📚 Behaviour of z-Transforms in the Complex Plane | 复平面上z变换的行为
are complex variables. Understanding the behaviour of these transformations geometrically — how they move, stretch, rotate, or invert sets of points — ticle explores the key transformations of the complex plane, from simple translations to Mö transformations, and explains their effects on shapes, angles, and the point at infinity.
在复数学习中，z变换是指从复平面到其自身的映射，通常表示为 w = f(z)，其中 z 和 w 均为复变量。从几何上理解这些变换的行为——它们如何移动、拉伸、旋转或反演点集——对于掌握复分析、保形映射及其在物理和工程中的应用至关重要。本文探讨复平面上从简单平移到莫比乌斯变换等关键变换，并解释它们对图形、角度以及无穷远点的作用。
1. The Complex Plane as a Dynamic Space | 作为动态空间的复平面
oint z. By studying how regions, curves, and entire figures are mapped onto the w-plane, we gain insight into the function' ex mappings are best visualised by observing the deformation of grids, circles, and lines.
复数 z = x + iy 可以表示为阿根平面上的点 (x, y)。变换 w = f(z) 为每个点 z 分配一个新的复数 w = u + iv。通过研究区域、曲线和整个图形如何映射到 w 平面，我们能深入理解函数的整体行为。与实函数不同，复映射的最佳可视化方法是观察网格、圆和直线的变形。
2. Translation: w = z + a | 平移：w = z + a
r example, if a = 3 + 2i, the entire plane slides 3 units to the right and 2 units upward.
最简单的变换是平移，由 w = z + a 给出，其中 a 是一个固定的复数。每个点 z 都会沿着对应于 a 的向量平移。这种映射保持距离、角度和方向不变，是一种等距变换。例如，若 a = 3 + 2i，整个平面将向右移动 3 个单位、向上移动 2 个单位。
w = z + a → u = x + Re(a), v = y + Im(a)
3. Rotation and Scaling: w = kz | 旋转与伸缩：w = kz
Multiplication by a fixed complex constant k = reⁱᶿ t at the origin when r = 0, and it sends circles centred at the origin to similar circles.
乘以固定复常数 k = reⁱᶿ（其中 r > 0，θ 是以弧度为单位的辐角）相当于将平面绕原点旋转角度 θ 并以因子 r 均匀缩放。如果 r = 1，变换是绕原点的纯旋转；如果 θ = 0，则是纯伸缩。这种映射除原点外处处保形，并将以原点为中心的圆映射为相似的圆。
| Form of k | Transformation |
|---|---|
| k = eⁱᶿ | Rotation by θ |
| k = r (real > 0) | Scaling by factor r |
| k = reⁱᶿ | Rotation by θ and scaling by r |
4. The Reciprocal Transformation: w = 1/z | 倒数变换：w = 1/z
The mapping w = 1/z is fundamental in complex analysis. Written as w = 1/z = z̄/|z| ide, and vice versa. The origin maps to the point at infinity, and infinity maps to the origin. This transformation has the remarkable property of mapping circles and lines to circles or lines.
映射 w = 1/z 是复分析中的基本变换。将其写为 w = 1/z = z̄/|z|²，可知它由关于单位圆的反演和关于实轴的反射组成。单位圆外的点映射到圆内，反之亦然。原点映射到无穷远点，无穷远点映射到原点。该变换的显著性质是能将圆和直线映射为圆或直线。
w = 1/z → |w| = 1/|z|, arg(w) = -arg(z)
5. Effect of w = 1/z on Lines and Circles | w = 1/z 对直线和圆的作用
ersely, a circle passing through the origin becomes a line that does not pass through the origin. Circles not passing through the origin map to other circles not passing through the origin. This unifying behaviour is best understood by treating lines as 'circles through infinity' in the extended complex plane.
在 w = 1/z 下，不经过原点的直线变换为经过原点的圆。反过来，经过原点的圆变成不经过原点的直线。不经过原点的圆映射为其他不经过原点的圆。如果把直线视为扩展复平面中“经过无穷远点的圆”，这一统一行为就更容易理解了。
Line: ax + by + c = 0 (c ≠ 0) → Circle through origin
6. The Extended Complex Plane and the Point at Infinity | 扩展复平面与无穷远点
z smoothly, we add a single point at infinity, denoted ∞, to the complex plane, forming the extended complex plane (Riemann sphere). All parallel lines 'meet' tional function can be extended continuously to infinity. This compactification allows us to say that Mö on the extended plane, mapping circles and lines onto circles and lines as a single family.
为了平滑地处理 w = 1/z 这类变换，我们在复平面上添加一个无穷远点（记作 ∞），形成扩展复平面（黎曼球面）。所有平行线在无穷远处“相交”，且任何多项式或有理函数都可以连续地延拓到无穷远点。这种紧致化使得我们可以说莫比乌斯变换在扩展平面上是双射的，并把圆和直线作为一个统一的族映射到圆和直线上。
7. General Bilinear (Möbius) Transformations | 一般双线性（莫比乌斯）变换
A Mö where a, b, c, d are complex constants and ad − bc ≠ 0. This family includes translations, rotations, dilations, and the reciprocal mapping as special cases. The condition ad − bc ≠ 0 ensures the map is invertible and non-degenerate. Under such a transformation, circles and lines in the z-plane are always mapped to circles or lines in the w-plane.
莫比乌斯变换是形如 w = (az + b)/(cz + d) 的有理函数，其中 a、b、c、d 为复常数且 ad − bc ≠ 0。该族包含了平移、旋转、伸缩和倒数映射作为特例。条件 ad − bc ≠ 0 确保映射可逆且非退化。在这样的变换下，z 平面上的圆和直线总是映射为 w 平面上的圆或直线。
w = (az + b)/(cz + d), ad - bc ≠ 0
8. Decomposition of Möbius Transformations | 莫比乌斯变换的分解
Any Mö − ad)/(c(cz + d)). This reveals that the mapping is a composition of z → cz + d (rotation/scaling + translation), then a reciprocal, then scaling by (bc − ad)/c, and finally translation by a/c. This decomposition clarifies why circles/lines are preserved.
任何莫比乌斯变换都可以分解为一系列更简单的变换：先平移，再倒数映射，再缩放/旋转，最后再平移。具体而言，若 c ≠ 0，可写为 w = a/c + (bc − ad)/(c(cz + d))。这表明该映射是由 z → cz + d（旋转/伸缩 + 平移）、接着倒数、再乘以 (bc − ad)/c 缩放、最后平移 a/c 复合而成。这种分解说明了为何圆/直线得以保持。
9. Conformal Mapping and Angle Preservation | 保形映射与角度保持
All the transformations discussed — translations, rotations, scaling, reciprocals, and Möbius maps — secting curves. For w = f(z), conformality holds at points where f is differentiable and f '(z) ≠ 0. In Mö = −d/c, though the map remains angle-preserving as a mapping on the Riemann sphere.
上面讨论的所有变换——平移、旋转、伸缩、倒数以及莫比乌斯映射——在解析且导数非零的点上都是保形的。这意味着它们保持相交曲线之间角的大小和方向。对于 w = f(z)，保形性在 f 可微且 f '(z) ≠ 0 的点处成立。在莫比乌斯变换中，唯一不保形的点是极点 z = −d/c，但作为黎曼球面上的映射，它依然保持角度不变。
10. Fixed Points of Transformations | 变换的不动点
z + a (a ≠ 0), there is no finite fixed point; the point at infinity is fixed. For a Mö p, solving (az + b)/(cz + d) = z yields a quadratic equation. In general, a non-identity Mö tations are around a fixed point, and hyperbolic transformations have two distinct fixed points.
变换的不动点满足 f(z) = z。对于平移 w = z + a（a ≠ 0），没有有限不动点；无穷远点是不动点。对于莫比乌斯映射，求解 (az + b)/(cz + d) = z 得到一个二次方程。通常，非恒等的莫比乌斯变换在扩展平面上有一个或两个不动点。这些不动点支配着映射的动力学行为；例如旋转绕一个不动点进行，双曲变换则有两个不同的不动点。
11. Summary and Geometric Insight | 总结与几何直观
The behaviour of z-transforms in the complex plane reveals a rich interplay between algebra and geometry. Starting with simple shifts, scaling, and rotation, we build up to the reciprocal map and the powerful class of Möbius transformations. Their ability to map circles and lines onto circles and lines while preserving angles makes them indispensable in conformal mapping, fluid dynamics, and electrical engineering. Visualising these maps not only solidifies understanding of complex functions but also illustrates the beauty of the extended plane as a compact, unified space.
复平面上 z 变换的行为揭示了代数与几何之间丰富的相互作用。从简单的平移、伸缩和旋转出发，我们逐步构建倒数映射以及强大的莫比乌斯变换类。它们能将圆和直线映射为圆或直线并保持角度不变，因而在保形映射、流体力学和电气工程中不可或缺。将这些映射可视化不仅能巩固对复变函数的理解，还展示了扩展平面作为一个紧致、统一的空间的优美之处。
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