commit 1f3fc0545725ededf992f084d5779dff774d7988 parent d45573342607ebfa2750df39eda5dce43249f0ed Author: minerva-jupiter <ryouturn@gmail.com> Date: Sat, 13 Jun 2026 16:36:00 +0900 docs: format math equations in README.md Convert inline LaTeX equations to block code format for improved rendering and readability. Diffstat:
| M | README.md | | | 20 | ++++++++++---------- |
1 file changed, 10 insertions(+), 10 deletions(-)
diff --git a/README.md b/README.md @@ -40,18 +40,18 @@ Let the sampling period be $T$, and consider the integral from continuous time t The essence of the bilinear transform is to approximate the integrand using a trapezoidal approximation (multiplying the mean value at both ends by the interval width $T$) for the integral from time $(n-1)T$ to $nT$. -####### Discretization of position $x[n]$ +Discretization of position $x[n]$ +```math +\int_{(n-1)T}^{nT} \frac{dx(t)}{dt} dt = \int_{(n-1)T}^{nT} v(t) dt -$$\int_{(n-1)T}^{nT} \frac{dx(t)}{dt} dt = \int_{(n-1)T}^{nT} v(t) dt$$ - -$$x[n] - x[n-1] = \frac{T}{2} \left( v[n] + v[n-1] \right) \quad \cdots \text{(Equation 1)}$$ - -####### Discretization of velocity $v[n]$ - -$$\int_{(n-1)T}^{nT} \frac{dv(t)}{dt} dt = \int_{(n-1)T}^{nT} \frac{1}{m} \left( F(t) - r v(t) - k x(t) \right) dt$$ - -$$v[n] - v[n-1] = \frac{T}{2m} \left( F[n] + F[n-1] - r(v[n] + v[n-1]) - k(x[n] + x[n-1]) \right) \quad \cdots \text{(Equation 2)}$$ +x[n] - x[n-1] = \frac{T}{2} \left( v[n] + v[n-1] \right) \quad \cdots \text{(Equation 1)} +``` +Discretization of velocity $v[n]$ +```math +\int_{(n-1)T}^{nT} \frac{dv(t)}{dt} dt = \int_{(n-1)T}^{nT} \frac{1}{m} \left( F(t) - r v(t) - k x(t) \right) dt +v[n] - v[n-1] = \frac{T}{2m} \left( F[n] + F[n-1] - r(v[n] + v[n-1]) - k(x[n] + x[n-1]) \right) \quad \cdots \text{(Equation 2)} +``` --- ###### 3. Resolving the Algebraic Loop and Deriving the Simulation Formula