📚 CCEA Pre-U Art: Quick-Reference Handbook of Formulas and Theorems | Pre-U CCEA 艺术：公式定理速查手册
This quick-reference handbook gathers the essential formulas and theorems that underpin artistic practice for CCEA Pre-U Art. From the geometry of perspective to colour mixing equations, these mathematical relationships provide a systematic foundation for composition, form, and expression. Use this guide to reinforce your understanding and apply these principles with confidence.
本速查手册汇集了 CCEA Pre-U 艺术课程所依托的核心公式与定理。从透视几何学到色彩混合方程，这些数学关系为构图、造型和表现提供了系统化的基础。运用本手册巩固你的理解，并自信地将这些原理付诸实践。
1. Linear Perspective: The Vanishing Point Theorem | 线性透视：消失点定理
In one-point perspective, all lines parallel to the viewer’s line of sight converge at a single vanishing point (VP). If the picture plane is at a distance D from the observer, an object of actual height h placed behind the plane at distance d projects an on-plane height h’ according to the formula:
在一点透视中，所有与观察者视线平行的线汇聚于单一消失点（VP）。若画面与观察者相距 D，实际高度为 h 的物体位于画后方距离 d 处，其在画面上的投影高度 h’ 由以下公式给出：
h’ = h × (D / (D + d))
For two-point perspective, each set of horizontal parallel lines has its own VP on the horizon line (HL). The angle between these sets and the picture plane determines the VPs’ positions. The distance between the two VPs is related to the station point distance and the angle of vision.
在两点透视中，每组水平平行线在视平线（HL）上各有一个消失点。这些线条与画面的夹角决定了消失点的位置。两消失点之间的距离与视点距离及视角大小相关。
2. The Golden Ratio in Composition | 构图中的黄金比例
The Golden Ratio φ (phi) is found by dividing a line into two parts so that the whole length divided by the long part equals the long part divided by the short part. Numerically:
黄金比例 φ（phi）是通过将线段分为两部分，使整体与较长部分的比值等于较长部分与较短部分的比值而得到的。数值上：
φ = (1 + √5) / 2 ≈ 1.618
Placing the focal point at a Golden Ratio divider of the canvas or using a Golden Rectangle (aspect ratio φ:1) creates aesthetically harmonious compositions. The Golden Spiral, built from quarter-circles inside successive Golden Rectangles, guides the viewer’s eye naturally through the work.
将主体或焦点置于画布的黄金分割点，或采用黄金矩形（宽高比 φ:1），都能营造出审美上和谐的构图。黄金螺旋由一系列黄金矩形内的四分之一圆弧连接而成，可自然地引导观者的视线穿行于作品之中。
3. Additive Colour Mixing Theorem | 加色混合定理
Additive mixing applies to coloured light, where the primary colours are Red, Green, and Blue (RGB). The fundamental equations are:
加色混合适用于色光，其三原色为红、绿、蓝（RGB）。基本方程式为：
R + G = Yellow
R + B = Magenta
G + B = Cyan
R + G + B = White
On the additive colour wheel, hues are positioned by angle: Yellow 60°, Cyan 180°, Magenta 300°. Pairs separated by 180° are complementary; when mixed as light, they sum to white.
在加色色环上，色相按角度定位：黄 60°、青 180°、品红 300°。相隔 180° 的颜色互为补色；作为色光混合时，它们相加得到白光。
4. Subtractive Colour Mixing Formula | 减色混合公式
When working with pigments, inks, or paints, subtractive mixing governs colour interaction. The primary colours are Cyan, Magenta, and Yellow (CMY). The mixing relationships are:
在使用颜料、墨水或涂料时，减色混合支配着色彩相互作用。原色为青、品红和黄（CMY）。混合关系为：
C + M = Blue
C + Y = Green
M + Y = Red
C + M + Y = Black (K)
On the artist’s colour wheel, complementary pairs sit opposite: Red–Cyan, Green–Magenta, Blue–Yellow. Mixing a hue with its complementary desaturates the colour, producing neutral greys or browns. The proportion of each component determines the resulting chroma and value.
在艺术家色环上，互补色对相对而立：红—青、绿—品红、蓝—黄。将一种色相与其补色混合会降低饱和度，产生中性灰或棕色。各组分的比例决定了最终的彩度和明度。
5. Colour Harmony: The Complementary Theorem | 色彩调和：互补定理
Harmonic colour schemes follow angular relationships on the colour wheel. The core theorem states that any hue angle θ has a complement at (θ + 180°) mod 360°. The main harmonic structures are summarised below:
调和色彩方案遵循色环上的角度关系。核心定理指出，任何色相角 θ 的补色为 (θ + 180°) mod 360°。主要调和结构总结如下：
| Scheme / 方案 | Angle interval / 角度间隔 | Example / 示例 |
|---|---|---|
| Complementary / 互补 | 180° | Red + Cyan / 红 + 青 |
| Split-complementary / 分裂互补 | Base ± (150°–165°) | Red + Blue-Green + Yellow-Green / 红 + 蓝绿 + 黄绿 |
| Analogous / 类似 | 30°–60° spread | Blue, Blue-Violet, Violet / 蓝、蓝紫、紫 |
| Triadic / 三色 | 120° | Red, Yellow, Blue / 红、黄、蓝 |
| Tetradic / 四色 | Two pairs, each 180° | Red–Green + Blue–Orange / 红–绿 + 蓝–橙 |
Applying these formulas ensures visual unity while controlling contrast and mood.
应用这些公式可确保视觉统一性，同时控制对比度和氛围。
6. The Rule of Thirds | 三分法则
The Rule of Thirds divides the picture plane into nine equal rectangles using two equally spaced horizontal lines and two equally spaced vertical lines. For a canvas of width W and height H, the lines occur at:
三分法则使用两条等距水平线和两条等距垂直线将画面划分为九个相等的矩形。对于宽为 W、高为 H 的画布，这些线的位置为：
Horizontal divisions: y = H/3 and y = 2H/3
Vertical divisions: x = W/3 and x = 2W/3
The four intersection points are known as power points. Placing the main subject on one of these points or aligning key elements along these lines yields dynamic yet balanced compositions. The Rule of Thirds approximates the Golden Ratio (φ ≈ 1.618) with the simpler fraction 2/3 ≈ 0.667 versus the golden section ratio of 0.618.
四个交点被称为视觉重点。将主体置于其中一点或让关键元素沿这些线对齐，可获得既动态又平衡的构图。三分法则用更简单的比例 2/3 ≈ 0.667 来近似黄金比例（φ ≈ 1.618），而黄金分割比约为 0.618。
7. Aerial Perspective Formula | 空气透视公式
Aerial perspective describes how contrast, saturation, and hue shift with distance due to atmospheric scattering. Distant objects appear lighter, bluer, and less distinct. A simplified model for saturation S and value V at distance x is:
空气透视描述了由于大气散射，对比度、饱和度和色相如何随距离变化。远处物体显得更亮、更蓝且细节模糊。距离 x 处饱和度 S 与明度 V 的简化模型为：
S’ ≈ S / (1 + kx) and V’ ≈ V + (1 − V)(1 − e−mx)
where k and m are positive constants. In practical terms, artists apply three shifts: value shifts to lighter tones in the background; chroma shifts to weaker intensity; and hue shifts towards cooler (blue) colours. This formula helps plan depth in landscape painting.
其中 k 和 m 为正常数。在实际运用中，艺术家实施三种偏移：明度向更亮的背景色调偏移；彩度向更弱的强度偏移；色相向偏冷（蓝）色调偏移。该公式有助于在风景画中规划深度。
8. Proportions of the Human Figure | 人体比例定理
Classical canons of proportion use the head as the unit of measurement. The average adult figure stands 7.5 heads tall, while the idealised heroic figure follows an 8-head canon. For a total height H, head height is H / 7.5 or H / 8. Key proportional constants include:
古典比例准则以头长为测量单位。成人平均身高为 7.5 头长，而理想化英雄式人体遵循 8 头身比例。对于总高度 H，头高为 H/7.5 或 H/8。关键比例常数包括：
- Midpoint of the body is at the pubic bone (approximately 4 heads from the top). / 身体中点位于耻骨处（从头顶起约 4 头长）。
- Legs from hip to sole ≈ 4 heads. / 髋至脚底约 4 头长。
- Fingertips reach mid-thigh; elbow aligns with waist. / 指尖达大腿中部；肘部与腰齐平。
- Shoulder width ≈ 2 heads (male) or 1.5 heads (female). / 肩宽男性约 2 头宽、女性约 1.5 头宽。
The navel often divides the body such that total height : navel–floor ≈ φ (1.618), linking human anatomy to the Golden Ratio.
肚脐通常以黄金比例 φ (1.618) 分割身体，即总身高与肚脐到地面距离之比接近 1.618，将人体解剖与黄金比例联系起来。
9. Shadow Projection Formula | 投影阴影公式
Cast shadows follow geometric rules determined by the light source. For a vertical object of height h illuminated by a distant light source (parallel rays) at an inclination angle α above the horizon, the shadow length L on flat ground is:
投影阴影遵循由光源决定的几何规则。对于高度为 h 的垂直物体，被仰角 α 的平行光源（如太阳）照射时，在水平地面上的阴影长度 L 为：
L = h / tan(α)
If the light source is a point with known coordinates, the shadow is constructed by projecting vertices from the light source onto the receiving surface. The shadow of a vertical pole always points directly away from the light source’s ground projection. In perspective drawing, shadows from multiple objects converge towards a single vanishing point on the horizon, known as the light-shadow VP.
若光源是已知坐标的点，则通过从光源向承影面投射物体顶点来构建阴影。垂直杆的阴影总是指向远离光源地面投影的方向。在透视绘画中，多个物体的阴影汇聚于视平线上的一个消失点，即光影消失点。
10. Rhythm and Repetition Theorem | 节奏与重复定理
Visual rhythm is created by repeating elements with controlled variation. The spacing d between identical elements defines spatial frequency f = 1/d. Types of rhythmic patterns can be expressed mathematically:
视觉节奏通过重复元素及受控的变化来创造。相同元素间的间距 d 定义了空间频率 f = 1/d。节奏模式的类型可数学表示为：
- Regular rhythm: constant spacing d. / 规则节奏：恒定间距 d。
- Progressive rhythm: spacing changes by a constant increment Δd, following dn = d1 + (n − 1)Δd. / 渐进节奏：间距以恒量 Δd 递增，遵循 dₙ = d₁ + (n − 1)Δd。
- Alternating rhythm: motifs follow a periodic sequence ABAB or ABCABC. / 交替节奏：母题遵循周期序列 ABAB 或 ABCABC。
Rhythm density affects tempo: closer spacing increases visual speed and tension. This theorem helps artists orchestrate movement and focal emphasis across a composition.
节奏密度影响速度感：间距越密，视觉速度和张力越强。该定理帮助艺术家在构图中安排运动感与焦点强调。
11. Texture Contrast Formula | 质感对比公式
Visual texture can be quantified by the mark size and distribution density. Define a Texture Roughness Index (TRI) as the average area of a mark divided by the unit area of the surface:
视觉质感可通过笔触大小和分布密度来量化。定义质感粗糙度指数（TRI）为笔触平均面积除以表面单位面积：
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