Racing · ECT demo赛车 · ECT 演示
Shapes, swept
形状，扫描
Each 2025 F1 circuit is a simple closed curve, represented here as a one-dimensional simplicial complex. Pick a direction, sweep a filtration line across it, and the Euler characteristic transform reads off how the sublevel set's topology changes. Drag the arrow; drag the slider; watch the right-hand curve.
2025 赛季每一条 F1 赛道都是一条简单闭合曲线，这里把它表示为一个一维单纯复形。选一个方向，拉一条滤过线扫过它， 欧拉示性数变换就会告诉你子水平集的拓扑如何变化。拖动箭头、拖动滑块，看右边的曲线。
θ = 0° t = 0.00 χ = 0
ECT curveECT 曲线
Initialising…初始化中……
SampEuler vectorisationSampEuler 向量化
One precomputed frame per circuit. Columns sweep filtration; rows bin the Euler characteristic over every direction on S¹; brighter yellow means more directions agree. The y-axis is the global χ range across all 21 circuits, so the images are directly comparable. Same construction as the SampEuler shape descriptor.
每条赛道预先生成一张。列表示滤过值的扫描；行是对 S¹ 上所有方向的欧拉示性数的分箱统计；黄色越亮，表示一致的方向越多。y 轴是 21 条赛道全局 χ 的取值范围，所以各张图可直接对比。构造方式与 SampEuler 形状描述子相同。
Why this page exists
为什么会有这一页
Formula 1 is the other thing I spend my weekends on: watching drivers hunt for the optimal racing line, balancing tyre degradation against lap time. It left me curious: can we extract useful shape information from the track layout itself, rather than running costly on‑track experiments?
F1 是我周末的另一件事——看车手寻找最佳走线，在轮胎磨损与单圈时间之间做权衡。这让我好奇：能不能从赛道布局本身提取有用的形状信息，而不必依赖代价高昂的实车试验？
It turns out we can. The Euler characteristic transform gives each circuit a compact shape descriptor computed directly from its geometry. That descriptor feeds prediction tasks (tyre strategy, degradation curves, stint lengths) using shape statistics rather than track time. Silverstone is my favourite example: its fourteen corners give fourteen distinct critical values, so the ECT curve is detailed enough to be interesting and small enough to stay legible.
结果是可以的。欧拉示性数变换直接从赛道几何出发，为每一条赛道算出一个紧凑的形状描述子。这个描述子可以用来做预测任务——轮胎策略、磨损曲线、单段圈数——用的是形状统计量，而不是实车计时。银石是我最喜欢的例子：十四个弯角成为十四个不同的临界值，所以它的 ECT 曲线细节丰富又不至于无法辨读。