Chapter 16: Analytic Geometry and the Cartesian Coordinate System

Gao De opened his eyes and subconsciously rubbed his temples.

Constructing spell models is extremely taxing on the mind.

His apprentice meditation technique has reached the five-petal stage; this level of mental power is sufficient for him to complete the model construction for cantrips, though it is a bit of a struggle.

If he could cultivate to the perfection of sixteen petals, with such powerful mental energy, constructing 0-circle spell models would be far more effortless.

The difficulty in constructing spell models lies, firstly, in the need for precision—a precision where a hair's breadth difference leads to a thousand-mile error—and secondly, in the need for the magic user to possess enough mental energy to consume and to conduct repeated attempts.

With Gao De's current mental energy, every time he fails to construct the Acid Splash spell model, he feels his head swell and becomes exhausted.

At most, after three failures, his brain begins to ache, his mental energy is over-consumed, and he must rest and wait for his mental energy to recover, unable to continue constructing spell models.

This is the drawback of insufficient mental energy. If a 1st-circle magic user were to construct a 0-circle spell model, not to mention that their efficiency would be over ten times that of Gao De, even if they failed, failing dozens of times a day would be no problem for them.

"Constructing spell models is indeed not simple. No wonder the predecessor took over a year to master the two cantrips, Mending and Mage Hand," Gao De muttered to himself.

If mastering even a 0-circle spell is this arduous, one can imagine how much effort is required to become a powerful magic user.

However, he did not complain.

It is said that magic users are the "Magic Lords."

Magic Lord, Magic Lord—if you don't act like a grandson first, how can you become a lord?

Failure is the mother of success.

Gao De closed his eyes to review the failure he just experienced, and soon found the problem: while focusing on controlling the movement of the third star-seed, the position of the second star-seed shifted slightly. One hair moves the whole body.

With the second star-track connecting the second and third star-seeds already extended, the moment the second star-seed's position shifted even a little, the entire spell model naturally collapsed.

This is another difficulty in constructing spell models: there cannot be even the slightest mistake, otherwise everything must be started over from the beginning, rather than just correcting where the error occurred. "This fault tolerance is too low," Gao De muttered to himself, subconsciously thinking: "Can the construction process of the spell model be optimized?"

If his current thoughts were known by other magic users, they would certainly mock him for not knowing the immensity of heaven and earth.

Not to mention how could there be room for optimization in a method of constructing spell models passed down for who knows how many years, even if there were, is it something a magic user apprentice could think of?

Gao De, however, did not have these miscellaneous misgivings.

In the world of mathematics, if a method doesn't work or is difficult to walk, changing one's line of thinking is a very common occurrence.

Could one fix the positions of all star-seeds first, and then connect the star-tracks?

Such a thought suddenly popped into Gao De's mind.

After this thought appeared, he felt as if he had been enlightened; the more he thought about it, the more feasible it seemed, and he even felt that this was the correct way to open the construction of spell models.

—This way, even if a star-seed shifts from its original position during the process of constructing the spell model, it would not lead to the collapse of the entire spell model and having to start everything over; one would only need to adjust the position of that star-seed in time.

Compared to the traditional method of spell model construction, how much would this increase efficiency?

It is simply the difference between an abacus and a computer.

Gao De's ability to act has always been strong; if he has an idea, he executes it.

The first thing to solve is how to determine the position of each star-seed.

The construction processes for spell models recorded in all spell formulas involve connecting star-tracks while simultaneously determining the position of each star-seed through relative displacement; they do not describe how to determine the position of star-seeds without connecting star-tracks.

But for Gao De, this is not a problem at all; the existing information is sufficient to use—isn't it just simple analytic geometry? By directly establishing a Cartesian coordinate system and then decomposing the vector coordinates of each star-seed, wouldn't he be able to determine the position of each star-seed?

First, an origin is needed.

The origin is the source of all vectors.

Only by determining the origin can one determine length and distance, and subsequently determine the vector coordinates of each node.

In the spell star-sea, there are no objects other than star-seeds and spell models; however, the star-seeds are constantly moving, so they are clearly not fixed reference points and cannot serve as an origin.

Although the spell model does not move, it is a model composed of multiple star-seeds; how can it serve as a reference object? If one were to use one of the star-seeds within the spell model as the origin, it would lead to overlapping spell model nodes or interference from intersecting star-tracks.

However, this is also easy to handle: just treat the position of the first star-seed as the origin.

With the origin as the center, establish a most classic xyz coordinate system.

Then use an ordered triplet to determine the position of each node in the spell model.

A triplet consists of three numbers; these three numbers are responsible for guiding how to reach its tip (the vector endpoint) starting from the origin (the vector starting point).

The first number represents how far to go along the x-axis; a positive number represents moving to the right, and a negative number represents moving to the left.

The second number represents how far to go along the direction parallel to the y-axis after that.

The third number represents how far to go along the z-axis direction.

Similarly, by using the star-seed paths recorded in the spell formula, one can deduce the coordinates of each star-seed.

Gao De stood up, took a charcoal pencil from the shelf beside him, and began to record directly on the blank space of the spell formula.

The first star-seed is the origin, with coordinates marked as (0, 0, 0).

"Advance one, right one and one-third, up one-quarter."

Left and right are the x-axis, front and back are the y-axis, and up and down are the z-axis.

The coordinates of the second star-seed are marked as (4/3, 1, 1/4).

"Advance one-half, right two-thirds, down one-half."

The third star-seed moves starting from the second star-seed; it cannot be directly compared to the origin for recording, but that is not a big problem either—isn't it just simple vector addition?

Through calculation, one can obtain the coordinates of the third star-seed as (2, 3/2, -1/4).

Just continue to deduce in this order.

Soon, Gao De decomposed the Acid Splash spell model into an xyz coordinate axis and nine vector coordinates, including the origin.

Afterward, Gao De looked at the nine triplets on the paper with burning eyes and began to try to memorize them.

Clearly, nine triplets are much simpler than the complex descriptions in the spell formula, not to mention that Gao De is naturally highly sensitive to numbers.

In just a few minutes, he had committed these nine coordinates to memory.

"Let's try it."

Since the preliminary work was done, Gao De acted as soon as he thought of it, and immediately began to try.

(End of chapter) ===== CHAPTER 17 =====