2.8 投影柱面

空间曲线
在xOy坐标面上的投影曲线是
投影柱面的参数方程为

例 2.8.1 将以下空间曲线投影到xOy坐标面，试作出投影柱面的图形。
解 输入以下代码：

with(plots):
x:=t->cos(t): y:=t->sin(t): z:=t->t:
S:=2: T:=6:K:=50:
quxian:=spacecurve([x(t),y(t),z(t)],t=1…T,thickness=4,color=red):
qumian:=plot3d([x(t),y(t),u*z(t)], t=1…T, u=0…1):
x_axis:=plot3d([u,0,0],u=-1.2…1.5,v=0…0.01,thickness=3):
y_axis:=plot3d([0,u,0],u=-1.2…1.5,v=0…0.01,thickness=3):
z_axis:=plot3d([0,0,u],u=-.2…2,v=0…0.01,thickness=3):
xyz:=display(x_axis,y_axis,z_axis,thickness=3):
text:=textplot3d([0,0,-2,“scuxuxz 2020”],color=grey):
display(qumian,quxian,text,xyz,orientation=[40,70],tickmarks=[4,4,4],
axes=none,grid=[54,76],lightmodel=light2,scaling=unconstrained);

输出图形：

曲面的交线

例 2.8.2 将以下交线投影到xOy坐标面，试作出投影柱面的图形。

解 输入以下代码：

with(plots):
x:=t->1.2* cos(t): y:=t->sin(t): z:=(x,y)->2-x^ 2-y^ 2:
T:=2*Pi:
quxian:=spacecurve([x(t),y(t),z(x(t),y(t))],t=0…T,thickness=3,color=red):
qumian:=plot3d(z(x,y),x=-1.5…1.5,y=-1.5…1.5):
zhumian:=plot3d([x(t),y(t),u],u=-3…2,t=0…T):
x_axis:=plot3d([u,0,0],u=-1.2…2,v=0…0.01,thickness=3):
y_axis:=plot3d([0,u,0],u=-1.2…2,v=0…0.01,thickness=3):
z_axis:=plot3d([0,0,u],u=-.2…3,v=0…0.01,thickness=3):
xyz:=display(x_axis,y_axis,z_axis,thickness=3):
text:=textplot3d([0,0,-5,“scuxuxz 2020”],color=grey):
display(zhumian,qumian,quxian,text,xyz,orientation=[40,70],tickmarks=[4,4,4],
axes=none,grid=[54,76],lightmodel=light2,scaling=unconstrained);

输出图形：
曲面的交线
例 2.8.3 将以下交线投影到xOy坐标面，试作出投影柱面的图形。
解 输入以下代码：

qumian1:=plot3d([u * cos(t)/sqrt(2),u* sin(t)/sqrt(3),u^2],u=0…2,t=0…2 * Pi,color=yellow):
qumian2:=plot3d([u* cos(t)/sqrt(2),u* sin(t),4-u^2],u=0…2,t=0…2* Pi,color=green):
quxian:=spacecurve([cos(t),sin(t),2+sin(t) * sin(t)],t=0…2*Pi,color=red,thickness=8):
x_axis:=plot3d([u,0,0],u=-1…1,v=0…0.01,thickness=3):
y_axis:=plot3d([0,u,0],u=-1…1,v=0…0.01,thickness=3):
z_axis:=plot3d([0,0,u],u=-.2…4.4,v=0…0.01,thickness=3):
xyz:=display(x_axis,y_axis,z_axis,thickness=3):
text:=textplot3d([0,0,-1.4,“scuxuxz 2020”],color=grey):
display(qumian1,qumian2,quxian,xyz,text,scaling=constrained,orientation=[40,60]);

输出图形（交线）：

输入以下代码：

with(plots):R:=1:
f:=(x,y)->2* x ^ 2+3* y^2:
qumian1:=plot3d([u* cos(t)/sqrt(2),u* sin(t)/sqrt(3),u^2],u=0…2,t=0…2* Pi,color=yellow):
qumian2:=plot3d([u* cos(t)/sqrt(2),u* sin(t),4-u^2],u=0…2,t=0…2* Pi,color=green):
quxian:=spacecurve([cos(t),sin(t),2+sin(t)* sin(t)],t=0…2* Pi,color=red,thickness=8):
zhumian:=plot3d([cos(t),sin(t),u],t=0…2*Pi,u=0…f(cos(t),sin(t))):
x_axis:=plot3d([u,0,0],u=-1…1,v=0…0.01,thickness=3):
y_axis:=plot3d([0,u,0],u=-1…1,v=0…0.01,thickness=3):
z_axis:=plot3d([0,0,u],u=-.2…4.4,v=0…0.01,thickness=3):
xyz:=display(x_axis,y_axis,z_axis,thickness=3):
text:=textplot3d([0,0,-1.4,“scuxuxz 2020”],color=grey):
display(zhumian,qumian1,quxian,xyz,text,scaling=constrained,orientation=[40,60]);

输出图形（投影柱面）：
返回《Maple图形与动画》目录