Above the mage

Chapter 16 Analytic Geometry and the Cartesian Coordinate System

Gao De opened his eyes and subconsciously rubbed his temples.

Constructing spell models is extremely mentally taxing.

His apprentice had cultivated his meditation technique to the point of having five petals, a level of mental strength sufficient for him to complete the model construction of the magic trick, though it was somewhat strenuous.

If one can cultivate to the point of having all sixteen petals, with that level of spiritual power, constructing a spell model for a 0-ring spell will become much easier.

The difficulty in constructing a spell model lies in two aspects: first, it requires precision, to the point that even the slightest error can lead to a completely different result; second, it requires the mage to possess sufficient mental energy to expend and to conduct repeated trials.

With Gao De's current mental strength, he tried to construct a magic model for acid splashing. Every time he failed, he felt his head throbbing and he was exhausted.

At most, after three failures, the brain will start to ache, the mental energy will be over-exhausted, and rest will be needed to wait for the mental energy to recover, making it impossible to construct spell models anymore.

This is the drawback of not having strong enough mental power. If a first-ring mage were to construct a spell model for a zero-ring spell, not to mention that the efficiency would be ten times that of a high-ranking mage, even if it failed, it wouldn't be a problem for the mage to fail dozens of times a day.

"Constructing a spell model is indeed not easy. No wonder it took my predecessor more than a year to master the two tricks, Healing and Mage's Hand," Gao De muttered to himself.

If mastering even a level 0 spell is this difficult, one can only imagine how much effort it takes to become a powerful mage.

However, he did not complain.

Everyone says that a Buddhist monk is a "master of law".

Master Lawyer, how can you be a master if you don't act like a grandson first?

Failure is the mother of success.

Gaode closed his eyes and reviewed the failed setup, quickly finding the problem—while focusing on controlling the movement of the third star, the position of the second star shifted slightly.

A single hair can affect the whole body.

With the second star orbit connecting the second and third stars already extended, any slight shift in the position of the second star would cause the entire spell model to collapse.

This is another challenge in constructing spell models:

There can be no room for error, otherwise everything will have to start from scratch, instead of just correcting the mistakes as they occur.

"This margin for error is too low," Gao De muttered to himself, subconsciously thinking, "Could we optimize the spell model construction process?"

If other mages knew what he was thinking at this moment, they would surely laugh at him for being ignorant and arrogant.

Not to mention that this method of constructing spell models, which has been passed down for countless years, could possibly still have room for optimization. Even if it did, how could a mage apprentice possibly think of it?

Gaode Maps wouldn't have these kinds of miscellaneous concerns.

In the world of mathematics, if one method doesn't work or is difficult, it's common to try a different approach.

Could we determine the positions of all the stars first, and then connect the star trails?

A thought suddenly popped into Gao De's mind.

After this idea came to him, it was like a sudden enlightenment; he suddenly understood and found it increasingly feasible, even believing that this was the correct way to construct a spell model.

In this way, even if any star shifts from its original position during the construction of the spell model, it will not cause the entire spell model to collapse. Everything can be started from scratch, and the position of the star can be adjusted in time.

Compared to traditional methods of constructing spell models, this method is far more efficient.

That's like the difference between an abacus and a computer.

Gaode has always been very proactive; if they have an idea, they will put it into action.

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

All spell recipes describe the spell model construction process as connecting star orbits while determining the position of each star through relative displacement, without explaining how to determine the position of stars without connecting star orbits.

But for Gaode Maps, this is not a problem at all; the existing information is sufficient—it's just simple analytic geometry.

Why not simply establish a Cartesian coordinate system, then decompose the vector coordinates of each star to determine its position?

First, we need an origin.

The origin is the source of all vectors.

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

In the Spell Star Sea, there are no other objects besides the stars and spell models. However, the stars are constantly moving, so they are obviously not fixed reference points and cannot be used as origins.

Although the spell model doesn't move, it's a model made up of multiple planets, so how can it be used as a reference point?

If one of the planets in the spell model is taken as the origin, there will be situations where the nodes of the two spell models overlap or the star orbits interfere with each other.

However, this is easy to solve; we can simply take the position of the first star as the origin.

Establish a classic xyz coordinate system centered at the origin.

Then, use an ordered triple array to determine the position of each node in the spell model.

A ternary array consists of three numbers that guide how to reach its tip (the end point of the vector) from the origin (the starting point of the vector).

The first number represents how far to move 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 travel along the z-axis.

Similarly, the coordinates of each star can be deduced by using the star paths recorded in the spell recipe.

Gao De got up, took a charcoal pencil from the shelf next to him, and began to write directly on the blank space of the spell recipe.

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

"Move forward one step, move one and a third to the right, move one-quarter upward."

The left and right sides are the x-axis, the front and back are the y-axis, and the up and down are the z-axis.

第二枚星子的坐标记为（4/3，1，1/4）。

"Advance one-half, advance two-thirds to the right, advance one-half down."

The third star moves from the second star, and cannot be directly compared to the origin for recording, but that's not a big problem—it's just a simple vector addition operation.

The coordinates of the third star can be calculated as (2, 3/2, -1/4).

Continue calculating in this manner.

Soon, Gaode decomposed the spell model of acid splashing into an xyz coordinate axis and nine vector coordinates including the origin.

Then, Gao De stared intently at the nine ternary arrays on the paper and began to try to memorize them.

Obviously, nine ternary arrays are much simpler than the complicated descriptions of spell recipes, not to mention that Gaode has an innate and extremely high sensitivity to numbers.

In just a few minutes, he memorized the nine coordinates.

"Give it a try."

With the preliminary work already done, Gaode decided to get right to it and immediately began testing it.

(End of this chapter)