📚 Pre-U CAIE Art: Formula & Theorem Quick Reference Handbook | Pre-U CAIE 艺术：公式定理速查手册
This handbook bridges the analytical and the aesthetic, compiling the measurable principles that underpin strong visual compositions. From the mathematics of perspective to the geometry of colour harmony, these formulas and theorems serve as a reliable toolkit for the Pre-U CAIE Art & Design candidate. Use them to sharpen your formal analysis, strengthen your practical work, and articulate the structural logic behind your creative decisions.
本手册连接分析性与审美性，梳理支撑有力视觉构图的量化原理。从透视数学到色彩调和的几何规则，这些公式与定理为 Pre-U CAIE 艺术与设计考生提供一套可靠的工具包。运用它们深化形式分析、强化实践作品，并清晰阐述创作决策背后的结构逻辑。
1. Golden Ratio | 黄金比例
The golden ratio φ arises when a line is divided so that the ratio of the whole to the larger segment equals the ratio of the larger to the smaller. Often found in classical architecture, Renaissance painting, and natural spirals, it can be applied to canvas proportions, focal‑point placement, and compositional grids.
当一条线段被分成两段，且全长与较长段之比等于较长段与较短段之比时，即得黄金比例 φ。它常见于古典建筑、文艺复兴绘画与自然螺旋中，可应用于画布比例、焦点定位和构图网格。
φ = (1 + √5) / 2 ≈ 1.618
The reciprocal of φ is approximately 0.618, a ratio frequently observed in the spacing of elements within a well‑balanced design.
φ 的倒数约 0.618，在均衡设计的元素间距中频繁出现。
2. Rule of Thirds | 三分法则
The rule of thirds divides the picture plane into nine equal rectangles by two equally spaced horizontal lines and two equally spaced vertical lines. Placing key subjects along these lines or at their intersections creates dynamic tension and avoids the static central composition.
三分法则用两条等距水平线和两条等距垂直线将画面分成九个均等矩形。将主要对象沿这些线条或在其交点处放置，能产生动态张力，避免呆板的中心构图。
Intersection points: (w/3, h/3), (2w/3, h/3), (w/3, 2h/3), (2w/3, 2h/3)
Where w is the canvas width and h the height. Though more a guideline than a strict formula, mathematically it leverages the high‑impact positions of a 3 × 3 grid.
其中 w 为画布宽度，h 为高度。虽更似指导原则而非严格公式，但从数学上它利用了 3 × 3 网格中视觉冲击最强的位置。
3. Fibonacci Spiral in Composition | 构图中的斐波那契螺旋
The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, …) generates quarter‑circle arcs within squares whose side lengths correspond to the sequence. Overlaying this spiral on an artwork guides the viewer’s eye in a natural, flowing path toward the focal point.
斐波那契数列（0, 1, 1, 2, 3, 5, 8, 13, …）在边长对应数列数值的正方形内生成四分之一圆弧。将此螺旋叠于作品之上，可引导视线沿自然流转的路径移向焦点。
Fₙ = Fₙ₋₁ + Fₙ₋₂, with F₀ = 0, F₁ = 1
The ratio Fₙ / Fₙ₋₁ converges to φ as n increases, linking the spiral to the golden ratio. Use this to position secondary elements so they spiral inward to the centre of interest.
当 n 增大，比值 Fₙ / Fₙ₋₁ 趋向 φ，将螺旋与黄金比例关联。以此安排次要元素，使之螺旋向内聚焦于兴趣中心。
4. One‑Point Perspective: Depth Projection | 一点透视：深度投影
In one‑point perspective, all receding parallel lines converge at a single vanishing point (VP) on the horizon line. The apparent height of an object decreases proportionally with distance. The mathematical formula for the image height y’ of an object of real height Y at distance Z from the viewer (with station point distance D) is as follows.
在一点透视中，所有后退的平行线汇聚于视平线上单一消失点。物体的视高度随距离按比例缩小。真实高度为 Y 的物体在距观察者距离 Z 处，其像高 y’ 的计算公式如下（观察点距离为 D）。
y’ = (D × Y) / Z
Set the vanishing point at coordinates (VPₓ, horizon) and measure spacing along receding lines using equal divisions that contract toward VP. This formula ensures accurately foreshortened spatial intervals.
将消失点设于坐标 (VPₓ, 地平线)，沿后退线度量等分间距，随靠近 VP 而收缩。该公式确保透视缩短的空间间隔准确。
5. Two‑Point Perspective: Measuring Points | 两点透视：测点法
Two‑point perspective uses two vanishing points on the horizon. To construct a cube correctly, you need measuring points (MP) derived from the station point (SP). For a 90° corner facing the viewer, the distance from each VP to its corresponding MP equals the distance from VP to SP.
两点透视使用视平线上两个消失点。为正确构建立方体，需要用测点，它由观察点导出。对于面向观众的 90° 转角，每个消失点至相应测点的距离等于消失点至观察点的距离。
VP₁ MP₁ = VP₁ SP and VP₂ MP₂ = VP₂ SP
Use these measuring points to transfer true widths into perspective depth. The geometry guarantees that the foreshortened square base obeys the law of similar triangles, preserving proportional accuracy across the composition.
利用这些测点将真实宽度转入透视深度。该几何关系确保透视缩短的方形底面遵循相似三角形法则，维持构图整体的比例准确性。
6. Vitruvian Proportions of the Human Figure | 维特鲁威人体比例
Leonardo da Vinci’s Vitruvian Man codified canonical body ratios: height equals arm span, the navel marks the centre of a circle encompassing the limbs, and the body can be inscribed in both a circle and a square. Many life‑drawing systems rely on the head‑height module.
达·芬奇的维特鲁威人将经典人体比例法典化：身高等于臂展，肚脐为包络四肢的圆心，人体可同时内接于圆与正方形。许多人体素描体系依赖头高模块。
Total height = 8 head‑lengths (idealised); foot = 1/6 of height; face height = 1/10 of height
These modular relationships help artists maintain consistency when scaling figures, designing fashion illustrations, or placing figures within architectural settings.
这些模块化关系帮助艺术家在缩放人物、设计时装画或将人物置于建筑场景时保持一致性。
7. Colour Harmony Equations (Complementary & Analogous) | 色彩调和方程（互补与类似色）
Colour wheels can be represented on a circular continuum of hues numbered 0° to 360°. Complementary colours lie 180° apart; split‑complementary uses ±150° offsets. Analogous harmonies sit within a 30°–60° sector. Here are the hue‑angle formulas.
色轮可表示为 0° 至 360° 的连续色相环。互补色相距 180°；分裂互补色使用 ±150° 偏移。类似色调和位于 30°–60° 扇区内。以下是色相角公式。
H₂ = (H₁ + 180°) mod 360° (complementary)
H₂ = (H₁ + 150°) mod 360°, H₃ = (H₁ + 210°) mod 360° (split‑complementary)
Analogous palettes: H₂ = H₁ ± 30°, H₃ = H₁ ± 60°. Adjusting saturation and brightness while maintaining these angular relationships yields cohesive colour schemes ideal for painting, graphic design, and textile work.
类似色盘：H₂ = H₁ ± 30°， H₃ = H₁ ± 60°。保持这些角度关系的同时调整饱和度与明度，可生成协调的配色方案，适用于绘画、平面设计与纺织品创作。
8. RGB to CMYK Conversion for Print | 印刷用 RGB 转 CMYK 公式
When preparing digital artwork for print, precise colour management requires converting additive RGB values (0–255) to subtractive CMYK percentages. First normalise R, G, B to the range 0–1, then compute K, C, M, Y using the standard formulas.
准备数字作品用于印刷时，准确的色彩管理需将加色 RGB 值（0–255）转换为减色 CMYK 百分比。首先将 R、G、B 归一化至 0–1，然后用标准公式计算 K、C、M、Y。
K = 1 − max(R’, G’, B’)
C = (1 − R’ − K) / (1 − K), M = (1 − G’ − K) / (1 − K), Y = (1 − B’ − K) / (1 − K)
Where R’ = R/255, etc. These formulas preserve the integrity of deep shadows and vivid hues when transferring artwork to physical media.
其中 R’ = R/255，等等。这些公式在将作品转移到实体媒介时保持深色阴影与鲜艳色彩的完整性。
9. Luminosity and Contrast Formula | 亮度与对比度公式
Luminosity (perceived brightness) of a colour can be estimated by weighted RGB components. For accurate tonal analysis, artists and photographers rely on the linear luminance Y. The contrast between two areas is then expressed as a ratio.
色彩的亮度（感知光度）可通过加权 RGB 分量估算。为精确进行调性分析，艺术家与摄影师依赖线性亮度 Y。两个区域间的对比度以比值表示。
Y = 0.2126 × R + 0.7152 × G + 0.0722 × B (sRGB linearised)
Contrast ratio = (Y₁ + 0.05) / (Y₂ + 0.05)
Maintaining a minimum contrast ratio of 3:1 for large elements and 4.5:1 for fine detail ensures legibility in graphic design and accessible art presentations.
大型元素保持最低 3:1 的对比度，精细细节保持 4.5:1，可确保平面设计中的可读性与无障碍艺术展示。
10. Visual Weight and Balance Equation | 视觉重量与平衡方程
Visual weight is a composite of size, colour intensity, texture, and position. A simple equilibrium model places elements on a see‑saw centred on the composition’s optical centre. The moment of an element equals its visual weight multiplied by its distance from the centre.
视觉重量是大小、色彩强度、质感与位置的复合体。一个简单的平衡模型将元素置于以构图光学中心为支点的跷跷板上。元素的视觉力矩等于其视觉重量乘以距中心的距离。
Σ (Wᵢ × dᵢ) = 0, where Wᵢ is visual weight and dᵢ is signed distance from centre
Achieving zero net moment produces a sense of equilibrium. Adjusting a bright red accent (high weight) further from centre can offset a large but neutral mass on the opposite side.
实现净力矩为零可产生平衡感。将明亮的红色点缀（高重量）置于稍远离中心的位置，可抵消对侧大块但中性的体量。
11. Typography x‑Height and Readability Ratios | 字体 x 高度与可读性比率
The x‑height is the height of the lowercase ‘x’ relative to the cap height. It significantly affects perceived type size and readability. The x‑height ratio formula helps typographers match typefaces harmoniously.
x 高度是小写字母 ‘x’ 相对于大写字母高度的高度。它明显影响感知字号与可读性。x 高度比率公式有助于排印师协调匹配字体。
x‑height ratio = x‑height / cap height
Ratios of 0.45–0.55 are typical for body text; display faces may range from 0.35 to 0.65. When pairing typefaces, choose similar x‑height ratios to maintain visual rhythm across multilingual layouts.
正文通常比率为 0.45–0.55；展示字体可在 0.35 至 0.65 间变化。搭配字体时，选择相近的 x 高度比率，可在多语言版式中保持视觉节奏。
12. Dynamic Symmetry and Root Rectangles | 动态对称与根号矩形
Dynamic symmetry constructs compositions using root rectangles (√2, √3, √4, √5) whose proportions generate endless harmonic subdivisions. A √2 rectangle, for instance, can be divided into a square and a smaller √2 rectangle, creating a natural nesting rhythm ideal for complex narrative layouts.
动态对称利用根号矩形（√2、√3、√4、√5）构建构图，其比例可产生无穷的调和细分。例如，√2 矩形可被分为一个正方形与一个更小的 √2 矩形，形成天然的嵌套韵律，适合复杂的叙事性布局。
√2 ratio = 1 : 1.414; √3 ratio = 1 : 1.732; √5 ratio = 1 : 2.236
These irrational‑proportioned canvases were fundamental to the design canons of classical Greek art and were revived by modernist painters. Use their internal diagonal‑reciprocal divisions to place horizon lines and key structural breaks with mathematical elegance.
这些无理数比例的画布是古希腊艺术设计准则的基础，并被现代主义画家复兴。利用其内部对角线‑互逆划分来设置地平线和关键结构转折，可获得数学式的优雅。
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