📚 Pre-U CIE Art: Formula & Theorem Quick Reference Handbook | Pre-U CIE 艺术：公式定理速查手册
In Pre-U CIE Art & Design, creative expression often benefits from a deep understanding of underlying visual principles that behave almost like mathematical formulas. This quick reference handbook compiles the essential ‘formulas’ and ‘theorems’ of composition, colour, perspective, and proportion—rules you can apply to strengthen your artwork and analytical vocabulary. Mastery of these concepts will enable you to construct visually compelling pieces and to articulate your decisions with precision in your supporting studies.
在 Pre-U CIE 艺术与设计课程中，深刻理解那些如同数学公式一般的底层视觉原理，往往能让创意表达如虎添翼。这份速查手册汇集了构图、色彩、透视与比例方面最关键的“公式”与“定理”——你可以运用这些法则强化作品的表现力，并丰富分析评述时的专业词汇。掌握这些概念，你就能构建出视觉张力十足的作品，并在创作研究中有理有据地陈述自己的设计决策。
1. The Golden Ratio in Composition | 构图中的黄金比例
The golden ratio, denoted by the Greek letter φ (phi), approximates 1.618 and is found widely in nature and classical art. In composition, a canvas can be divided using the ratio a/b = (a+b)/a. Placing the focal point at the golden section line or using a golden rectangle often creates a sense of balance that feels instinctively harmonious to the viewer. Many iconic paintings and architectural designs adhere to this proportion.
黄金比例用希腊字母 φ 表示，约等于 1.618，广泛存在于自然界和古典艺术中。在构图中，画布可以按照 a / b = (a + b) / a 的关系进行分割。将视觉焦点放置在黄金分割线上，或采用黄金矩形构图，往往能营造出一种让观者本能感受到的平衡与和谐。许多经典绘画和建筑设计都严格遵循这一比例。
2. The Rule of Thirds | 三分法则
Divide the image frame into nine equal rectangles with two equally spaced horizontal lines and two equally spaced vertical lines. The rule of thirds states that important compositional elements should be placed along these lines or at their intersections. This simple grid offsets the subject from the centre, introducing dynamic tension and a more natural viewing flow. It is a practical simplification of the golden ratio for rapid compositional checks.
用两条等距水平线和两条等距垂直线将画面等分为九个矩形，这就是三分法则的网格。法则指出，重要的构图元素应当放置在这些线条上或它们的交叉点上。这一简易网格让主体偏离中心，增添了动感张力与更自然的视觉流动感。它是黄金比例在快速构图检查中的实用简化版。
| Element Placement | Horizon on upper or lower third line |
| Focal Point | At any of the four intersection points |
元素放置：地平线置于上三分之一或下三分之一线。焦点：任意一个交叉点。
3. Fibonacci Spiral in Visual Flow | 视觉流动中的斐波那契螺旋
The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, …) gives rise to a logarithmic spiral that appears in seashells and galaxies. By superimposing this spiral onto a canvas, artists can guide the viewer’s eye smoothly from the outermost curve toward the centre of interest. Overlaying a Fibonacci spiral helps in arranging elements along a sweeping, natural path that enhances narrative and rhythm.
斐波那契数列 (0,1,1,2,3,5,8,13,21, …) 衍生出对数螺旋，常见于贝壳与星系的形态。将这一螺旋叠放在画布上，艺术家可以引导观者的视线从外周弧线平滑地游向兴趣中心。叠加斐波那契螺旋有助于沿一条流畅的自然路径安排元素，从而强化叙事感与节奏变化。
Fₙ = Fₙ₋₁ + Fₙ₋₂ with F₀ = 0, F₁ = 1
4. Colour Wheel & Complementary Colours | 色环与互补色
The traditional 12-part colour wheel is a formulaic tool built from three primary colours (red, yellow, blue). Secondary colours are obtained by mixing two primaries: orange (red + yellow), green (yellow + blue), violet (blue + red). Complementary colours sit directly opposite each other on the wheel (e.g., red and green). Placing complements side by side maximises contrast and visual vibrancy, while mixing them neutralises intensity.
传统的十二色相环是由三原色（红、黄、蓝）构建的公式化工具。间色由两种原色混合而成：橙（红+黄）、绿（黄+蓝）、紫（蓝+红）。互补色位于色环的正对两端（例如红与绿）。将互补色并置能最大程度增强对比和视觉活力，而将它们相互混合则会削弱色彩纯度。
5. Colour Mixing Formulas | 色彩混合公式
Artists use precise mixing ratios to achieve desired hues. For instance, a warm green can be mixed as 3 parts yellow to 1 part blue, while a cool violet may require 2 parts blue to 1 part red. Tints are created by adding white (value increase with no hue shift), shades by adding black (value decrease), and tones by adding grey (both value and saturation change). Digital colour spaces also follow formulas: RGB additive mixing yields white light, whereas CMYK subtractive mixing yields black.
艺术家利用精确的混合比例来获得想要的色调。例如，暖调绿色可能用 3 份黄与 1 份蓝调出，而冷调紫色则需 2 份蓝加 1 份红。加白产生明调（明度提升而色相不变），加黑产生暗调（明度降低），加灰产生浊调（明度与饱和度同时改变）。数字色彩空间同样遵循公式：RGB加色模式混合出白光，而CMYK减色模式混合出黑。
| Target Hue | Suggested Ratio |
| Orange | 1 Red : 1 Yellow |
| Warm Green | 3 Yellow : 1 Blue |
| Violet | 2 Blue : 1 Red |
6. Colour Harmony Schemes | 色彩和谐公式
Colour harmony can be systematically derived from geometric relationships on the colour wheel. An analogous scheme uses three adjacent hues (e.g., yellow-green, green, blue-green) for unity. A triadic scheme selects three colours evenly spaced by 120° (e.g., red, yellow, blue) for vibrant balance. A split-complementary scheme takes one base colour and the two hues adjacent to its complement to soften contrast while retaining dynamics. These formulas enable intentional mood-setting and visual coherence.
色彩和谐可以从色环上的几何关系系统推导。类似色方案运用三个相邻色相（例如黄绿、绿、蓝绿）来营造统一感。三色组方案选取彼此间隔 120° 的三种颜色（如红、黄、蓝）以实现生动平衡。分裂补色方案取一基准色以及其补色两侧的两种色相，在保持动感的同时柔化对比。这些公式有助于有意识地塑造情绪和视觉连贯性。
7. Linear Perspective: One-Point | 线性透视：一点透视
One-point perspective relies on a single vanishing point (VP) on the horizon line. All receding parallel lines orthogonal to the picture plane converge at this VP. The formula for apparent size reduction with distance is that an object’s perceived height is inversely proportional to its distance from the viewer. This system is ideal for scenes facing the viewer directly, such as a corridor or a road disappearing into the distance.
一点透视依赖于地平线上的一个消失点。所有与画面垂直、向远方退去的平行线都汇聚于这一消失点。物体因距离产生的视觉尺寸缩减遵循这样的规律：感知高度与物体到观察者的距离成反比。这种系统最适合表现正对观者的场景，比如走廊或向远方消失的道路。
Apparent size ∝ 1 / Distance
8. Linear Perspective: Two-Point | 线性透视：两点透视
Two-point perspective uses two vanishing points located on the horizon line. Edges recede toward either the left or right VP, while vertical lines remain true. This formula is essential for depicting objects viewed at an angle, such as the corner of a building. The relative position of the VPs determines the angle of view; placing them far apart minimises distortion, while close VPs create dramatic angularity.
两点透视使用地平线上的两个消失点。棱边分别向左消失点或右消失点后退，而垂直线保持不变。这一公式对于表现成角度的物体至关重要，例如建筑物的转角。消失点的相对位置决定了视角；将它们设置得较远可以减少畸变，而较近则产生强烈的角度感。
9. Linear Perspective: Three-Point | 线性透视：三点透视
Three-point perspective adds a third vanishing point either high above (bird’s-eye view) or far below (worm’s-eye view). All vertical lines now converge to this third VP, dramatically exaggerating scale and depth. This formula is used for skyscraper views, towering cliffs, or extreme foreshortening. The three VPs must be carefully managed to avoid unrealistic warping unless intentional distortion is desired.
三点透视增加了第三个消失点，或在高远的空中（鸟瞰），或在深低的地面（虫视）。此时所有垂直线都汇聚于第三个消失点，极大地夸张了尺度和纵深感。这种公式用于摩天大楼场景、高耸的悬崖或极端的前缩透视。三个消失点必须仔细把控，以免产生失真，除非刻意追求变形效果。
10. Atmospheric Perspective Formula | 大气透视公式
Atmospheric perspective describes how colour and value shift with distance. As objects recede, contrast diminishes, colours become cooler and lighter, and edges soften. A practical formula is to mix a small amount of the sky colour (often a pale blue-grey) into distant objects. The rule ‘Value lightens by about 10% for every hundred metres of simulated depth’ helps create convincing spatial recession in landscapes.
大气透视描述了颜色与明度如何随距离变化。当物体向远方退去，对比度降低，色彩变冷变亮，边缘变柔。一个实用的混合公式是将少量天空颜色（通常为淡蓝灰）调入远景物体中。“模拟深度每增加一百米，明度约提升 10%”的规律有助于在风景画中营造出令人信服的空间退后感。
11. Light and Shadow: The Chiaroscuro Ratio | 光影：明暗对照比例
Chiaroscuro relies on a controlled ratio of light to shadow to model form. A common formula for a spotlight effect is a key light : fill light : reflected light intensity ratio of approximately 8:2:1. The core shadow is the darkest area on the form itself, and its edge marks the transition from light. The position of the highlight follows the law of reflection: angle of incidence equals angle of reflection (θᵢ = θᵣ). Mastering these luminance relationships turns flat shapes into volumetric solids.
明暗对照法依赖受控的光影比例来塑造形体。聚光效果常见的公式为，主光：补光：反射光的光强比例约为 8:2:1。明暗交界线是形体上最暗的区域，其边缘标示出光影转折。高光的位置遵循反射定律：入射角等于反射角（θᵢ = θᵣ）。掌握这些亮度关系，就能将平面图形转化为立体实体。
12. Human Proportions: The Vitruvian Formulas | 人体比例：维特鲁威公式
Leonardo da Vinci’s Vitruvian Man encodes proportional formulas: the height of the entire body equals eight head lengths; the distance from the top of the head to the chin is one-eighth of the total height; the span of the outstretched arms equals the height. The face can be subdivided: eyes sit at the halfway line, the base of the nose is one-third from eyes to chin, and the mouth is two-thirds down. These proportional rules provide a reliable scaffold for life drawing and figurative composition.
达芬奇的《维特鲁威人》编码了一套比例公式：全身高度等于八个头长；头顶到下巴的距离为总身高的八分之一；双臂平伸的宽度等于身高。面部可进一步细分：眼睛位于中线上，鼻底在眼到下巴的下三分之一处，嘴巴位于下三分之二处。这些比例法则为素描写生和人物构图提供了可靠的支架。
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