生成函数运动图像---5. 对数螺旋线 (Logarithmic Spiral)
公式 (极坐标) : r = ae^(bθ)
运动特点:
动点旋转时, 半径按指数增长, 轨迹与极径的夹角始终不变 (等角螺旋) , 如鹦鹉螺壳、星系旋臂。
约束条件:
* a > 0, b ≠ 0 (a 为 θ = 0 时的半径, b 决定螺旋紧密程度)
* θ ∈ R (可正负, 对应顺时针 / 逆时针旋转) 。
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The logarithmic spiral is defined by the equation r equals a times e to the power of b theta. As the angle theta increases, the radius grows exponentially, creating a beautiful spiral pattern that maintains constant angles with radial lines.
The parameters a and b control the spiral's behavior. Parameter a determines the initial radius when theta equals zero, while parameter b controls how tightly the spiral winds. When b is positive, the spiral expands outward. When b is negative, it contracts inward.
The most remarkable property of the logarithmic spiral is that it maintains a constant angle with every radius vector. This angle is determined by the parameter b and equals arctan of one over b. This constant angle property makes the spiral self-similar at all scales.
The logarithmic spiral is ubiquitous in nature. Nautilus shells grow in perfect logarithmic spirals, with each new chamber maintaining the same proportional relationship. Galaxy arms follow this pattern as they rotate and expand. Hurricanes form spiral eye walls, and even sunflower seeds arrange themselves in logarithmic spiral patterns for optimal packing efficiency.
The logarithmic spiral demonstrates beautiful bidirectional motion. When theta is positive, the spiral expands outward in a counterclockwise direction. When theta is negative, it contracts inward in a clockwise direction. This complete range of motion shows how the same mathematical formula can describe both growth and decay patterns in nature.