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5 April 2012,Santa II, p.166,四川大学数学学院,徐小湛,分针绕时针针尖旋转时,分针针尖的轨迹,成都熊猫基地,06APR12,一位网友提出以下问题: 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?,设时针的长为,a,,分针的长为,b,。,当时针绕原点旋转时,时针针尖的位置为:,当分针绕原点旋转时,分针针尖的位置为:,当分针绕时针针尖旋转时,分针针尖的位置为:,所以,分针针尖的参数方程为:,作图的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);,a,=1,b,=1.2 的图形,a,=1,b,=0.2 的图形,a,=1,b,=0.1 的图形,动画1,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);,K:=180:,for i from 1 to K do,ti:=i*0.5*Pi/K:,quxiani:=plot(x(t),y(t),t=0.ti,thickness=2):,shizheni:=plot(0,0,a*cos(-ti),a*sin(-ti),color=blue,thickness=3):,fenzheni:=plot(x(ti),y(ti),a*cos(-ti),a*sin(-ti),color=gray,thickness=3) od:,quxian:=display(seq(quxiani,i=1.K),insequence=true):,shizhen:=display(seq(shizheni,i=1.K),insequence=true):,fenzhen:=display(seq(fenzheni,i=1.K),insequence=true):,display(quxian,shizhen,fenzhen,scaling=constrained,tickmarks=5,3);,动画2,with(plots):,a:=1:b:=0.2:k:=60:,x:=t-a*cos(-t)+b*cos(-k*t);,y:=t-a*sin(-t)+b*sin(-k*t);,K:=360:,for i from 1 to K do,ti:=i*1*Pi/K:,quxiani:=plot(x(t),y(t),t=0.ti,thickness=2):,shizheni:=plot(0,0,a*cos(-ti),a*sin(-ti),color=blue,thickness=3):,fenzheni:=plot(x(ti),y(ti),a*cos(-ti),a*sin(-ti),color=gray,thickness=3) od:,quxian:=display(seq(quxiani,i=1.K),insequence=true):,shizhen:=display(seq(shizheni,i=1.K),insequence=true):,fenzhen:=display(seq(fenzheni,i=1.K),insequence=true):,display(quxian,shizhen,fenzhen,scaling=constrained,tickmarks=5,3);,动画3,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:,quxiani:=plot(x(t),y(t),t=0.ti,thickness=2):,shizheni:=plot(0,0,a*cos(-ti+Pi/2),a*sin(-ti+Pi/2),color=blue,thickness=3):,fenzheni:=plot(x(ti),y(ti),a*cos(-ti+Pi/2),a*sin(-ti+Pi/2),color=gray,thickness=3) od:,quxian:=display(seq(quxiani,i=1.K),insequence=true):,shizhen:=display(seq(shizheni,i=1.K),insequence=true):,fenzhen:=display(seq(fenzheni,i=1.K),insequence=true):,display(quxian,shizhen,fenzhen,scaling=constrained,tickmarks=5,3);,
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