📚 Year 11 Cambridge Art: Formula & Theorem Quick-reference Handbook | Year 11 剑桥艺术：公式定理速查手册
This handbook brings together the must-know formulas, rules, and theorems from the Cambridge IGCSE Art & Design syllabus. Whether you are analysing artworks or creating your own compositions, these principles provide a reliable toolkit for controlling colour, space, proportion, and visual impact.
本手册汇集了剑桥 IGCSE 艺术与设计课程中必须掌握的公式、规则和定理。无论是分析艺术作品还是独立创作，这些原理为你提供一套可靠的工具，帮助你掌握色彩、空间、比例与视觉张力。
1. Colour Mixing Formulae | 色彩混合公式
The subtractive primaries used in painting are cyan, magenta, and yellow, but in studio practice most artists work with red, yellow, and blue as practical primaries. Every other hue is created by mixing these.
绘画中使用的减色三原色为青、品红、黄，但在实际画室中，多数艺术家将红、黄、蓝作为实用三原色。其他所有颜色都由它们混合而成。
Red + Yellow → Orange
红 + 黄 → 橙
Yellow + Blue → Green
黄 + 蓝 → 绿
Blue + Red → Purple
蓝 + 红 → 紫
Mixing a primary with an adjacent secondary produces a tertiary colour, such as yellow-green or red-violet. When all three primaries are combined in balanced amounts, the result is a neutral dark brown or grey, often called the neutrality rule.
将一种原色与相邻的间色混合可得到复色，例如黄绿或红紫。当三种原色以平衡比例混合时，结果是一种中性深棕色或灰色，这常被称为中和定律。
A high proportion of white added to any hue creates a tint; adding black creates a shade; adding grey produces a tone. The tint formula reduces saturation while raising value.
在任意颜色中加入大量白色可得到浅色调（tint）；加入黑色得到暗色调（shade）；加入灰色得到中间调（tone）。浅色调的公式在降低饱和度的同时提升明度。
2. The Colour Wheel Theorem | 色轮定理
The 12-step colour wheel organises hues by their visual relationships. The angle between two colours on the wheel predicts their harmony or contrast.
12 色色轮按照视觉关系排列颜色。两种颜色在色轮上的角度可以预测它们的调和或对比程度。
Complementary pair: angle = 180°
互补色对：角度 = 180°
Analogous colours: angle ≤ 30°
类似色：角度 ≤ 30°
Triadic scheme: three colours separated by 120°
三角色方案：三种颜色间隔 120°
Split-complementary: base colour + two colours adjacent to its complement (approx. ±30° from 180°)
分裂互补色：基色 + 其互补色两侧相邻的两色（距 180° 约 ±30°）
The theorem states that colours opposite each other (complements) intensify each other when placed side by side, while analogous neighbours create a restful, unified mood.
该定理指出，彼此相对的颜色（互补色）并列时会相互增强，而相邻的类似色则营造出宁静、统一的感觉。
3. Complementary Contrast & Afterimage Law | 互补对比与残像定律
When two complementary colours are juxtaposed, each appears more vivid. This simultaneous contrast effect follows a neurological law: the eye seeks balance and generates the complement as an afterimage.
当两种互补色并置时，每一种都显得更加鲜艳。这种同时对比效应遵循一条神经生理定律：眼睛寻求平衡，并产生补色作为残像。
Staring at red → afterimage appears cyan
注视红色 → 残像呈现青色
Staring at blue → afterimage appears yellow
注视蓝色 → 残像呈现黄色
This law is used deliberately in painting to give shadows extra life: a shadow on a yellow object can be tinted with violet to make the lit area sing.
该定律被画家刻意运用，使阴影更具生命力：在黄色物体的暗部加入紫色，可让受光区域更加夺目。
4. Rule of Thirds & Golden Ratio | 三分法与黄金比例
Dividing a picture plane into nine equal rectangles with two horizontal and two vertical lines creates four power points. Placing key elements on these intersections yields a dynamic but balanced composition.
用两条水平线和两条垂直线将画面分成九个相等的矩形，就产生了四个力量点。将关键元素放置在这些交点上，可以获得动态而平衡的构图。
The golden ratio provides a timeless proportional system. The ratio φ (phi) ≈ 1.618 appears in nature and classical art. When a line is divided so that the ratio of the smaller part to the larger part equals the ratio of the larger part to the whole, the division follows the golden section.
黄金比例提供了一种永恒的比例系统。比值 φ ≈ 1.618 出现在自然界和古典艺术中。当一条线段被分割，使得较短部分与较长部分之比等于较长部分与整段之比，该分割就符合黄金分割。
φ = (1 + √5) ÷ 2 ≈ 1.618
φ = (1 + √5) ÷ 2 ≈ 1.618
A golden rectangle’s sides are in the ratio φ:1. By removing a square, the remaining rectangle retains the same proportions. Artists often superimpose a golden spiral, constructed from quarter-circles inside the sequence of golden rectangles, to guide the viewer’s eye.
黄金矩形的边长比为 φ:1。移除一个正方形后，剩下的矩形仍保持相同比例。艺术家常在连续的黄金矩形内绘制四分之一圆弧，构成黄金螺旋，用以引导观者的视线。
5. Linear Perspective Principles | 线性透视定理
Linear perspective relies on the convergence of parallel lines at vanishing points on the horizon line. The horizon line always sits at the viewer’s eye level.
线性透视依赖于平行线向地平线上的灭点汇聚。地平线始终位于观者的视平线处。
One-point perspective: all depth lines converge to a single vanishing point
一点透视：所有纵深线汇聚于单一灭点
Two-point perspective: vertical edges remain vertical; horizontal edges converge to left and right vanishing points
两点透视：垂直边保持垂直；水平边分别向左、右灭点汇聚
Equal-sized objects appear smaller as their distance increases. The apparent height h’ is inversely proportional to the distance d from the viewer: h’ ∝ 1/d. This diminishing scale creates a realistic sense of depth.
同样大小的物体随距离增大而显得更小。外观高度 h’ 与观者距离 d 成反比：h’ ∝ 1/d。这种尺度缩减能营造逼真的深度感。
Foreshortening applies the same principle to figures: a limb pointing toward the viewer appears compressed, and its length is reduced according to the angle it makes with the picture plane.
前缩法应用相同原理于人体：指向观者的肢体显得压缩，其长度根据与画面平面的夹角而减小。
6. Aerial Perspective Formula | 空气透视公式
Aerial perspective, also called atmospheric perspective, describes how distance alters colour and tone. The further an object recedes, the more it is veiled by the intervening atmosphere.
空气透视，也称大气透视，描述了距离如何改变色彩与影调。物体越远，越被中间大气所笼罩。
Contrast reduction: distant darks become lighter, lights become slightly darker → tonal range narrows
对比度降低：远处的暗部变亮，亮部微暗 → 影调范围缩小
Colour shift: warm hues recede toward cool (blue-grey) tones; saturation decreases
色彩偏移：暖色向冷色（蓝灰）消退；饱和度降低
Edge softening: outlines become less distinct with distance
边缘柔化：轮廓随距离变得不清晰
An approximate practical formula: for every unit of depth, mix a small amount of sky colour (typically a greyed blue) into the object’s local colour and reduce its contrast by one step. This simulates atmospheric scattering.
一条近似实用公式：每增加一个深度单位，就在物体固有色中混入少量天空色（通常是灰蓝色），并将对比度降低一级，以模拟大气散射。
7. Human Proportions Canon | 人体比例准则
The classical canon of human proportions, popularised by Leonardo da Vinci’s Vitruvian Man, uses the head length as the basic unit of measurement. An idealised standing figure is 7.5 to 8 heads tall.
以达·芬奇的《维特鲁威人》为代表的古典人体比例准则，使用头长作为基本测量单位。理想化的站立人体身高为 7.5 至 8 个头长。
Total Height = 8 × Head Length
身高 = 8 × 头长
Shoulder Width ≈ 2 × Head Length
肩宽 ≈ 2 × 头长
Pelvis height = 4 heads from top (midpoint of the body)
耻骨高度 = 从头顶向下 4 个头长（身体中点）
Arm length from shoulder to fingertip ≈ 3.5 heads; hand length ≈ ¾ head
臂长从肩到指尖 ≈ 3.5 个头长；手长 ≈ ¾ 头长
These modular relationships help artists draw figures convincingly and check proportion errors. The face itself can be divided into thirds: hairline to brow, brow to nose base, nose base to chin.
这些模块化关系有助于艺术家令人信服地绘制人体，并检查比例错误。面部本身可分为三等份：发际线至眉毛、眉毛至鼻底、鼻底至下巴。
8. Tonal Value & Shading Models | 明度值与阴影模型
A successful rendering of form depends on a logical tonal structure. The five-value system organises light and shadow into highlight, light, mid-tone, shadow core, and cast shadow. Reflected light lifts the edge of the shadow core.
成功的形体塑造依赖于合乎逻辑的明度结构。五调子系统将光与影组织为高光、亮部、中间调、明暗交界线和投影。反光会提亮暗部边缘。
Highlight → Light → Mid-tone → Core Shadow → Reflected Light → Cast Shadow
高光 → 亮部 → 中间调 → 明暗交界线 → 反光 → 投影
The core shadow, or terminator, occurs where the surface turns away from the light source. The angle of reflectance follows the law that the angle of incidence equals the angle of reflection, determining where the brightest highlight appears on a glossy surface.
明暗交界线出现在表面背离光源的转折处。反射角遵循入射角等于反射角的定律，决定了光滑表面上最亮高光的位置。
Values can be arranged on a 9-step scale from white (value 1) to black (value 9). A practical formula for shading a sphere: the lightest area is the highlight; immediately below it is the light-tone; the core shadow sits opposite the light source; reflected light softens the core shadow on the side facing the ground.
明度可按照从白（明度 1）到黑（明度 9）的 9 级色阶来排列。绘制球体阴影的实用公式：最亮处为高光；紧接下方为亮部；明暗交界线位于背对光源的一侧；朝下的一面则因反光而使暗部柔化。
9. Visual Balance & Composition Weight | 视觉平衡与构图重量
Visual weight is not physical mass but the perceptual pull an element exerts. Large, dark, highly textured, or complex shapes appear heavier; small, light, smooth, or simple shapes appear lighter. A composition feels balanced when the sum of visual moments about the centre is zero.
视觉重量并非物理质量，而是元素所产生的感知吸引力。大面积、暗色、纹理丰富或形状复杂的物体显得更重；小而亮、平滑或简洁的形状显得更轻。当关于画面中心的视觉力矩之和为零时，构图便达到平衡。
Visual moment = Visual weight × Distance from centre
视觉力矩 = 视觉重量 × 到中心距离
A dark, small shape placed far from the centre can balance a large, pale shape placed closer to the centre. This lever principle allows asymmetric balance, which is more dynamic than formal symmetry.
一个远离中心的暗色小图形可以平衡一个靠近中心的亮色大图形。这一杠杆原理允许不对称平衡，比形式上的对称更富动感。
Radial balance arranges elements around a central point, creating a sense of swirling energy, while symmetrical balance distributes equal weights on both sides of a central axis for stability.
辐射式平衡将元素围绕中心点排列，产生旋转的能量感；对称式平衡则在中心轴两侧分布等量视觉重量，以营造稳定感。
10. Colour Harmony Schemes | 色彩调和方案
Colour harmony follows predictable geometric relationships on the colour wheel. Choosing a scheme before starting a painting helps unify the emotional tone and visual coherence of the work.
色彩调和遵循色轮上可预测的几何关系。在动笔前选定一种调合方案，有助于统一作品的情感基调与视觉连贯性。
| Scheme | 方案 | Formula | 公式 | Mood | 氛围 |
|---|---|---|
| Monochromatic | 单色 | One hue + its tints, shades and tones | Calm, unified, subtle |
| Analogous | 类似色 | Three adjacent hues (e.g. yellow, yellow-green, green) | Harmonious, serene, natural |
| Complementary | 互补色 | Hue + hue at 180° opposite | Vibrant, high contrast, energetic |
| Split-complementary | 分裂互补色 | Base hue + two colours adjacent to its complement | Strong but less tense than straight complement |
| Triadic | 三角色 | Three hues evenly spaced at 120° | Balanced, lively, colourful |
| Tetradic (rectangle) | 矩形配色 | Two pairs of complements (e.g. red-green + blue-orange) | Published by TutorHao | Year 11 艺术 Revision Series | aleveler.com
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