📚 Pre-U Cambridge Art: Formulas & Theorems Quick Reference Handbook | 剑桥Pre-U艺术：公式定理速查手册
This quick reference handbook distils essential quantifiable principles, geometric relationships and compositional ‘theorems’ that underpin the Pre-U Cambridge Art & Design syllabus. From harmonic proportion to colour mixing equations, perspective depth rules and gestalt laws, these formulas arm practitioners with structural awareness for both analysis and studio practice.
本速查手册提炼了支撑剑桥Pre-U艺术与设计课程的基本可量化原理、几何关系和构图“定理”。从和谐比例到色彩混合方程、透视深度法则和格式塔定律，这些公式为艺术家的分析和创作实践提供了结构意识。
1. Golden Ratio & Divine Proportion | 黄金比例与神圣比例
The golden ratio, denoted by the Greek letter φ (phi), is an irrational number approximately equal to 1.618. It arises when a line is divided into two parts so that the whole length divided by the long part is equal to the long part divided by the short part.
黄金比例用希腊字母 φ 表示，是一个约等于1.618的无理数。当一条线段被分成两部分，使得整体与较长部分之比等于较长部分与较短部分之比时，就产生了这个比例。
φ = (1 + √5) / 2 ≈ 1.618
The golden rectangle, with a side ratio of 1:φ, has been employed in works such as the Parthenon and Le Corbusier’s Modulor system. The Fibonacci sequence approximates φ, and its associated spiral can be used to place focal points.
边比为1:φ的黄金矩形被应用于帕特农神庙和勒·柯布西耶的模度系统。斐波那契数列近似于 φ，与之相关的螺旋线可用于放置画面焦点。
- Golden rectangle proportions: width = a, height = a/φ ≈ 0.618a
- 黄金矩形比例：宽度 = a，高度 = a/φ ≈ 0.618a
- Fibonacci spiral grows by quarter-turns following squares of side lengths 1,1,2,3,5,8…
- 斐波那契螺旋按1,1,2,3,5,8…边长的正方形四分之一圆弧依次增长。
2. Colour Theory: Additive & Subtractive Mixing Formulas | 色彩理论：加色与减色混合公式
Additive colour mixing applies to light (e.g. screens, stage lighting). The primaries are Red, Green, Blue. Combining all three produces white.
加色混合适用于光（如屏幕、舞台灯光），三原色为红、绿、蓝，三者叠加产生白光。
R + G → Y G + B → C B + R → M R + G + B → W
Subtractive mixing applies to pigments, inks and dyes. The primaries are Cyan, Magenta, Yellow. In theory, mixing all three yields black (K).
减色混合适用于颜料、油墨和染料，三原色为青、品红、黄。理论上三者混合得到黑色(K)。
C + M + Y → K C + M → B (Blue) M + Y → R (Red) C + Y → G (Green)
| Additive primaries: R G B | 加色原色：红 绿 蓝 |
| Subtractive primaries: C M Y | 减色原色：青 品红 黄 |
3. One-Point Perspective Depth Formula | 一点透视深度公式
In linear perspective, objects of equal size appear smaller as their distance from the viewer increases. For one-point perspective, the apparent height h’ of an object can be expressed as a function of its real height h and its depth distance D from the picture plane.
在线性透视中，同样大小的物体随着离观看者距离的增加而显得更小。对于一点透视，物体的视觉高度 h’ 可表示为其实际高度 h 和距画面深度 D 的函数。
h’ ≈ h × (d₀ / D) (where d₀ is the distance from eye to picture plane)
A more practical studio rule: if an object moves twice as far from the viewer, its projected height halves. Use convergence lines meeting at the vanishing point to determine proportional reductions.
更实用的画室法则：如果物体离观看者的距离增加一倍，其投影高度减半。利用汇聚于消失点的透视线来确定比例缩减。
Depth scaling between two identical objects at distances D₁ and D₂ from the station point: h₂ = h₁ × (D₁ / D₂).
两个相同物体位于距视点D₁和D₂时的深度缩放：h₂ = h₁ × (D₁ / D₂)。
4. Rule of Thirds & Compositional Balance | 三分法则与构图平衡
The rule of thirds divides the picture frame into a 3×3 grid, generating four power points where the lines intersect. Placing key elements on these intersections or along the grid lines enhances dynamic balance.
三分法则将画面平分成3×3网格，产生四个交叉点。将关键元素放置在这些交叉点或网格线上，可以增强动态平衡。
| Intersection coordinates (width w, height h): | 交叉点坐标（宽度 w，高度 h）： |
| (w/3, h/3) (2w/3, h/3) | 左下、右下附近 |
| (w/3, 2h/3) (2w/3, 2h/3) | 左上、右上附近 |
Visual weight can be balanced by distributing tonal contrast, colour saturation and detail density. A small dark shape can counterbalance a large light area.
通过分配色调对比、色彩饱和度和细节密度来平衡视觉重量。一小块暗色形状可以平衡一大片亮色区域。
5. Human Proportions: Vitruvian Canon | 人体比例：维特鲁威标准
Leonardo da Vinci’s ‘Vitruvian Man’ illustrates idealised human proportions derived from Vitruvius. The height of a well-proportioned figure is commonly about 7.5 to 8 times the height of the head.
达芬奇的《维特鲁威人》展示了源自维特鲁威的理想人体比例。一个比例良好的形象高度通常是头高的7.5到8倍。
H ≈ 7½h to 8h (H = total height, h = head height)
Other canonical ratios: arm span equals body height; the navel divides the body at about φ; the face is roughly three nose-lengths high. These modular relationships aid figure drawing and sculpture.
其他规范比例：臂展等于身高；肚脐将身体大致分为黄金比；面部约三鼻长。这些模数关系有助于人物素描和雕塑。
- Height = 8 heads; Pubic bone at 4 heads; Knees at 6 heads.
- 身高=8头高；耻骨在4头高处；膝盖在6头高处。
6. Light Intensity & Shading Laws | 光强与明暗定律
The apparent brightness of a surface depends on both the distance from the light source and the angle of incidence. The inverse-square law governs point-source illumination.
表面视觉亮度取决于与光源的距离以及入射角度。平方反比定律支配点光源照明。
I = I₀ / d² (I = intensity, d = distance from source)
For a surface tilted relative to the light, Lambert’s cosine law applies: the intensity of light reflected to the viewer is proportional to the cosine of the angle between the light ray and the surface normal.
对于相对于光源倾斜的表面，适用朗伯余弦定律：反射到观察者的光强与光线和表面法线夹角的余弦成正比。
Iₛ = I₀ × cos θ (θ = angle of incidence)
Artists use these principles to model chiaroscuro and to predict the fall-off of cast shadows. The darkest shadows often follow a core shadow zone, transitioning with a terminator line.
艺术家运用这些原理塑造明暗对比，并预测投影的衰减。最暗的阴影通常遵循明暗交界线，形成过渡。
7. Colour Harmony: Hue Angles & Schemes | 色彩和谐：色相角度与配色方案
On a standard colour wheel, the hue circle can be treated as a 360° spectrum. Harmonious colour relationships are often defined by specific angular separations.
在标准色轮上，色相环可视为360°光谱。和谐的配色关系通常由特定的角度间隔定义。
| Complementary (ΔH = 180°) | 互补色，色相差180°，如红与青 |
| Split-complementary (ΔH ≈ 150° & 210°) | 分裂互补，两侧各偏移约30° |
| Analogous (ΔH ≤ 60°, adjacent) | 类似色，相邻60°以内 |
| Triadic (ΔH = 120°, 120°, 120°) | 三角色，间隔120°均匀分布 |
These hue-angle formulas help create balanced palettes. The addition of neutrals (black, white, grey) modifies saturation and value without altering the hue harmony.
这些色相角公式有助于创建平衡的调色板。添加中性色（黑、白、灰）可改变饱和度和明度而不破坏色相和谐。
8. Gestalt Principles of Visual Grouping | 格式塔视觉组合原则
Gestalt psychology provides a set of ‘laws’ that describe how the human brain organises visual elements into unified wholes. These principles function as perceptual theorems in composition.
格式塔心理学提供了一套“定律”，描述人脑如何将视觉元素组织成统一的整体。这些原则在构图中充当知觉定理。
- Law of Proximity: Elements close together are perceived as a group.
- 接近律：靠得近的元素被视为一个组。
- Law of Similarity: Similar shapes, colours or textures group perceptually.
- 相似律：相似的形状、颜色或质感在知觉上成组。
- Law of Closure: The mind fills missing information to complete a familiar form.
- 闭合律：大脑会补全缺失的信息以构成熟悉形状。
- Law of Continuity: The eye follows smooth paths and lines.
- 连续律：视线会沿着光滑路径和线条移动。
- Figure-Ground: Separation of the main subject from the background.
- 图底关系：主体与背景的分离。
Applying these principles allows artists to direct attention, create tension, and achieve clarity without relying on explicit outlines.
运用这些原则可使艺术家引导视线、制造张力并实现清晰度，而无需依赖明确的轮廓线。
9. Dynamic Symmetry & Root Rectangles | 动态对称与根号矩形
Dynamic symmetry is a compositional method based on root rectangles, where the ratio of the longer side to the shorter side is an irrational square root. These proportions were used extensively in classical and Renaissance art.
动态对称是一种基于根号矩形的构图方法，其中长边与短边之比为无理数平方根。这些比例在古典和文艺复兴艺术中被广泛使用。
√2 rectangle: 1 : √2 ≈ 1 : 1.414
√3 rectangle: 1 : √3 ≈ 1 : 1.732
√4 rectangle: 1 : 2 (double square)
√5 rectangle: 1 : √5 ≈ 1 : 2.236 (contains golden section)
A √2 rectangle retains its proportion when folded in half, forming the basis of the ISO A-series paper sizes (A3, A4 etc.). The internal subdivisions generate harmonic diagonals and reciprocal lines that guide placement.
√2矩形对折后比例不变，是ISO A系列纸张（A3、A4等）的基础。其内部分割产生的对角线和互反线可引导元素放置。
| √2 rectangle splits into two √2 rectangles | √2矩形可分割为两个小√2矩形 |
更多咨询请联系16621398022（同微信）
CommentsMore posts |
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply