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分针绕时针针尖旋转时，分针针尖的轨迹

2012-04-05 16:55:31| 分类：高等数学图形动画 | 标签： |举报 |字号大中小订阅

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问题：机械钟的时针和分针，都是围绕钟的中心轴顺时针旋转的。如果我们以时针的针尖为参考系，那么分针的针尖围绕时针的针尖运动轨道，是不是外摆线？

答：这个问题很有趣。下面写出其参数方程，并做出图形。

分针绕时针针尖旋转时，分针针尖的轨迹 - calculus - 徐小湛的博客

由于分针比时针更长，设时针长为1，分针长为1.2 。图形如下：

分针绕时针针尖旋转时，分针针尖的轨迹 - calculus - 徐小湛的博客

这是3点钟到6点钟的动画（完整的动画太大了。）

设时针长为1，分针长为0.2 。则图形如下：

分针绕时针针尖旋转时，分针针尖的轨迹 - calculus - 徐小湛的博客

动画的Maple程序（帧数：540）：

with(plots): with(plottools):
a:=1:b:=0.2:k:=60:
x:=t->a*cos(-t+Pi/2)+b*cos(-k*t+Pi/2);
y:=t->a*sin(-t+Pi/2)+b*sin(-k*t+Pi/2);
K:=540: for i from 1 to K do
ti:=i*2*Pi/K:
quxian[i]:=plot([x(t),y(t),t=0..ti],color=gray):
shizhen[i]:=plot([[0,0],[a*cos(-ti+Pi/2),a*sin(-ti+Pi/2)]],color=blue,thickness=3):
fenzhen[i]:=plot([[x(ti),y(ti)],[a*cos(-ti+Pi/2),a*sin(-ti+Pi/2)]],color=black,thickness=3):
dongyuan[i]:=circle([a*cos(-ti+Pi/2),a*sin(-ti+Pi/2)],b,color=red,thickness=2) od:
quxian:=display(seq(quxian[i],i=1..K),insequence=true):
shizhen:=display(seq(shizhen[i],i=1..K),insequence=true):
fenzhen:=display(seq(fenzhen[i],i=1..K),insequence=true):
dongyuan:=display(seq(dongyuan[i],i=1..K),insequence=true):
yuan:=circle([0,0],a,color=brown,thickness=3):
display(yuan,dongyuan,shizhen,quxian,fenzhen,scaling=constrained,tickmarks=[0,0]);

设时针长为1，分针长为0.1。则图形如下：

分针绕时针针尖旋转时，分针针尖的轨迹 - calculus - 徐小湛的博客

图形的Maple程序：

with(plots):
a:=1:b:=1.2:k:=60:
x:=t->a*cos(-t)+b*cos(-k*t);
y:=t->a*sin(-t)+b*sin(-k*t);
curve:=plot([x(t),y(t),t=0..2*Pi],thickness=2):
display(curve,scaling=constrained,tickmarks=[4,4]);

动画的Maple程序：

with(plots):
a:=1:b:=0.1:k:=60:
x:=t->a*cos(-t+Pi/2)+b*cos(-k*t+Pi/2);
y:=t->a*sin(-t+Pi/2)+b*sin(-k*t+Pi/2);
K:=360:
for i from 1 to K do
ti:=i*2*Pi/K:
quxian[i]:=plot([x(t),y(t),t=0..ti],thickness=2):
shizhen[i]:=plot([[0,0],[a*cos(-ti+Pi/2),a*sin(-ti+Pi/2)]],color=blue,thickness=3):
fenzhen[i]:=plot([[x(ti),y(ti)],[a*cos(-ti+Pi/2),a*sin(-ti+Pi/2)]],color=gray,thickness=3) od:
quxian:=display(seq(quxian[i],i=1..K),insequence=true):
shizhen:=display(seq(shizhen[i],i=1..K),insequence=true):
fenzhen:=display(seq(fenzhen[i],i=1..K),insequence=true):
display(quxian,shizhen,fenzhen,scaling=constrained,tickmarks=[5,3]);

分针绕时针针尖旋转时，分针针尖的轨迹 - calculus - 徐小湛的博客

成都大熊猫基地 2012.4.6

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