Analytic Geometry and the Cartesian Coordinate System

Gao De opened his eyes and instinctively rubbed his temples.

Constructing a Spell Model was tremendously taxing.

He had cultivated his Apprentice Meditation Art to five petals. This level of Mental Strength was enough for him to complete the construction of a cantrip's model, though it was somewhat strenuous.

If he could cultivate it to the full sixteen petals, then with Mental Strength that powerful, constructing the Spell Model of a Zero Circle Spell would be far more effortless.

There were two difficulties in constructing a Spell Model. First, it required precision—precision where the slightest error could lead a thousand miles astray. Second, it required a mage to possess enough Mental Strength to expend on attempt after attempt.

With Gao De's current Mental Strength, every failed attempt at constructing the Spell Model for Acid Splash left his head feeling swollen, tired, and exhausted.

At most, after three failures, his brain would begin to ache. His Mental Strength would be overly depleted, and he would need to rest and wait for it to recover before he could no longer continue constructing Spell Models.

This was the drawback of insufficiently powerful Mental Strength. If a First Circle Mage were constructing the Spell Model for a Zero Circle Spell, not only would their efficiency be more than ten times Gao De's, but even failing dozens of times a day would not be a problem.

"Constructing a Spell Model really isn't simple. No wonder my predecessor spent over a year mastering the two cantrips Repair and Mage Hand," Gao De muttered to himself.

Even mastering a Zero Circle Spell was this difficult. One could only imagine how much effort it took to become a powerful mage.

But he did not complain.

They all said mages were grand masters.

Grand master, grand master—how could you become the master without first being the grandson?

Failure was the mother of success.

Gao De closed his eyes and reviewed the failed construction just now. He quickly found the problem—while he had focused on controlling the third Starseed's movement, the second Starseed's position had shifted slightly.

Pull one hair, and the whole body moved.

Once the second star track connecting the second and third Starseeds had extended out, the slightest shift in the second Starseed's position would naturally cause the entire Spell Model to collapse.

This was another difficulty in constructing a Spell Model:

There could not be the slightest mistake. Otherwise, everything had to be started over from scratch; one could not simply correct whatever part had gone wrong.

"The margin for error is way too low," Gao De muttered. He subconsciously thought, Could I optimize the Spell Model construction process?

If the other mages had learned of his thoughts at that moment, they would surely have laughed at his ignorance of the immensity of heaven and earth.

Leaving aside how a method for constructing Spell Models passed down for countless years could possibly have room for optimization, even if there were such room, how could a mere Mage Apprentice possibly think of it?

Gao De did not have such miscellaneous concerns.

In the world of mathematics, if one approach did not work or was difficult to follow, changing one's line of thought was an exceedingly common thing.

Could he first fix the positions of all the Starseeds, then connect the star tracks?

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

The moment that thought appeared, it was as though enlightenment had been poured over his head. Everything suddenly became clear. The more he considered it, the more feasible it seemed, and he even felt that this was the proper way to construct a Spell Model.

—This way, even if a Starseed shifted from its original position during the construction process, it would not cause the entire Spell Model to collapse and force everything to start over. He would only need to promptly adjust that Starseed's position.

Compared to the traditional method of constructing Spell Models, how could this be merely a slight increase in efficiency?

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

Gao De had always been highly decisive. If he had an idea, he put it into action.

The first thing to solve was how to determine the position of each Starseed.

Every Spell Model construction process recorded in spell formulas involved connecting star tracks while using relative displacement to determine each Starseed's position. None explained how to determine a Starseed's position without connecting star tracks.

But to Gao De, this was not a problem at all. The existing information was enough—it was nothing more than simple analytic geometry.

He only needed to directly establish a Cartesian Coordinate System, then break down the vector coordinates of every Starseed. Would that not determine each Starseed's position?

First, he needed an origin.

The origin was the source of all vectors.

Only after determining the origin could he determine lengths and distances, and then determine the vector coordinates of every node.

There were no objects in the Spell Starsea other than Starseeds and Spell Models. Yet the Starseeds were constantly moving, so they clearly could not serve as a fixed reference point or an origin.

Although the Spell Model itself did not move, it was a model formed from multiple Starseeds. How could it serve as a reference point?

If one of the Spell Model's Starseeds were used as the origin, there would also be cases where two Spell Model nodes overlapped or star tracks crossed and interfered with one another.

But that was easy enough to solve. He only needed to treat the position of the first Starseed as the origin.

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

Then he would use an ordered triple to determine the position of every node in the Spell Model.

An ordered triple consisted of three numbers. Those three numbers instructed how to travel from the origin, the vector's starting point, to its tip, the vector's endpoint.

The first number represented how far to travel along the x-axis. A positive number meant moving right, while a negative number meant moving left.

The second number represented how far to travel afterward in the direction parallel to the y-axis.

The third number represented how far to travel along the z-axis.

Likewise, through the paths recorded for the Starseeds in the spell formula, he could reverse-engineer the coordinates of every Starseed.

Gao De stood, took a charcoal pencil from the shelf beside him, and began making notes directly in a blank area of the spell formula.

The first Starseed was the origin, marked with the coordinates (0,0,0).

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

Left and right were the x-axis, forward and back were the y-axis, and up and down were the z-axis.

The second Starseed's coordinates were recorded as (4/3,1,1/4).

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

The third Starseed moved with the second Starseed as its starting point, so it could not be directly recorded relative to the origin. But that was no problem either—it was merely simple vector addition.

Through calculation, the third Starseed's coordinates came out to (2,3/2,-1/4).

He calculated them one after another in this fashion.

Very soon, Gao De had broken down Acid Splash's Spell Model into an xyz coordinate axis and nine vector coordinates, including the origin.

Then Gao De stared intently at the nine ordered triples on the paper and began trying to memorize them.

Obviously, nine ordered triples were far simpler than the convoluted descriptions in the spell formula. Not to mention, Gao De had an innate sensitivity to numbers.

In only a few minutes, he had firmly committed those nine coordinates to memory.

"Let's try it."

Since the preliminary work was done, Gao De immediately got to work and began his attempt.