scumath

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In a recent blog entry, I proposed an easy way to plot the region between two curves. Later I read from an earlier blog entry that “filled=true” in implicitplot can produce amazing effects for plotting regions. Inspired by the blog entry, I’d like to introduce another easy way to plot the region between two curves.

To plot the region between y=f(x) and y=g(x) (x=a..b), we just need the following code:
with(plots):

f:=x->f(x): g:=x->g(x):

implicitplot(y=0, x=a..b, y=f(x)..g(x), filled=true, coloring=[green,green]);


The key to the success of this code is that Maple 8 allows
varying range for the second variable y (i.e. y=f(x)..g(x)). However I was sorry to find that this is not allowed in Maple 11 (This will be addressed later.) .

 

Example 1  The region between y=x and y=x^2.

with(plots):

f:=x->x:g:=x->x^2:

a:=0: b:=1:

region:=implicitplot(y=0,x=a..b,y=f(x)..g(x),filled=true,coloring=[yellow,yellow]):

F:=plot(f(x),x=a-0.2..b+0.2,thickness=3,color=red):

G:=plot(g(x),x=a-0.2..b+0.2,thickness=3,color=blue):

display(F,G,region,scaling=constrained);

 

 

Example 2 The region between y=sin(x) and y=cos(x).

with(plots):

f:=x->sin(x):g:=x->cos(x):

a:=0: b:=6:

region:=implicitplot(y=0,x=a..b,y=f(x)..g(x),filled=true,coloring=[grey,grey]):

F:=plot(f(x),x=a-0.2..b+0.2,thickness=3,color=red):

G:=plot(g(x),x=a-0.2..b+0.2,thickness=3,color=blue):

display(F,G,region,scaling=constrained);



 

Now if we reverse the order of the range options from “x=a..b, y=f(x)..g(x)” to “y=f(x)..g(x), x=a..b”, some strange but interesting thing will happen (See Example 3.

Example 3

with(plots):

f:=x->sin(x):g:=x->cos(x):

a:=-1: b:=6:

region:=implicitplot(y=0, y=f(x)..g(x),x=a..b, filled=true,coloring=[grey,grey]):

F:=plot(f(x),x=a-0.2..b+0.2,thickness=3,color=red):

G:=plot(g(x),x=a-0.2..b+0.2,thickness=3,color=blue):

display(F,G,region,scaling=constrained);
 


It can be seen that the region in Example 2 has been reflected with respect to the line y=x. But this is not bad because can use this phenomenon to plot regions between curves x=f(y) and x=g(y) (See Example 4).

 

Example 4  The region between x=y^2/2 and x=y^4/4-y^2/2.

with(plots):

f:=y->y^4/4-y^2/2: g:=y->y^2/2:

region:=implicitplot(y=0,x=f(y)..g(y),y=0..2,filled=true,coloring=[grey,grey]):

F:=plot([f(y),y,y=-1..2.3],thickness=3):

G:=plot([g(y),y,y=-1..2.3],thickness=3,color=blue):

display(region,F,G,scaling=constrained);

 

 

 


Finally some questions to be answered or discussed.

    1. Is “coloring” used in the examples an option in the package Plots? But I cannot find it in the Help (Typing ? coloring produces no results.) .
   2. Why the strange but interesting thing happens in Example 3 ?

   3. Why the above method cannot be realized in Maple 11?
   If we input the following code in Maple 11,
with(plots):

implicitplot(y=0,x=0..1,y=x..x^2,filled=true,coloring=[yellow,yellow]);with(plots);
An error warning will occur:
Error, (in plots/implicitplot) invalid input: invalid range for second variable

This means varying range for the second variable y (eg. y=x..x^2) is not allowed in Maple 11 , but which is allowed in Maple 8. If this is true, then I doubt if Maple 11 is really stronger than its earlier versions in all respects.

 

I wonder if there are size limits for uploaded files. And what are they? How Thanks.

In the blog  Plots of twisted ribbons, the author gave an interesting description of plotting twisted ribbons. In this blog , we give a similar description of twisted ribbons and give the geometrical interpretations of this definition.

 

Let r(phi)=[a*cos(phi), a*sin(phi), 0] (phi=0..2*Pi) be a circle in the xy-plane, and P be a point on the circle. Let QR be a line segment (with length of 2) passing through the point P and let P be the middle point of QR. Also, QR is coplanar with the z-axis.

Now let P rotate about the z-axis at the angular velocity of phi, where phi is the angle between OP and the x-axis. At the same time, the line segment QR is rotating about its middle point, P, at the angular velocity of theta (where theta is the angle between PQ and the z-axis and theta is dependent on phi, eg, theta=k*phi). In the whole process, QR will remain coplanar with the z-axis.

 

Apparently, the locus of the line segment QR is a twisted ribbon. When theta=phi/2, we have the Mobius strip.


2. The equation of the twisted ribbon

 

Now we try to find the equation of the surface. Clearly,

vector(OP)= [a*cos(phi), a*sin(phi), 0].

And with some geometrical manipulations, we have

vector(PQ)=[sin(theta)*cos(phi), sin(theta)*sin(phi), cos(theta)].

So the vector equation of the twisted ribbon is

V(phi, t)=vector(OP)+t*vector(PQ)

And the parametric equation is
x=a*cos(phi)+t*sin(theta)*cos(phi),
y=a*sin(phi)+t*sin(theta)*sin(phi)

z=t*cos(theta)

(where theta=k*phi (k a constant), phi=0..2*Pi and t=-b..b (b determines the width of the ribbon.))

Or
x=(a+t*sin(k*phi))*cos(phi),
y=(a+t*sin(k*phi))*sin(phi)

z=t*cos(k*phi)

(where k a constant, phi=0..2*Pi and t= -b..b)

When we take k=1/2, 1, 3/2, 2,…, we have different twisted ribbons.

When k=1/2, we have the equation of the Mobius strip:
x=(a+t*sin(phi/2))*cos(phi),
y=(a+t*sin(phi/2))*sin(phi)

z=t*cos(ph)/2

(phi=0..2*Pi and t= -b..b) .

 

k=1

 

k=2

I want to plot the normal vector of the Mobius strip represented by the parametric equations:
x=(2+u*cos(v/2))*cos(v):

In Maple we can easily plot a space curve if it is given by a parametric form:

x=x(t), y=y(t), z=z(t) (t=a..b).

However, Maple does not give a easy way to plot a space curve if it is given by the intersection of two surfaces:
F(x,y,z)=0, G(x,y,z)=0.

 So, I wonder if there are easy ways to solve this plotting problem?

For example, I'd like to know how to plot the following space curve In Maple :
x^2+y^2+3*z^2=1, 2*x+3*y+z=0.

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