[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"origin-above-the-mage":3,"chapter-above-the-mage-above-the-mage-chapter-16":6},{"origin":4,"title":5},"chinese","Above the Mage!",{"chapter":7,"nextChapterSlug":19,"prevChapterSlug":20,"totalChapters":21,"novelImage":22},{"id":8,"novel_id":9,"title":10,"slug":11,"index":12,"content":13,"wordcount":14,"created_at":15,"updated_at":15,"volume":16,"translator":17,"content_hash":18},2257020,4404,"Chapter 16: Analytic Geometry and Cartesian Coordinates","above-the-mage-chapter-16",16,"\u003Cp>Gao De opened his eyes and unconsciously rubbed his temples.\u003C\u002Fp>\n\u003Cp>Constructing a spell model is extremely draining.\u003C\u002Fp>\n\u003Cp>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.\u003C\u002Fp>\n\u003Cp>If he could cultivate it to the full sixteen petals, such immense mental energy would make constructing a 0-level spell model far easier.\u003C\u002Fp>\n\u003Cp>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.\u003C\u002Fp>\n\u003Cp>With Gao De’s current mental energy, each failure in constructing the acid splash spell model left his head throbbing and his mind exhausted.\u003C\u002Fp>\n\u003Cp>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.\u003C\u002Fp>\n\u003Cp>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.\u003C\u002Fp>\n\u003Cp>“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.\u003C\u002Fp>\n\u003Cp>If even mastering a 0-level spell is this grueling, imagine the effort required to become a powerful mage.\u003C\u002Fp>\n\u003Cp>Yet he did not complain.\u003C\u002Fp>\n\u003Cp>They say mages are spell lords.\u003C\u002Fp>\n\u003Cp>Spell lords, spell lords—how can you become a lord without first being a servant?\u003C\u002Fp>\n\u003Cp>Failure is the mother of success.\u003C\u002Fp>\n\u003Cp>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.\u003C\u002Fp>\n\u003Cp>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.\u003C\u002Fp>\n\u003Cp>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?”\u003C\u002Fp>\n\u003Cp>If other mages knew his thought now, they would laugh at his arrogance.\u003C\u002Fp>\n\u003Cp>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?\u003C\u002Fp>\n\u003Cp>Gao De had no such superstitious reservations.\u003C\u002Fp>\n\u003Cp>In the world of mathematics, if a method is unworkable or too difficult, changing your approach is common.\u003C\u002Fp>\n\u003Cp>Could he first fix the positions of all stars, then connect the star-trails?\u003C\u002Fp>\n\u003Cp>This idea suddenly flashed into Gao De’s mind.\u003C\u002Fp>\n\u003Cp>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.\u003C\u002Fp>\n\u003Cp>—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.\u003C\u002Fp>\n\u003Cp>Compared to the traditional method of constructing spell models, the efficiency gain was not merely slight.\u003C\u002Fp>\n\u003Cp>It was the difference between an abacus and a computer.\u003C\u002Fp>\n\u003Cp>Gao De had always been action-oriented: if he had an idea, he acted on it.\u003C\u002Fp>\n\u003Cp>The first problem to solve: how to determine the position of each star.\u003C\u002Fp>\n\u003Cp>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.\u003C\u002Fp>\n\u003Cp>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?\u003C\u002Fp>\n\u003Cp>First, you need an origin.\u003C\u002Fp>\n\u003Cp>The origin is the starting point of all vectors.\u003C\u002Fp>\n\u003Cp>Only by establishing the origin can you determine distances and thus the vector coordinates of each node.\u003C\u002Fp>\n\u003Cp>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.\u003C\u002Fp>\n\u003Cp>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.\u003C\u002Fp>\n\u003Cp>But this was easily solved: treat the position of the first star as the origin.\u003C\u002Fp>\n\u003Cp>Centered on the origin, establish the classic xyz coordinate system.\u003C\u002Fp>\n\u003Cp>Then use an ordered triplet to define the position of each node in the spell model.\u003C\u002Fp>\n\u003Cp>A triplet consists of three numbers, guiding how to reach the vector’s endpoint (tip) from the origin (starting point).\u003C\u002Fp>\n\u003Cp>The first number indicates distance along the x-axis: positive means right, negative means left.\u003C\u002Fp>\n\u003Cp>The second number indicates distance along the parallel y-axis after that.\u003C\u002Fp>\n\u003Cp>The third number indicates distance along the z-axis.\u003C\u002Fp>\n\u003Cp>Likewise, by analyzing the star movements recorded in the spell recipe, he could reverse-calculate each star’s coordinates.\u003C\u002Fp>\n\u003Cp>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.\u003C\u002Fp>\n\u003Cp>The first star is the origin, coordinates: (0, 0, 0).\u003C\u002Fp>\n\u003Cp>“Forward one, right one and one-third, up one-quarter.”\u003C\u002Fp>\n\u003Cp>Left-right is the x-axis, front-back is the y-axis, up-down is the z-axis.\u003C\u002Fp>\n\u003Cp>The second star’s coordinates: (4\u002F3, 1, 1\u002F4).\u003C\u002Fp>\n\u003Cp>“Forward one-half, right two-thirds, down one-half.”\u003C\u002Fp>\n\u003Cp>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?\u003C\u002Fp>\n\u003Cp>After calculation, the third star’s coordinates were (2, 3\u002F2, -1\u002F4).\u003C\u002Fp>\n\u003Cp>Continue this process step by step.\u003C\u002Fp>\n\u003Cp>Soon, Gao De had decomposed the acid splash spell model into an xyz coordinate axis and nine vector coordinates, including the origin.\u003C\u002Fp>\n\u003Cp>Then, Gao De stared intently at the nine triplets on the paper and began trying to memorize them.\u003C\u002Fp>\n\u003Cp>Clearly, nine triplets were far simpler than the convoluted descriptions in the spell recipe—especially since Gao De had an innate sensitivity to numbers.\u003C\u002Fp>\n\u003Cp>Within just a few minutes, he had memorized all nine coordinates.\u003C\u002Fp>\n\u003Cp>“Let’s try.”\u003C\u002Fp>\n\u003Cp>Since the preparatory work was done, Gao De acted immediately and began his attempt.\u003C\u002Fp>\n\u003Cp>(End of chapter)\u003C\u002Fp>",1055,"2026-06-19T14:28:48.629Z",1,"Qwen3-Next 80B","75a6d56ef017acae765ae2c80fe5a1bcc16c1c0fcde66399ce69d7ddba51df9c","above-the-mage-chapter-17","above-the-mage-chapter-15",529,"https:\u002F\u002Fnovelzhen.com\u002Fimages\u002Fcovers\u002Fabove-the-mage-cover.jpg"]