Build the idea, one step at a time一步步建立理解
Need a math refresher?需要补一点数学基础？
Open only the tool you need; return here to use it in the next step.按需打开一个小工具，再回到本课把它用起来。
Vectors & weighted sums向量与加权求和Exponentials & logarithms指数与对数STEP步骤 1 / 5
Agree on the clock first先统一时钟
Normalize capture time with a documented start and duration. A file using t in 0–1 cannot be evaluated with seconds directly. Playback speed changes the rate at which t advances, not the learned motion coefficients. Our browser fixture maps its capture interval to the normalized model time.用明确记录的起点与时长归一化时间。模型若使用 0–1 的 t，就不能直接传入秒数。播放速度改变的是 t 推进的速度，不是学到的运动系数。浏览器真实模型将采集区间映射到模型的归一化时间。
STEP步骤 2 / 5
Position is a short polynomial位置是一段短多项式
Start at μ₀ when t = τ. Add a term proportional to elapsed time, another to its square and another to its cube. Each coefficient is a 3D vector. The square and cube allow bending and changing speed. These are fitted coefficients; calling m₂ acceleration is inaccurate unless you include the derivative’s factor of two.t = τ 时从 μ₀ 出发，再加与时间差、时间差平方、时间差立方成比例的三项。每个系数都是三维向量，平方与立方使轨迹能弯、速度能变。它们是拟合系数；若把 m₂ 直接叫加速度，就漏掉求导后的系数 2。
STEP步骤 3 / 5
Being present is separate from moving是否出现与怎么移动是两件事
Multiply base opacity by a time window centered at τ. For this STG convention the exponent has no one-half factor. At one width ρ from the center, the multiplier is e⁻¹ ≈ 0.368. A narrow window makes a primitive fade away outside its useful time interval; it does not stop its polynomial from being evaluable.基准 opacity 再乘以 τ 为中心的时间窗。本 STG 约定的指数前没有二分之一。离中心一倍时间宽度 ρ 时，倍率为 e⁻¹ ≈ 0.368。窄窗口使点在有效时间外淡出，但不意味着它的运动多项式无法求值。
STEP步骤 4 / 5
Orientation changes before projection旋转也要在投影前求出
In the current record format, form q₀ + Δtω and normalize this four-component vector to get orientation. It is not a general physical angular-velocity integrator. Use the resulting rotation and scales to build covariance, then run the same projection and compositing pipeline as in earlier lessons.在当前格式中，先算 q₀ + Δtω，再把这个四分量向量归一化得到方向。它不是通用物理角速度积分器。用得到的旋转与尺度构造协方差，再走前面相同的投影和合成流程。
STEP步骤 5 / 5
Other 4D methods put time elsewhere其他 4D 方法把时间放在别处
A deformation model predicts changes to canonical 3D Gaussians using a network and time-dependent features. A joint 4D Gaussian stores a spacetime covariance and conditions on t to obtain a 3D slice. In that second case, spatial–temporal covariance controls the conditional center shift. These models share a rendering destination but not a parameter layout or decoder.形变方法用网络与随时间变化的特征，预测标准三维 Gaussian 的变化。联合四维高斯存时空协方差，在给定 t 后条件化得到三维切片；空间与时间的交叉协方差控制条件中心的偏移。它们最后都可走三维渲染，但参数布局和解码器不同。
PUT NUMBERS IN代入数字
For the existing teaching point A, μ₀ = (1, 2, 10), m₁ = (2, 0, 0), m₂ = (0, 4, 0), m₃ = (0, 0, 8). At t = 0.75 with τ = 0.5, Δt = 0.25 and μ = (1.5, 2.25, 10.125).现有教学 A 点中，μ₀ = (1, 2, 10)、m₁ = (2, 0, 0)、m₂ = (0, 4, 0)、m₃ = (0, 0, 8)。t = 0.75、τ = 0.5 时，Δt = 0.25，μ = (1.5, 2.25, 10.125)。