Question

证明该曲线是一般螺线

Answer 1

其实计算不是很难.

r(t) = (cos³(t),sin³(t),cos(2t)),

切向量r'(t) = (-3cos²(t)sin(t),3sin²(t)cos(t),2sin(2t)) = cos(t)sin(t)(-3cos(t),3sin(t),4),

|r'(t)| = cos(t)sin(t)√((-3cos(t))²+(3sin(t))²+4²) = cos(t)sin(t)√(3²+4²) = 5cos(t)sin(t),

因此单位切向量r'(t)/|r'(t)| = (-3cos(t),3sin(t),4)/5.

要说明r'(t)与固定方向夹角为定值,只需说明r'(t)/|r'(t)|与某个向量内积为定值,

取单位向量u = (0,0,1),易见u·r'(t)/|r'(t)| = 4/5,

可知r'(t)与u夹角 = arccos(4/5)为定值,即所求证.