https://sichen-wang.github.io/tags/analytic-geometry/Analytic GeometryHugoBlox Kit (https://hugoblox.com)en-usMon, 13 Jan 2025 20:20:13 +0000https://sichen-wang.github.io/media/icon_hu_9dfcb674aa885f38.pnghttps://sichen-wang.github.io/tags/analytic-geometry/https://sichen-wang.github.io/blog/%E4%B8%89%E7%A7%8D%E6%9B%B2%E7%BA%BF%E6%96%B9%E7%A8%8B%E5%88%BB%E7%94%BB%E6%96%B9%E5%BC%8F%E7%9A%84%E7%9B%B8%E4%BA%92%E8%BD%AC%E6%8D%A2/Mon, 13 Jan 2025 20:20:13 +0000https://sichen-wang.github.io/blog/%E4%B8%89%E7%A7%8D%E6%9B%B2%E7%BA%BF%E6%96%B9%E7%A8%8B%E5%88%BB%E7%94%BB%E6%96%B9%E5%BC%8F%E7%9A%84%E7%9B%B8%E4%BA%92%E8%BD%AC%E6%8D%A2/<p><strong>直角坐标方程</strong>形如 \(y = f(x)\) 或 \(f(x, y) = 0\)，<strong>参数方程</strong>形如 </p> \[ \begin{cases} x = f(t) \\ y = g(t) \end{cases} \]<p> <strong>极坐标方程</strong>形如 \(\ell = \rho(\theta)\)。</p> <h3 id="直角坐标方程转参数方程">直角坐标方程转参数方程</h3> \[ y = f(x) \Longrightarrow \begin{cases} x = t \\ y = f(t) \end{cases} \]<h3 id="参数方程转直角坐标方程">参数方程转直角坐标方程</h3> <ul> <li>代入法: 直接从 \(x = f(t)\) 解出 \(t = f^{-1}(x)\), 然后将其带入 \(y = g(t)\) 解得 \(x\) 和 \(y\) 关系.</li> <li>三角法: 如果方程涉及三角函数, 则可以利用三角恒等式解出直角坐标方程. \[ \begin{cases} x = r \cos t \\ y = r \sin t \end{cases} \Longrightarrow x^2 + y^2 = r^2 (\sin^2 t + \cos^2 t) = r^2 \]</li> </ul> <h3 id="极坐标方程转参数方程">极坐标方程转参数方程</h3> \[ \ell = \rho(\theta) \Longrightarrow \begin{cases} y = \rho(t) \sin t \\ x = \rho(t) \cos t \end{cases} \]<h3 id="参数方程转极坐标方程">参数方程转极坐标方程</h3> <p>注意到在极坐标系中, 点的表示为 </p> \[ \begin{aligned} \ell &= \sqrt{x^2 + y^2} \\ \theta &= \arctan \frac{y}{x} \end{aligned} \]<p>于是将参数方程带入即可得出 \(\ell, \theta\) 与 \(t\) 的关系, 设 \(\ell = p(t), \theta = \arctan q(t)\), 则 </p> \[ \ell = p \circ q^{-1}(\tan \theta) \]