檢視貝茲曲線的原始碼

[[分類:數學]][[分類:HTML]]
==二次貝茲曲線==
#二次貝茲曲線畫出的是拋物線，無法畫出橢圓和雙曲線。故無法畫出正圓。
#所有拋物線都「相似」(不是相等)，所有曲率的微線段都有。
#兩端點外只有一個控制點。
#拋物線方程式 ax<sup>2</sup>+bxy+cy<sup>2</sup>+dx+ey+f=0 則 b<sup>2</sup> - 4ac=0 ，即前三項為完全平方式。
===參考===
#[http://www.kut.com.tw/Upload//ProductProbation/File/數學高三甲上第二章第3節主題4觀念一.pdf 二元二次方程式圖形判別的例題]<br/>---------- 擬合圓 ----------
#[https://stackoverflow.com/questions/1734745/how-to-create-circle-with-b%c3%a9zier-curves/1734859 How to create circle with Bézier curves?]
#[https://oomake.com/question/343212 如何使用Bézier曲線創建圓？]
#[https://www.jianshu.com/p/5198d8aa80c1 用三阶贝塞尔曲线拟合圆]

===一、===
<table class=nicetable><tr>
	<td>Q or q<br/>(quadratic<br/>Bézier<br/>curve)</td>
	<td colspan=2>x1 y1 x y<br/><img src='http://jendo.org/wiki1231/images/a/a8/SvgPathQ.png' width='200px' height='*'/><br/><span style='font-size:90%'>&lt;path d='M0,0 Q50,50 100,0' style='stroke:black'/&gt;<br/>&lt;path d='m0,0 q50,50 100,0' style='stroke:black'/&gt;</span></td>
	<td>從目前點的座標畫條二次貝茲曲線到指定點的 x,y 座標：其中 x1,y1 為控制點</td>
</tr></table>
二次貝茲曲線的參數方程為：
<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>t</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>×</mo><mi>P</mi><mn>0</mn><mo>+</mo><mn>2</mn><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo><mi>t</mi><mo>×</mo><mi>P</mi><mn>1</mn><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup><mo>×</mo><mi>P</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">B(t) = (1-t)^2 \times P0 + 2(1-t)t \times P1 + t^2 \times P2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal" style="margin-right: 0.05017em;">B</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right: 0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right: 0.2778em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 1.0641em; vertical-align: -0.25em;"></span><span class="mord mathnormal">t</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8141em;"><span style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.7667em; vertical-align: -0.0833em;"></span><span class="mord mathnormal" style="margin-right: 0.13889em;">P</span><span class="mord">0</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord">2</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 1em; vertical-align: -0.25em;"></span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.7667em; vertical-align: -0.0833em;"></span><span class="mord mathnormal" style="margin-right: 0.13889em;">P</span><span class="mord">1</span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.8974em; vertical-align: -0.0833em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height: 0.8141em;"><span style="top: -3.063em; margin-right: 0.05em;"><span class="pstrut" style="height: 2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right: 0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right: 0.2222em;"></span></span><span class="base"><span class="strut" style="height: 0.6833em;"></span><span class="mord mathnormal" style="margin-right: 0.13889em;">P</span><span class="mord">2</span></span></span></span></span>

==參考文章==
*[https://zh.wikipedia.org/wiki/貝茲曲線 貝茲曲線]
*[http://blog.iderzheng.com/continuous-and-smooth-bezier-curve/ 連續平滑的貝塞爾曲線]