4.2 参数曲线的动画

例 4.2.1 作出椭圆 x=3cost, y=2sint 的动画。
解 输入以下命令：

with(plots):
x:=3* cos(t):y:=2* sin(t):
a:=0: b:=2* Pi: K:=30:
for i from 1 to K do ti:=a+i*(b-a)/K:
M[i]:=plot([x(t), y(t), t=a…ti], thickness=2) od:
display(seq(M[i], i=1…K), insequence=true,
tickmarks=[4,4], scaling=constrained);

输出动画：

例 4.2.2 作下列旋轮线的动画。
解 输入以下命令：

with(plots):R:=1:
x:=R*(t-sin(t)): y:=R*(1-cos(t)):
a:=0: b:=4* Pi: K:=30:
for i from 1 to K do
ti:=a+i*(b-a)/K:
M[i]:=plot([x(t),y(t),t=a…ti], thickness=2) od:
display(seq(M[i],i=1…K), insequence=true,
tickmarks=[4,2], scaling=constrained);

输出动画：
输入以下命令：

with(plots): with(plottools):
R:=1: x:=t->R*(t-sin(t)):y:=t->R*(1-cos(t)): K:=60: a:=0: b:=4* Pi:
for i from 1 to K do
ti:=a+i*(b-a)/K:
quxian[i]:=plot([x(t),y(t),t=a…ti]):
yuan[i]:=circle([ti,R],R,color=blue):
xian[i]:=plot([[ti,R],[x(ti),y(ti)]],color=brown):
M[i]:=display(quxian[i],yuan[i],xian[i]) od:
L:=display(seq(M[i],i=1…K),insequence=true):
display(L,L,scaling=constrained,tickmarks=[3,2],thickness=2);

输出动画（有旋轮）：

例 4.2.3 椭圆的轨迹。
解 椭圆是到两个定点的距离之和为常数的点的轨迹。
输入以下命令：

with(plots): with(plottools):
a:=4: b:=3:F:=sqrt(a^ 2-b^2):
jiaodian1:=circle([-F,0],0.1):
jiaodian2:=circle([F,0],0.1):
x:=t->a* cos(t): y:=t->b* sin(t):
K:=40: for i from 1 to K do ti:=i2Pi/K:
tuoyuan[i]:=plot([x(t),y(t),t=0…ti]):
zuo[i]:=plot([[-F,0],[x(ti),y(ti)]],color=blue):
you[i]:=plot([[F,0],[x(ti),y(ti)]],color=blue):
M[i]:=display(tuoyuan[i], you[i], zuo[i]) od:
L:=display(seq(M[i],i=1…K),insequence=true):
display(L, jiaodian1, jiaodian2, scaling=constrained,
tickmarks=[3,2],thickness=2);

输出图形：

例 4.2.4 椭圆的参数方程的几何解释。
解 椭圆的参数方程:
输入以下命令：

with(plots): with(plottools): a:=5: b:=3:
yuan_a:=circle([0,0],a,color=red,thickness=2):
yuan_b:=circle([0,0],b,color=red,thickness=2):
x:=t->a* cos(t): y:=t->b* sin(t):
K:=60: for i from 1 to K do ti:=i2Pi/K:
tuoyuan[i]:=plot([x(t),y(t),t=0…ti],color=blue,thickness=3):
La[i]:=plot([[0,0],[x(ti),a* sin(ti)]], color=green,thickness=3):
Lb[i]:=plot([[x(ti),0], [x(ti),a* sin(ti)]], color=brown,thickness=2):
Lc[i]:=plot([[b* cos(ti),y(ti)], [a* cos(ti),y(ti)]], color=brown,thickness=2):
dian[i]:=circle([x(ti),y(ti)],0.1,color=blue,thickness=3):
M[i]:=display(tuoyuan[i],La[i],Lb[i],Lc[i],dian[i]) od:
L:=display(seq(M[i],i=1…K),insequence=true):
display(L,yuan_a,yuan_b,scaling=constrained,tickmarks=[3,2]);

输出图形：

例 4.2.5 圆的渐开线。
解 输入以下命令：

with(plots): with(plottools): a:=1: yuan:=circle([0,0],a,color=blue):
x:=t->a*(cos(t)+t* sin(t)): y:=t->a*(sin(t)-t* cos(t)):
K:=40:
for i from 1 to K do ti:=i* 3* Pi/K:
quxian[i]:=plot([x(t), y(t), t=0…ti], color=red):
xian[i]:=plot([[a* cos(ti), a* sin(ti)], [x(ti), y(ti)]], color=brown):
M[i]:=display(quxian[i], xian[i]) od:
L:=display(seq(M[i], i=1…K), insequence=true):
display(L, yuan, scaling=constrained, tickmarks=[3,2], thickness=3);

输出图形：

例 4.2.6 作出下列曲线的动画。

解 输入以下命令：

with(plots): with(plottools):
x:=t->cos(t)-sin(t)* cos(80* t): y:=t->2* sin(t)-sin(80* t):
K:=20: for i from 1 to K do
ti:=i2Pi/K:
quxian[i]:=plot([x(t),y(t),t=0…ti],color=red) od:
L:=display(seq(quxian[i],i=1…K),insequence=true):
display(L,L,tickmarks=[2,2]);

输出图形：

返回《Maple图形与动画》目录