Chapter 16: Chapter 16: Analytical Geometry and the Cartesian Coordinate System
Gao De opened his eyes and instinctively rubbed his temples.
Constructing spell models was mentally taxing.
His Apprentice Meditation Technique had reached the five-petal stage. This level of mental power was sufficient for him to complete the construction of cantrip models, albeit with some effort.
If he could cultivate it to the full sixteen-petal stage, with that level of mental power, constructing 0-Circle spell models would become much easier.
The difficulty in constructing spell models lay in two aspects: firstly, it required extreme precision, the kind where even the slightest deviation could lead to catastrophic failure. Secondly, it demanded that the mage possess sufficient mental power to expend and endure repeated attempts.
With Gao De's current mental power, each failed attempt at constructing the Acid Splash spell model left him with a headache and mental fatigue.
After failing three times at most, his brain would start to ache, his mental power would be depleted, and he would need to rest and wait for it to recover before he could continue.
This was the drawback of insufficient mental power. If a 1-Circle mage were to construct a 0-Circle spell model, not only would their efficiency be dozens of times higher than Gao De's, but even if they failed, they could easily withstand dozens of failures in a single day.
"Constructing spell models is truly not simple. No wonder it took my predecessor over a year to master the two cantrips, Repair and Mage Hand," Gao De muttered to himself.
If even mastering a 0-Circle spell was this arduous, one could imagine the effort required to become a powerful mage.
However, he didn't complain.
It was said that mages were "spell masters."
Spell masters, spell masters – how could one become a master without first being an apprentice?
Failure was the mother of success.
Gao De closed his eyes and replayed the failed construction attempt, quickly identifying the problem. While he was focused on controlling the movement of the third star, the position of the second star had shifted slightly.
A single hair can affect the whole body.
With the second star trail connecting the second and third stars already extended, even the slightest deviation in the position of the second star would naturally cause the entire spell model to collapse.
This was another difficulty in constructing spell models:
There was no room for error. Any mistake meant starting over from scratch, instead of simply correcting the error.
"This fault tolerance is too low," Gao De muttered to himself, instinctively thinking, "Is there a way to optimize the spell model construction process?"
If other mages were to hear his thoughts at this moment, they would undoubtedly mock his audacity.
Not to mention that the method of constructing spell models, passed down for who knows how many years, was unlikely to have room for optimization. Even if it did, how could a mere mage apprentice come up with it?
Gao De wouldn't be bothered by such concerns.
In the world of mathematics, if one method didn't work or was difficult to implement, it was common to change one's approach.
Could he determine the positions of all the stars first and then connect the star trails?
This thought suddenly popped into Gao De's mind.
As soon as this idea emerged, it was as if he had been struck by enlightenment. The more he thought about it, the more feasible it seemed, even feeling that this was the correct way to construct spell models.
– This way, even if a star deviated from its original position during the construction process, it wouldn't lead to the collapse of the entire spell model, requiring a complete restart. He would only need to adjust the position of that particular star in time.
Compared to the traditional method of spell model construction, the efficiency improvement wouldn't be just a little bit.
It would be like the difference between an abacus and a computer.
Gao De was always a man of action. If he had an idea, he would put it into practice.
The first problem to solve was how to determine the position of each star.
All the spell formulas recorded the spell model construction process as connecting star trails while simultaneously determining the position of each star through relative displacement. They didn't explain how to determine the position of a star without connecting star trails.
But for Gao De, this wasn't a problem at all. The existing information was sufficient – it was just simple analytical geometry.
Couldn't he simply establish a Cartesian coordinate system and then decompose the vector coordinates of each star to determine their positions?
Firstly, he needed an origin.
The origin was the starting point of all vectors.
Only by determining the origin could he determine lengths and distances, and subsequently determine the vector coordinates of each node.
Apart from stars and spell models, there were no other objects in the Spell Star Sea. However, the stars were constantly moving, so they couldn't serve as a fixed reference point and couldn't be used as the origin.
Although spell models didn't move, they were composed of multiple stars. How could they be used as a reference point?
If he were to use one of the stars in the spell model as the origin, there would be instances where two spell model nodes overlapped or star trails intersected, causing interference.
However, this was also easy to solve. He could simply treat the position of the first star as the origin.
With the origin as the center, he could establish a classic xyz coordinate system.
Then, he could use an ordered triple to determine the position of each node in the spell model.
An ordered triple consisted of three numbers. These three numbers instructed how to reach the tip of the vector (vector endpoint) starting from the origin (vector starting point).
The first number represented how far to move along the x-axis, with positive numbers indicating movement to the right and negative numbers indicating movement to the left.
The second number represented how far to move along the y-axis after that.
The third number represented how far to move along the z-axis.
Similarly, by using the star movements recorded in the spell formula, he could deduce the coordinates of each star.
Gao De got up, took a charcoal pencil from the nearby shelf, and began writing directly on the blank space of the spell formula.
The first star was the origin, with coordinates recorded as (0, 0, 0).
"Forward one, right one and one-third, up one-fourth…"
Left-right was the x-axis, forward-backward was the y-axis, and up-down was the z-axis.
The coordinates of the second star were recorded as (4/3, 1, 1/4).
"Forward one-half, right two-thirds, down one-half…"
The third star moved relative to the second star as its starting point, so it couldn't be directly recorded by comparing it to the origin. But this wasn't a big problem either – it was just a simple vector addition operation.
Through calculation, he could determine the coordinates of the third star as (2, 3/2, -1/4).
He continued to calculate in this manner.
Soon, Gao De had decomposed the Acid Splash spell model into an xyz coordinate system and nine vector coordinates, including the origin.
Then, Gao De stared intently at the nine ordered triples on the paper and began to memorize them.
Clearly, nine ordered triples were much simpler than the complex descriptions in the spell formula, not to mention Gao De's natural sensitivity to numbers.
In just a few minutes, he had memorized these nine coordinates.
"Let's give it a try."
Since the preliminary work was done, Gao De wasted no time and immediately began his attempt.
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