The latest articles on DEV Community by Suehiro Saito (@_4a5ba89166772d2). https://dev.to/_4a5ba89166772d2 https://media2.dev.to/dynamic/image/width=90,height=90,fit=cover,gravity=auto,format=auto/https:%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Fuser%2Fprofile_image%2F3676158%2F52226ead-f3b6-4851-9289-fb1516fff3e0.jpg https://dev.to/_4a5ba89166772d2 en Suehiro Saito Fri, 09 Jan 2026 01:37:00 +0000 https://dev.to/_4a5ba89166772d2/high-school-math-with-python-quadratic-curve-3n1h https://dev.to/_4a5ba89166772d2/high-school-math-with-python-quadratic-curve-3n1h <p><a href="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F76gbrp8ftffk557mofdv.png" class="article-body-image-wrapper"><img src="https://media2.dev.to/dynamic/image/width=800%2Cheight=%2Cfit=scale-down%2Cgravity=auto%2Cformat=auto/https%3A%2F%2Fdev-to-uploads.s3.amazonaws.com%2Fuploads%2Farticles%2F76gbrp8ftffk557mofdv.png" alt=" " width="800" height="377"></a></p> <h2> Circles, Parabolas, Ellipses, and Hyperbolas as Conic Sections </h2> <p>Circles, parabolas, ellipses, and hyperbolas are curves that appear as <strong>cross sections of a cone</strong>.</p> <p>When a cone is cut by a plane, the curve formed on the cutting surface depends on the <strong>angle between the plane and the cone</strong>.<br> By changing this angle, <strong>all four types of conic sections</strong> appear.</p> <p>If we list their standard forms (equations), we get:</p> <ul> <li><p><strong>Circle</strong><br> x^2 + y^2 = r<br> r is the radius</p></li> <li><p><strong>Parabola</strong><br> y^2 = 4px, y = x^2/4p</p></li> <li><p><strong>Ellipse</strong><br> (x/a)^2 + (y/b)^2 = 1 </p></li> <li><p><strong>Hyperbola</strong><br> (x/a)^2 - (y/b)^2 = 1 </p></li> </ul> <h2> What This Program Shows </h2> <p>In the program below, the <strong>equation of the cross section</strong> appears in the <strong>legend</strong> of the 2D graph.</p> <p>Try moving the slider to interactively change the cutting angle and observe how the curve changes.</p> <p>This program was generated with the help of <strong>AI</strong>.<br> It goes <strong>beyond standard high school mathematics</strong>, so the implementation itself is not meant as a manual exercise.</p> <h3> Key Point </h3> <p><strong>Circles, parabolas, ellipses, and hyperbolas all belong to the same family: conic sections.</strong></p> <h2> Executable Program (Google Colab) </h2> <p><a href="https://colab.research.google.com/drive/1QFEermnXAl1-DDMgpzlZ-HCNSrIZr0H7" rel="noopener noreferrer">https://colab.research.google.com/drive/1QFEermnXAl1-DDMgpzlZ-HCNSrIZr0H7</a></p> <h2> Python Code </h2> <div class="highlight js-code-highlight"> <pre class="highlight python"><code><span class="kn">import</span> <span class="n">numpy</span> <span class="k">as</span> <span class="n">np</span> <span class="kn">import</span> <span class="n">matplotlib.pyplot</span> <span class="k">as</span> <span class="n">plt</span> <span class="kn">from</span> <span class="n">ipywidgets</span> <span class="kn">import</span> <span class="n">interact</span><span class="p">,</span> <span class="n">FloatSlider</span> <span class="k">def</span> <span class="nf">draw_conic_with_equation</span><span class="p">(</span><span class="n">beta_deg</span><span class="p">):</span> <span class="n">alpha_deg</span> <span class="o">=</span> <span class="mi">45</span> <span class="n">alpha</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="nf">radians</span><span class="p">(</span><span class="n">alpha_deg</span><span class="p">)</span> <span class="n">beta</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="nf">radians</span><span class="p">(</span><span class="n">beta_deg</span><span class="p">)</span> <span class="n">d</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="p">.</span><span class="nf">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">14</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span> <span class="c1"># --- Left: 3D cone and cutting plane --- </span> <span class="n">ax1</span> <span class="o">=</span> <span class="n">fig</span><span class="p">.</span><span class="nf">add_subplot</span><span class="p">(</span><span class="mi">121</span><span class="p">,</span> <span class="n">projection</span><span class="o">=</span><span class="sh">'</span><span class="s">3d</span><span class="sh">'</span><span class="p">)</span> <span class="n">r</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="nf">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">40</span><span class="p">)</span> <span class="n">theta</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="nf">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="p">.</span><span class="n">pi</span><span class="p">,</span> <span class="mi">40</span><span class="p">)</span> <span class="n">R</span><span class="p">,</span> <span class="n">THETA</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="nf">meshgrid</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="n">theta</span><span class="p">)</span> <span class="n">X_cone</span> <span class="o">=</span> <span class="n">R</span> <span class="o">*</span> <span class="n">np</span><span class="p">.</span><span class="nf">cos</span><span class="p">(</span><span class="n">THETA</span><span class="p">)</span> <span class="n">Y_cone</span> <span class="o">=</span> <span class="n">R</span> <span class="o">*</span> <span class="n">np</span><span class="p">.</span><span class="nf">sin</span><span class="p">(</span><span class="n">THETA</span><span class="p">)</span> <span class="n">Z_cone_up</span> <span class="o">=</span> <span class="n">R</span> <span class="o">/</span> <span class="n">np</span><span class="p">.</span><span class="nf">tan</span><span class="p">(</span><span class="n">alpha</span><span class="p">)</span> <span class="n">Z_cone_down</span> <span class="o">=</span> <span class="o">-</span><span class="n">R</span> <span class="o">/</span> <span class="n">np</span><span class="p">.</span><span class="nf">tan</span><span class="p">(</span><span class="n">alpha</span><span class="p">)</span> <span class="n">X_p</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="nf">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="n">Y_p</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="nf">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="n">XP</span><span class="p">,</span> <span class="n">YP</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="nf">meshgrid</span><span class="p">(</span><span class="n">X_p</span><span class="p">,</span> <span class="n">Y_p</span><span class="p">)</span> <span class="n">ZP</span> <span class="o">=</span> <span class="n">XP</span> <span class="o">*</span> <span class="n">np</span><span class="p">.</span><span class="nf">tan</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span> <span class="o">+</span> <span class="n">d</span> <span class="n">ax1</span><span class="p">.</span><span class="nf">plot_surface</span><span class="p">(</span><span class="n">X_cone</span><span class="p">,</span> <span class="n">Y_cone</span><span class="p">,</span> <span class="n">Z_cone_up</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.15</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="sh">'</span><span class="s">cyan</span><span class="sh">'</span><span class="p">)</span> <span class="n">ax1</span><span class="p">.</span><span class="nf">plot_surface</span><span class="p">(</span><span class="n">X_cone</span><span class="p">,</span> <span class="n">Y_cone</span><span class="p">,</span> <span class="n">Z_cone_down</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.15</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="sh">'</span><span class="s">cyan</span><span class="sh">'</span><span class="p">)</span> <span class="n">ax1</span><span class="p">.</span><span class="nf">plot_surface</span><span class="p">(</span><span class="n">XP</span><span class="p">,</span> <span class="n">YP</span><span class="p">,</span> <span class="n">ZP</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.4</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="sh">'</span><span class="s">orange</span><span class="sh">'</span><span class="p">)</span> <span class="n">ax1</span><span class="p">.</span><span class="nf">set_title</span><span class="p">(</span><span class="sh">"</span><span class="s">3D View</span><span class="sh">"</span><span class="p">)</span> <span class="c1"># --- Right: 2D cross section --- </span> <span class="n">ax2</span> <span class="o">=</span> <span class="n">fig</span><span class="p">.</span><span class="nf">add_subplot</span><span class="p">(</span><span class="mi">122</span><span class="p">)</span> <span class="c1"># Expanded equation of the cross section: </span> <span class="c1"># y^2 = (tan^2(beta) - 1)x^2 + (2d tan(beta))x + d^2 </span> <span class="n">x_range</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="nf">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span> <span class="n">tan_b</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="nf">tan</span><span class="p">(</span><span class="n">beta</span><span class="p">)</span> <span class="n">y_sq</span> <span class="o">=</span> <span class="p">(</span><span class="n">x_range</span> <span class="o">*</span> <span class="n">tan_b</span> <span class="o">+</span> <span class="n">d</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="n">x_range</span><span class="o">**</span><span class="mi">2</span> <span class="n">valid</span> <span class="o">=</span> <span class="n">y_sq</span> <span class="o">&gt;=</span> <span class="mi">0</span> <span class="n">x_plot</span> <span class="o">=</span> <span class="n">x_range</span><span class="p">[</span><span class="n">valid</span><span class="p">]</span> <span class="n">y_plot</span> <span class="o">=</span> <span class="n">np</span><span class="p">.</span><span class="nf">sqrt</span><span class="p">(</span><span class="n">y_sq</span><span class="p">[</span><span class="n">valid</span><span class="p">])</span> <span class="c1"># Coefficients for legend display </span> <span class="n">A</span> <span class="o">=</span> <span class="n">tan_b</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="mi">1</span> <span class="n">B</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">d</span> <span class="o">*</span> <span class="n">tan_b</span> <span class="n">C</span> <span class="o">=</span> <span class="n">d</span><span class="o">**</span><span class="mi">2</span> <span class="c1"># Equation label </span> <span class="n">eq_label</span> <span class="o">=</span> <span class="sa">rf</span><span class="sh">'</span><span class="s">$y^2 = (</span><span class="si">{</span><span class="n">A</span><span class="si">:</span><span class="p">.</span><span class="mi">2</span><span class="n">f</span><span class="si">}</span><span class="s">)x^2 + (</span><span class="si">{</span><span class="n">B</span><span class="si">:</span><span class="p">.</span><span class="mi">2</span><span class="n">f</span><span class="si">}</span><span class="s">)x + </span><span class="si">{</span><span class="n">C</span><span class="si">:</span><span class="p">.</span><span class="mi">2</span><span class="n">f</span><span class="si">}</span><span class="s">$</span><span class="sh">'</span> <span class="n">ax2</span><span class="p">.</span><span class="nf">plot</span><span class="p">(</span><span class="n">x_plot</span><span class="p">,</span> <span class="n">y_plot</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="sh">'</span><span class="s">blue</span><span class="sh">'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">eq_label</span><span class="p">)</span> <span class="n">ax2</span><span class="p">.</span><span class="nf">plot</span><span class="p">(</span><span class="n">x_plot</span><span class="p">,</span> <span class="o">-</span><span class="n">y_plot</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="sh">'</span><span class="s">blue</span><span class="sh">'</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># Determine conic type </span> <span class="k">if</span> <span class="n">beta_deg</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span> <span class="n">shape_type</span> <span class="o">=</span> <span class="sh">"</span><span class="s">Circle</span><span class="sh">"</span> <span class="k">elif</span> <span class="n">beta_deg</span> <span class="o">&lt;</span> <span class="n">alpha_deg</span><span class="p">:</span> <span class="n">shape_type</span> <span class="o">=</span> <span class="sh">"</span><span class="s">Ellipse</span><span class="sh">"</span> <span class="k">elif</span> <span class="n">beta_deg</span> <span class="o">==</span> <span class="n">alpha_deg</span><span class="p">:</span> <span class="n">shape_type</span> <span class="o">=</span> <span class="sh">"</span><span class="s">Parabola</span><span class="sh">"</span> <span class="k">else</span><span class="p">:</span> <span class="n">shape_type</span> <span class="o">=</span> <span class="sh">"</span><span class="s">Hyperbola</span><span class="sh">"</span> <span class="n">ax2</span><span class="p">.</span><span class="nf">set_title</span><span class="p">(</span><span class="sa">f</span><span class="sh">"</span><span class="s">2D Cross Section: </span><span class="si">{</span><span class="n">shape_type</span><span class="si">}</span><span class="sh">"</span><span class="p">)</span> <span class="n">ax2</span><span class="p">.</span><span class="nf">set_xlabel</span><span class="p">(</span><span class="sh">'</span><span class="s">x</span><span class="sh">'</span><span class="p">)</span> <span class="n">ax2</span><span class="p">.</span><span class="nf">set_ylabel</span><span class="p">(</span><span class="sh">'</span><span class="s">y</span><span class="sh">'</span><span class="p">)</span> <span class="n">ax2</span><span class="p">.</span><span class="nf">set_xlim</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="n">ax2</span><span class="p">.</span><span class="nf">set_ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="n">ax2</span><span class="p">.</span><span class="nf">grid</span><span class="p">(</span><span class="bp">True</span><span class="p">)</span> <span class="n">ax2</span><span class="p">.</span><span class="nf">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="sh">'</span><span class="s">black</span><span class="sh">'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="n">ax2</span><span class="p">.</span><span class="nf">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="sh">'</span><span class="s">black</span><span class="sh">'</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span> <span class="c1"># Legend in the lower-left corner </span> <span class="n">ax2</span><span class="p">.</span><span class="nf">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="sh">'</span><span class="s">lower left</span><span class="sh">'</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span> <span class="n">plt</span><span class="p">.</span><span class="nf">tight_layout</span><span class="p">()</span> <span class="n">plt</span><span class="p">.</span><span class="nf">show</span><span class="p">()</span> <span class="nf">interact</span><span class="p">(</span> <span class="n">draw_conic_with_equation</span><span class="p">,</span> <span class="n">beta_deg</span><span class="o">=</span><span class="nc">FloatSlider</span><span class="p">(</span><span class="nb">min</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="nb">max</span><span class="o">=</span><span class="mi">85</span><span class="p">,</span> <span class="n">step</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">value</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">description</span><span class="o">=</span><span class="sh">'</span><span class="s">Angle β</span><span class="sh">'</span><span class="p">)</span> <span class="p">)</span> </code></pre> </div> python math highshcool