基本原理
通過橢圓方程旋轉45°,基礎為平面直角坐標系,不同於笛卡爾 的極坐標系 ;
⑵軸方向取絕對值,成為偶函式;
⑶新的偶函式為閉合心形線 。
推導過程
![茹茹心形線](/img/9/e44/wZwpmLwAzMyIDNwMzN5kzM0UTMyITNykTO0EDMwAjMwUzLzczL0QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
⑴原始橢圓方程以a=,b=1的參數為例:
![茹茹心形線](/img/1/d34/wZwpmLzMDN2MTMxEjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzL1IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![原始橢圓方程](/img/9/6b8/wZwpmLwAjN4cjMygTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL4EzL0czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![茹茹心形線](/img/3/50e/wZwpmLzMjMykjM1EDNxMDN0UTMyITNykTO0EDMwAjMwUzLxQzL2YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
⑵坐標系變換(逆時針旋轉 ),變換後:
![茹茹心形線](/img/5/988/wZwpmLxMTO2IDNzUjM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1IzL4YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
① ;
![茹茹心形線](/img/6/41d/wZwpmL2ITM1QTMygzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL4MzLyUzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
② ;
⑶帶入橢圓:
![茹茹心形線](/img/7/a24/wZwpmLyYjN5QDN2IDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLyQzL3QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![茹茹心形線](/img/4/e71/wZwpmLyEDN3UTM4MzM2EzM1UTM1QDN5MjM5ADMwAjMwUzLzMzL3czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![茹茹心形線](/img/5/970/wZwpmL0MzM4kTNzQTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzLzczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![茹茹心形線](/img/5/557/wZwpmL2IjN0cDOwIDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLyQzLyYzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![茹茹心形線](/img/c/39f/wZwpmL4gDNyMzN4EjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzLwYzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![茹茹心形線](/img/e/3c1/wZwpmL2MjN3ITO5QzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0MzLyAzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
化簡得:
![茹茹心形線](/img/5/c81/wZwpmL0MjN3UDN0QzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0MzL4QzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![旋轉後的橢圓方程](/img/d/dab/wZwpmL3UjMzgDM5MDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLzQzLzIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
⑷x軸取絕對值得:
![茹茹心形線](/img/1/d6f/wZwpmLzEDNwEjNxEjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzL1IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![x取絕對值圖形](/img/a/b93/wZwpmL3YTO3MTN4AjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLwIzL0EzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
⑸調整原始橢圓係數,可得一系列心形線方程,下圖為單位心形方程:
![茹茹心形線](/img/1/82c/wZwpmLwMDO1MTOwYzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL2MzL0gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![單位RC心形線](/img/0/c3a/wZwpmLzYjNyAzM0MjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLzIzL3gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
適用範圍
興趣數學
茹茹心形線,又名RC心形線,是多種心形線方程中較簡單的一系列心形線方程,相比於笛卡爾極坐標心形線以及其他數學愛好者推導的心形線方程。RC心形線方程推導過程簡單,形式變換多樣,只需簡單三角函式及橢圓方程即可推導,且可調整橢圓方程中a、b、c參數,即可獲得不同大小、形式的心形線。
通過橢圓方程旋轉45°,基礎為平面直角坐標系,不同於笛卡爾 的極坐標系 ;
⑵軸方向取絕對值,成為偶函式;
⑶新的偶函式為閉合心形線 。
⑴原始橢圓方程以a=,b=1的參數為例:
⑵坐標系變換(逆時針旋轉 ),變換後:
① ;
② ;
⑶帶入橢圓:
化簡得:
⑷x軸取絕對值得:
⑸調整原始橢圓係數,可得一系列心形線方程,下圖為單位心形方程:
興趣數學