θ)."

The formulas are written on the blackboard, like mysterious incantations, guiding a wonderful world.

"If we take it into the original functional equation, we can easily see that the two are equivalent, but they are different mathematical expressions of the same parabola in different coordinate systems."

And obviously, the polar function equation is very simple, even Dana can quickly calculate the value.

When he looked up the mathematical data of the world, he found that unexpectedly, the development of mathematics here is much behind that of other aspects. Although the development of various curve equations and trigonometric functions has been very fast, and most of the mathematical concepts have been determined, few people discuss the knowledge related to calculus and number theory, and the field of imaginary number is even more important It doesn't exist yet.

The legendary Wizard of the law system, sir isaris Alberton, is the founder of calculus, but at the beginning, he only used it to describe his three laws of motion, and he did not expect to carry them forward.

It was only a few years later that the popularity of calculus came to pass. The school where Mr. Alberton, who had just become a high-level mage, was facing a financial crisis. He thought of calculus as a compulsory course for law students. In that year, the income of the school's tuition fee increased by more than 500%, and he survived the crisis smoothly. Calculus also began to become a time for middle and high-level mages to build magic models For reference.

There are two reasons.

First, this is a magical world after all. Ancient mages developed brilliant civilization without any mathematical theory. For the vast majority of mages, experience intuition is far more convenient than calculation. The more advanced mages are, the more obvious this is.

To illustrate with a simple example is to measure the volume of an irregular bucket. People can choose to decompose it and continuously integrate it to get the final answer, or they can choose to fill it with magic directly to get the answer. The latter is obviously much simpler and more crude.

High level mages are like machines with powerful computing power. They can complete most of the calculation of magic models even with simple exhaustive method.

In the final analysis, mathematics is just a shortcut in this world. The strong don't need a shortcut, and the weak don't have enough knowledge to find a new shortcut. Therefore, the development of this subject has not been promoted.

Nowadays, the progress of mathematical achievements mostly depends on the problems that are difficult to solve in reality, so people will turn to seek the help of mathematics.

The second and most important point is that the development of mathematics can not get feedback from the world.

Even though Lena proposed the polar coordinate system, the feedback of the world almost does not exist. One thousand and eight hundred years ago, Thales anarchy proposed the anarchy theorem of triangles, but this important discovery could not get the feedback of the world, which once made him think he was wrong.

The calculus founded by Mr. Alberton did not help him build magic models and harvest Students' complaints. Therefore, up to now, there is no school specializing in Mathematics in the mage's school, let alone mathematicians. Most of the researchers are distributed in the law department and element department, focusing on optimizing magic arrays and magic models with mathematical knowledge, More inclined to applied mathematics.

The reason why the academic system of the world is booming and people are thirsty for the truth is in large part that the exploration of the real world can obtain feedback and strength, while mathematics, which seems to be "worthless", is naturally ignored.

"It's amazing."

Dana whispered that even she could quickly get the trajectory equation of the magic channel based on Lena's formula. Before today, she had never realized the wonderful power of mathematics.

Clare was lost in thought. She thought about it, then raised her hand and asked.

"But this can only explain the trajectory of parabola. There are more and more complex curves in the magic model, such as ellipse and hyperbola. What should we do?"

"That's the problem."

Lena smiles, draws an ellipse on the blackboard, establishes polar coordinates, and begins to deduce.

"The definition of an ellipse is a set of points whose distance from a plane to two fixed points is equal to a constant and greater than the distance between two fixed points. There is also a collimator and a focal point. The definition can be transformed into a set of points whose ratio of the distance from a plane to a fixed point to the collimator is a constant, which can be brought in in the same way as a parabola..."Lena's writing on the blackboard is regular, simple and clear, and Dana can understand it quickly.

Finally, after introducing polar coordinates, the ellipse obtains a formula r = e (1-e * cos θ), e = B ^ 2a, e = ca. A is the general of the major axis of the ellipse, B is half of the minor axis, and C is the distance between two focal points.

"These two formulas are very similar."

Dana was aware of some problems, but she couldn't come to a conclusion.

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

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

r=E（1-e*cosθ）。

The three equations are surprisingly consistent in form, which makes Claire and Dana speechless.

"In fact, we can assume that there is an e for parabola, but the value of E is 1, and the length of focus and major and minor axis can also be unified. In this way, ellipse, hyperbola and parabola can actually be expressed by the same polar coordinate equation, and what determines their difference is this e, which I define as eccentricity."

Looking at three very different curves and a bunch of formulas on the blackboard, Lehner said.

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

His conclusion seems to be difficult to accept, but the step-by-step derivation process is so clear that Claire and Dana can not find any fault.

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

Put down the chalk, Lena said softly.

"It's over."

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