The late high-ranking mage Andre Loire of the Law System defined a parabola as the trajectory of a point on a plane whose distance from a fixed point is equal to the distance from a fixed straight line passing through this point, and that fixed point is the focus of the parabola. , that fixed straight line is the directrix of the parabola.
"The directrix equation of this parabola is y=-p/2, and the focus is (0, p/2). 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 before, so now he was just reciting it.
"Then, the distance from point A on this parabola to the directrix is r*cosθ+p, and the distance to the focus is r. According to the definition, the two should be the same, that is, r=r*cosθ+p, slightly Simplify it and use θ as the independent variable, you can get an expression r=p/(1-cosθ).”
Calculation formulas are constantly written on the blackboard, like mysterious spells, guiding a wonderful world.
"Putting this into the original functional equation, it is easy to see that the two are equivalent, just different mathematical expressions of the same parabola in different coordinate systems."
It is obvious that the functional equation of polar coordinates is very simple, and even Dana can quickly calculate the value.
When Reiner checked the mathematical information of this world, he found that unexpectedly, the development of mathematics here was much behind the development of other aspects. Although the development of various curve equations and trigonometric functions has been rapid, most mathematical concepts have been determined. However, 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, His Excellency Isaris Eberton, was the founder of calculus, but at first he only used it to describe his three laws of motion, and had no idea of carrying it forward.
Calculus was only popularized a few years later. The school where Mr. Alberton, who had just become a high-level mage, was facing a funding crisis, only then did he think of making calculus a compulsory course for law students. The school's revenue from re-teaching fees increased that year. More than 500%, the crisis was successfully overcome, and calculus began to become a reference for mid-to-high-level mages when constructing spell models.
The reason for this, Reiner believes, is twofold.
First point, this is a magical world after all. Ancient mages developed a splendid civilization without any mathematical theory. For most mages, experience and intuition are far more convenient than calculation, and the more and more The more you are a high-level mage, the more obvious this becomes.
A simple example to illustrate is to measure the volume of an irregular bucket. People can either choose to decompose it and continuously integrate it to get the final answer, or they can choose to directly fill it with magic to get the answer, and the latter is obviously much simpler and more crude. .
High-level mages are like machines with powerful computing power. They can complete the calculations of most spell models even by using 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 no one has been promoting the development of this subject.
Nowadays, most of the progress in mathematical achievements relies on encountering difficult-to-solve problems in reality, and people will turn to seek help from mathematics.
The second point, and the 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 was almost non-existent. One thousand eight hundred years ago, Thales Anaxie proposed the Anakshi theorem of the triangle. This major discovery did not receive any feedback from the world. The feedback once made him think that he had made a mistake.
The calculus founded by Sir Alberton did not help him in constructing magic models and harvesting the resentment of his students. Therefore, until now, there is no faction among the mage factions that specializes in mathematics, let alone mathematics. Most of the researchers are in the law system and element system. They focus on using mathematical knowledge to optimize magic circles and spell models, and are more inclined to applied mathematics.
A large part of the reason why the academic system in this world is booming and why people are thirsty for the truth is that the real exploration of the world can get feedback and gain strength. Mathematics, which seems to be "useless", is naturally useless. People are interested.
"This is amazing."
Dana whispered that if the formula derived by Reiner was used, 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 deep in thought. She thought for a while before raising her hand and asking.
"But this can only explain the trajectory of a parabola. There are more complex curves in the spell model, such as ellipses and hyperbolas. What should we do with these?"
"This is where the problem lies."
Reiner smiled slightly, then drew an ellipse on the blackboard, established polar coordinates, and began to deduce.
"The definition of an ellipse is a set of points on a plane whose distance to two fixed points is equal to a constant and greater than the distance between the two fixed points. There are also directrix and focus. The definition can be transformed into the distance to the fixed points on the plane and the distance to The set of points whose distance ratio between directrix is a constant is brought into it in a similar way to the parabola... "
Reiner's writing on the blackboard is very neat, simple and clear, and Dana can understand it quickly.
Finally, after the introduction of polar coordinates, the ellipse obtained a formula r=E/(1-e*cosθ), E=b^2/a, e=c/a, a is the long axis of the ellipse, and b is is half the minor axis, and c is the distance between the two foci.
"These two formulas are very similar."
Dana realized something was wrong, but couldn't come to a conclusion.
Without waiting for them to think carefully, Reiner began to derive the polar coordinate equation of the hyperbola.
A hyperbola is a set of points where the absolute value of the difference between the distances to two fixed points is equal to a constant and less than the distance between the two points. Reiner had already derived the polar coordinate equations of parabolas and ellipses, so he quickly obtained the double curve. Polar equation of a curve.
r=E/(1-e*cosθ).
The forms of these three equations were surprisingly consistent, leaving Claire and Dana speechless in surprise.
"Actually, we can assume that a parabola also has an e, but the value of this e is 1, and the lengths of the focus and the major and minor axes can also be unified. From this point of view, ellipses, hyperbolas, and parabolas can actually use the same pole. Coordinate equations represent it, and what determines their difference is this e, which I define as eccentricity."
Looking at the three completely different curves and a large series of derivation formulas on the blackboard, Reiner said.
"When the eccentricity is less than 1, it is a hyperbola; when the eccentricity is greater than 1, it is an ellipse; when the eccentricity is equal to 1, it is a parabola; when the eccentricity is equal to 0, then it is a perfect circle."
His conclusion seems hard to accept, but the step-by-step derivation process is so clear that Claire and Dana can't find any faults.
“Thus, we can prove that these types of curves are actually variations of the same curve under different circumstances, and at the same time give these types of curves a more streamlined and unified definition: on the plane, the distance from a fixed point is the same as a fixed straight line. A set of points whose distance ratio is a constant, and this constant is the eccentricity e!"
Reiner said softly as he put down the chalk.
"Proof completed."