已经椭圆长半径、短半径、椭圆上任意一点到椭圆圆心的直线与长半径的夹角为α,求椭圆上任意一点到椭圆圆心的距离?

6个回答

  • 参数方程:

    x = a*cost

    y = b*sint

    注意,t 不是 α

    y/x = tg(α) = b/a * tg(t)

    所求为:

    r^2 = x^2 + y^2 = a^2 * (cost)^2 + b^2 * (sint)^2 =

    (cost)^2 * [a^2 + b^2 * (tgt)^2] =

    (cost)^2 * [a^2 + a^2 * tg(α)^2] =

    (cost)^2 / (cosα)^2 * a^2 =

    另一方面,

    a^2/b^2 * tg(α)^2 = tg(t)^2 ====>

    a^2/b^2 * tg(α)^2 + 1 = 1/(cost)^2 ====>

    [ a^2 * (sinα)^2 + b^2 * (cosα)^2 ] / b^2 = (cosα)^2 /(cost)^2 ====>

    r^2 = a^2 * b^2 / [ a^2 * (sinα)^2 + b^2 * (cosα)^2 ]

    再开方就得到距离.