Chapter 16: Analytic Geometry and Cartesian Coordinates

Gao De opened his eyes and unconsciously rubbed his temples.

Constructing a spell model is extremely draining.

His apprentice’s meditation technique had reached five petals; this level of mental energy was sufficient to complete the model for a cantrip, though it was somewhat taxing.

If he could cultivate it to the full sixteen petals, such immense mental energy would make constructing a 0-level spell model far easier.

The challenges in constructing a spell model are twofold: first, extreme precision—down to the slightest deviation, where a hair’s breadth error leads to a thousand-mile mistake—and second, the need for sufficient mental energy to expend through repeated attempts.

With Gao De’s current mental energy, each failure in constructing the acid splash spell model left his head throbbing and his mind exhausted.

After no more than three failures, his brain would begin to ache, his mental energy depleted beyond recovery—he would need rest to regain it before attempting again.

This is the drawback of insufficient mental energy: if a 1st-level mage were to construct a 0-level spell model, their efficiency would be dozens of times higher than Gao De’s—even if they failed, they could fail dozens of times in a single day without issue.

“Constructing spell models is indeed not simple—no wonder the previous owner spent over a year mastering the repair and mage hand cantrips,” Gao De muttered.

If even mastering a 0-level spell is this grueling, imagine the effort required to become a powerful mage.

Yet he did not complain.

They say mages are spell lords.

Spell lords, spell lords—how can you become a lord without first being a servant?

Failure is the mother of success.

Gao De closed his eyes and reviewed his recent failed attempt, quickly identifying the issue—while focusing on moving the third star, the second star had shifted slightly. A tiny disturbance triggered a chain reaction.

Since the second star-trail had already extended from the second star to the third, even the slightest deviation in the second star’s position caused the entire spell model to collapse.

This is another challenge in constructing spell models: no room for error whatsoever—if one mistake occurs, everything must start over; you cannot simply correct the flawed part. “This tolerance is far too low,” Gao De muttered, unconsciously thinking: “Could the construction process be optimized?”

If other mages knew his thought now, they would laugh at his arrogance.

Not only is the method for constructing spell models ancient beyond count, with no room for improvement—but even if there were, how could an apprentice possibly conceive it?

Gao De had no such superstitious reservations.

In the world of mathematics, if a method is unworkable or too difficult, changing your approach is common.

Could he first fix the positions of all stars, then connect the star-trails?

This idea suddenly flashed into Gao De’s mind.

As soon as it appeared, it struck him like a sudden revelation—he grew increasingly convinced it was viable, even feeling this was the correct way to construct a spell model.

—In this method, if any star drifted from its original position during construction, the entire model would not collapse; he would only need to adjust that single star’s position.

Compared to the traditional method of constructing spell models, the efficiency gain was not merely slight.

It was the difference between an abacus and a computer.

Gao De had always been action-oriented: if he had an idea, he acted on it.

The first problem to solve: how to determine the position of each star.

All spell recipes recorded the construction process as connecting star-trails while simultaneously determining each star’s position through relative displacement—they never explained how to fix a star’s position without connecting star-trails.

But for Gao De, this was no problem at all—the existing information was sufficient: wasn’t this simply analytic geometry? Establish a Cartesian coordinate system, then extract the vector coordinates of each star—wouldn’t that determine their positions?

First, you need an origin.

The origin is the starting point of all vectors.

Only by establishing the origin can you determine distances and thus the vector coordinates of each node.

The spell star-sea contained nothing but stars and spell models, yet the stars were constantly moving—they could not serve as fixed reference points.

The spell model itself does not move, but it is composed of multiple stars—how could it serve as a reference? If one star within the model were chosen as the origin, overlapping nodes or intersecting star-trails would occur.

But this was easily solved: treat the position of the first star as the origin.

Centered on the origin, establish the classic xyz coordinate system.

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

A triplet consists of three numbers, guiding how to reach the vector’s endpoint (tip) from the origin (starting point).

The first number indicates distance along the x-axis: positive means right, negative means left.

The second number indicates distance along the parallel y-axis after that.

The third number indicates distance along the z-axis.

Likewise, by analyzing the star movements recorded in the spell recipe, he could reverse-calculate each star’s coordinates.

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

The first star is the origin, coordinates: (0, 0, 0).

“Forward one, right one and one-third, up one-quarter.”

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

The second star’s coordinates: (4/3, 1, 1/4).

“Forward one-half, right two-thirds, down one-half.”

The third star’s movement begins from the second star’s position—it cannot be measured directly from the origin, but this was no major issue—wasn’t this just simple vector addition?

After calculation, the third star’s coordinates were (2, 3/2, -1/4).

Continue this process step by step.

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

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

Clearly, nine triplets were far simpler than the convoluted descriptions in the spell recipe—especially since Gao De had an innate sensitivity to numbers.

Within just a few minutes, he had memorized all nine coordinates.

“Let’s try.”

Since the preparatory work was done, Gao De acted immediately and began his attempt.

(End of chapter)