CutSurface.html

Clipping of surfaces

Example from the PscFunctions package

Using routines of the PscFunctions package it is possible to execute transformation of equations of surfaces.
Here we give some examples of clipping of surfaces. Let's create a surface .

`>`	restart; with(PscFunctions): f:=(x,y)->3/(1+x^2+y^2): plot3d(f(x,y),x=-2..2,y=-2..2,axes=boxed);

Using the base routines QL , QR and PR of the package it is possible to create a parametric equation of a clipped surface.
Let us cut off the surface from two sides along the curves f0(x) and f1(x).

`>`	f0:=t->-1-sqrt(1-(-1+2PR(t,-1,2))^2): f1:=t->1+sqrt(1-(-1+2PR(t,-1,2))^2): plot({f0(x),f1(x)},x=-2..2,thickness=3,scaling=CONSTRAINED,color=black);

> dlt:=1: # increment of parameter along a vertical shear
x:=(u,v)->QL(u,f0(v)-dlt)+f0(v)+(f1(v)-f0(v))*\
PR(u,f0(v),f1(v)-f0(v))+QR(u,f1(v)+dlt):
y:=(u,v)->v:
z:=(u,v)->f(f0(v)+(f1(v)-f0(v))*PR(u,f0(v),f1(v)-f0(v)),v)-\
f(f0(v),v)*(1-PR(u,f0(v)-dlt,dlt))-f(f1(v),v)*PR(u,f1(v),dlt):
plot3d([x(u,v),y(u,v),z(u,v)],u=-3..3,v=-2..2,scaling=CONSTRAINED,axes=BOXED,grid=[37,25],shading=ZHUE);

The SCone routine generates a parametric equation of a cone surface or a cylinder (a surface with bases). Using equation of the cylinder surface we can obtain a parametric equation of the truncated cylinder. To clipping a parametric surface by oblique plane it is possible to use PscMax , PscMin , PscMinMax routines of the package.

`>`	cl:=SCone(2,1,1): Zplane:=(x,y)->x/2-y/2+4/3: # equation of truncating plane x:=(u,v)->cl[1](u,v): y:=(u,v)->cl[2](u,v): z:=(u,v)->PscMin(cl[3](u,v),Zplane(x(u,v),y(u,v))): plot3d([x(u,v),y(u,v),z(u,v)],u=0..2*Pi,v=0..4,scaling=CONSTRAINED,grid=[21,21],axes=BOXED);

Here is the equation of the surface.

`>`	'X'=x(u,v); 'Y'=y(u,v); 'Z'=z(u,v);

Here is the equation of the sphere clipped from different sides.

> xs:=(R,phi,theta)->R*cos(phi)*cos(theta):
ys:=(R,phi,theta)->R*sin(phi)*cos(theta):
zs:=(R,phi,theta)->R*sin(theta):
X:=(u,v)->PscMinMax(xs(1,u,v),-3/4,3/4):
Y:=(u,v)->PscMinMax(ys(1,u,v),-3/4,3/4):
Z:=(u,v)->PscMinMax(zs(1,u,v),-3/4,3/4):
plot3d([X(u,v),Y(u,v),Z(u,v)],u=0..2*Pi,v=-Pi/2..Pi/2,scaling=CONSTRAINED,axes=BOXED,grid=[36,22]);
'X'=X(u,v);
'Y'=Y(u,v);
'Z'=Z(u,v);