Spring 2004 Math 253/501–503
Surface Graphics
c
Wed, 31/Mar
2004,
Art Belmonte
Solution
When the paraboloids intersect, their z-coordinates are equal.
Summary
4 + x 2 + y2
4x 2 + 4y 2
=
=
12 − 3x 2 − 3y 2
8
4r 2
r2
=
=
8
2
r
=
hvsd
My routine hvsd (“Horizontally or Vertically Simple Data”) is
principally designed to create rectangular coordinate data that is
used to graph surfaces. It allows for both rectangluar and nonrectangular parameter regions; e.g, regions that are rectangular,
triangular, bounded by curves, etc. Type “help hvsd” in the
MATLAB Command Window for exhaustive details. That said,
you’ll probably find the examples in this handout much easier to
understand (because they’re shorter).
√
√
± 2; pick 2.
Using cylindrical coordinates, we saw in class that the volume is
Z
ZZZ
1 dV =
V =
0
E
√ Z
2
12−3r 2
Z
2π
4+r 2
0
1·r dz dr dθ = 8π ≈ 25.13 cm3 .
As shown in class, here is the code that produced the plot.
The hvsd routine provides a way to produce surface graphics that
is both easier to use and more general than qsurf.
%
% Example Zero: A surface plot of intersecting
% circular paraboloids with curve of intersection
%
r = linspace(0, sqrt(2), 10);
t = linspace(0, 2*pi, 37);
[R,T] = meshgrid(r,t);
X = R .* cos(T);
Y = R .* sin(T);
Z1 = 4 + R.ˆ2;
Z2 = 12 - 3.*R.ˆ2;
surf(X,Y,Z1, ’FaceColor’, ’c’, ’EdgeColor’, ’k’)
hold on
surf(X,Y,Z2, ’FaceColor’, ’y’, ’EdgeColor’, ’k’)
x = sqrt(2).*cos(t); y = sqrt(2).*sin(t);
z = 0*x + 6;
plot3(x,y,z, ’m’, ’LineWidth’, 4)
%
Like many of the routines I’ve written, hvsd can be pushed
beyond its design specs to do some pretty unusual graphics.
More on this later.
Parametric surfaces
Cooper does a good job in Section 6.4 of A MATLAB Companion
for Multivariable Calculus in explaining how to graph parametric
surfaces using MATLAB. I’ll give several examples from our
lecture handouts that show the actual code used to produce the
graphics.
echo off; diary off
1. Explicit surface with a rectangular parameter region
MATLAB Examples
Graph z = x 2 + y 2 over the region −1 ≤ x ≤ 2, 0 ≤ y ≤ 3.
0. The “need-to-know” example for Exam 3
10
2
y
z
3
10
5
5
0
3
0
3
2
2
1
1
y
Find the volume of the solid bounded below by the circular
paraboloid z = 4 + x 2 + y 2 and above by the circular paraboloid
z = 12 − 3x 2 − 3y 2 .
4
15
z
At the end of class on Tuesday, 30/Mar, we graphed intersecting
circular paraboloids and found the volume between them. For
your benefit, we kick the party off by replicating this example.
hvsd / Example 1: Parameter region
hvsd / Example 1:
Solid color with black mesh
hvsd / Example 1:
Standard color map
15
1
0
2
2
−1
x
1
1
0
0
y
0
0
−1
x
−1
−2
−1
0
1
2
3
x
Solution
Example 0: "When paraboloids intersect!"
We could do this one the conventional way with linspace and
meshgrid. However, hvsd mimicks the way you’d set up a
mutiple integral (inner and outer ranges), so it helps you in that
way as well. In the code below, note that the “receiving” matrices
X , Y , and Z on the left-hand side of the assignment statement in
which hvsd appears occur in the same order as the arguments
passed to the routine. You follow up the data generation with a
surface graphics command: usually surf, but others are possible
(mesh, contour, pcolor, etc.).
z = 12 − 3x2 − 3y2
12
z
10
8
Curve of intersection
6
4
2
0
y
−2
−2
0
2
2
2
z=4+x +y
x
1
By default, a standard multicolor color map (“jet”) is used. You
may alternatively specify a solid color. As you saw in Example
Zero, this is useful when you have separate faces that you’d like to
distinguish. (These are called surface “patches.”) Here are two
codes that produced the respective plots.
t = linspace(1, 3, 20);
x1 = t; y1 = 0*t+1; x1 = fliplr(x1); y1 = fliplr(y1);
x2 = t; y2 = 5/2 - t/2;
x = [x1 x2]; y = [y1 y2];
fill(x,y,’g’); grid on;
axis equal; axis([0.5 3.5 0.5 2.5])
%
hold on
plot([0.5 3.5], [0 0], ’k’, ’LineWidth’, 2)
plot([0 0], [0.5 2.5], ’k’, ’LineWidth’, 2)
%
%==========
%
% hvsd / Example 1: surface pix
%
syms x y
[Z X Y] = hvsd(xˆ2 + yˆ2, [x -1 2 20], [y 0 3 20]);
surf(X,Y,Z)
figure
surf(X,Y,Z, ’FaceColor’, ’m’, ’EdgeColor’, ’k’)
%
echo off; diary off
%==========
%
% hvsd / Example 1: parameter region in xy-plane
%
M = [-1 0; 2 0; 2 3; -1 3];
x = M(:,1); y = M(:,2);
fill(x,y,’g’); grid on;
axis equal; axis([-2 3 -1 4])
%
hold on
plot([-2 3], [0 0], ’k’, ’LineWidth’, 2)
plot([0 0], [-1 4], ’k’, ’LineWidth’, 2)
set(gca, ’Ytick’, -1:4)
%
echo off; diary off
echo off; diary off
3. Explicit surface with a curved parameter region
Graph z = sin y + 18 x 2 over −y ≤ x ≤ 2 − y 2 , −1 ≤ y ≤ 2.
hvsd / Example 3:
Solid color with black mesh
hvsd / Example 3:
Standard color map
hvsd / Example 3: Parameter region
2
2
1.5
0.5
0.5
1
0
0.5
0
−0.5
−0.5
−1
2
−1
2
2
1
x = −y
0
2
1
0
0
−1
y
y
1
z
z
x=2−y
1
−2
x
0
0
y
−1
−2
−0.5
−1
−2
−1
x
0
x
1
Solution
2. Explicit surface with a triangular parameter region
Graph z =
+
x2
over the region 1 ≤ y ≤
y3
hvsd / Example 2:
Standard color map
5
2
−
hvsd / Example 2:
Solid color with black mesh
1
2 x, 1
Curved boundaries on your parameter region? No problemo!
≤ x ≤ 3.
%==========
%
% hvsd / Example 3: surface pix
%
syms x y
[Z X Y] = hvsd(sin(y + xˆ2/8), [x -y 2-yˆ2 10], [y -1 2 20]);
surf(X,Y,Z)
figure
surf(X,Y,Z, ’FaceColor’, ’m’, ’EdgeColor’, ’k’)
%
echo off; diary off
%==========
%
% hvsd / Example 3: parameter region in xy-plane
%
t = linspace(-1, 2, 20);
x1 = -t; y1 = t; x1 = fliplr(x1); y1 = fliplr(y1);
x2 = 2-t.ˆ2; y2 = t;
x = [x1 x2]; y = [y1 y2];
fill(x,y,’g’); grid on;
axis equal; axis tight
%
hold on
plot([-2 2], [0 0], ’k’, ’LineWidth’, 2)
plot([0 0], [-2 2], ’k’, ’LineWidth’, 2)
%
hvsd / Example 2: Parameter region
2.5
8
8
6
6
2
y
z
10
z
10
4
y = 5/2 − x/2
1.5
4
2
2
2
2
1
3
1.5
y
2
1
1
x
3
1.5
y
2
1
1
x
0.5
0.5
y=1
1
1.5
2
x
2.5
3
3.5
Solution
Note the order of the receiving matrices in the code below. The
specification of the parameter region is just like you’d set up the
the inner and outer ranges for a multiple integral; very natural, eh?
%==========
%
% hvsd / Example 2: surface pix
%
syms x y
[Z Y X] = hvsd(xˆ2 + yˆ3, [y 1 5/2-x/2 20], [x 1 3 20]);
surf(X,Y,Z)
figure
surf(X,Y,Z, ’FaceColor’, ’m’, ’EdgeColor’, ’k’)
%
echo off; diary off
%==========
%
% hvsd / Example 2: parameter region in xy-plane
%
% Here we trace the boundary of the filled region in a
% clockwise fashion. ‘The tangled webs we weave...’
%
echo off; diary off
4. Parametric surface (HCFT: half-cone + flat top, 853/4)
Sketch the solid whose volume is given by the integral
Z 2π Z π/3 Z sec φ
ρ 2 sin φ dρ dφ dθ.
0
2
0
0
Stewart 853/4: HCFT
(half−cone, flat top)
X = 4 * sin(P) .* cos(T);
Y = 4 * sin(P) .* sin(T);
Z = 4 * cos(P);
%mesh(X,Y,Z, ’FaceColor’, ’r’, ’EdgeColor’, ’k’)
surf(X,Y,Z)
axis equal
%
1
0.8
z
0.6
echo off; diary off
0.4
0.2
0
−1
0
1
1
6. Parametric surface (cone + sphere, 849/62)
−1
0
x
y
Sketch the solid described by the inequalities 0 ≤ φ ≤
ρ ≤ 2.
Stewart 849/62: Boy, howdy! Ice cream!
%
% Stewart 853/4g
%
p = linspace(0, pi/3, 7);
r = linspace(0, 2, 11);
t = linspace(0, 2*pi, 28);
[P,T] = meshgrid(p,t);
rho = sec(P);
X = rho .* sin(P) .* cos(T);
Y = rho .* sin(P) .* sin(T);
Z = rho .* cos(P);
mesh(X,Y,Z, ’FaceColor’, ’r’, ’EdgeColor’, ’k’)
hold on
%
[R,T] = meshgrid(r,t);
X = R .* sin(pi/3) .* cos(T);
Y = R .* sin(pi/3) .* sin(T);
Z = R .* cos(pi/3);
mesh(X,Y,Z, ’FaceColor’, ’g’, ’EdgeColor’, ’k’)
% axis equal
%
2
z
1.5
1
0.5
0
−1
0
1
1
0
−1
x
y
%
% Stewart 849/62
%
p = linspace(0, pi/3, 7);
r = linspace(0, 2, 11);
t = linspace(0, 2*pi, 28);
[P,T] = meshgrid(p,t);
X = 2 * sin(P) .* cos(T);
Y = 2 * sin(P) .* sin(T);
Z = 2 * cos(P);
mesh(X,Y,Z, ’FaceColor’, ’r’, ’EdgeColor’, ’k’)
hold on
%
[R,T] = meshgrid(r,t);
X = R .* sin(pi/3) .* cos(T);
Y = R .* sin(pi/3) .* sin(T);
Z = R .* cos(pi/3);
mesh(X,Y,Z, ’FaceColor’, ’y’, ’EdgeColor’, ’k’)
axis equal
%
echo off; diary off
5. Parametric surface (concentric spheres, 848/50)
Identify the surface whose equation is ρ 2 − 6ρ + 8 = 0.
Stewart 848/50:
Mother sphere with
baby sphere inside
4
2
echo off; diary off
0
−2
−4
−4
7. Parametric surface (cylinder + hemisphere, 849/65)
−2
0
2
y
4
4
2
0
−2
−4
Draw a silo consisting of a cylinder of radius 3 and height 10
surmounted by a hemisphere.
x
Stewart 849/62: Grain silo
%
% Stewart 848/50
%
p = linspace(0, pi, 19);
t = linspace(0, 2*pi, 37);
[P,T] = meshgrid(p,t);
X = 2 * sin(P) .* cos(T);
Y = 2 * sin(P) .* sin(T);
Z = 2 * cos(P);
mesh(X,Y,Z, ’FaceColor’, ’m’, ’EdgeColor’, ’k’)
hold on
%
p = linspace(0, pi, 19);
t = linspace(pi/2, 2*pi, 28);
[P,T] = meshgrid(p,t);
12
10
8
z
z
π,
3
6
4
2
0
2
3
0 −2
y
0
−2
2
x
%
% Stewart 849/65
%
p = linspace(0, pi/2, 10);
r = linspace(0, 3, 7);
t = linspace(0, 2*pi, 28);
z = linspace(0, 10, 6);
%
[T,Z] = meshgrid(t,z);
X = 3*cos(T);
Y = 3*sin(T);
mesh(X,Y,Z, ’FaceColor’, ’b’, ’EdgeColor’, ’k’)
hold on
%
[P,T] = meshgrid(p,t);
X = 3 .* sin(P) .* cos(T);
Y = 3 .* sin(P) .* sin(T);
Z = 3 .* cos(P) + 10;
mesh(X,Y,Z, ’FaceColor’, ’r’, ’EdgeColor’, ’k’)
axis equal
%
t = linspace(0, pi/2, 19);
[R,T] = meshgrid(r,t);
X1 = R .* sin(pi/4) .* cos(T);
Y1 = R .* sin(pi/4) .* sin(T);
Z1 = R .* cos(pi/4);
mesh(X1,Y1,Z1, ’FaceColor’, ’g’,
p = linspace(0, pi/4, 10);
[P,T] = meshgrid(p,t);
rho = sqrt(18);
X2 = rho .* sin(P) .* cos(T);
Y2 = rho .* sin(P) .* sin(T);
Z2 = rho .* cos(P);
mesh(X2,Y2,Z2, ’FaceColor’, ’r’,
%
p = linspace(0, pi/4, 10);
r = linspace(0, sqrt(18), 10);
[P,R] = meshgrid(p,r);
X3 = R .* sin(P) .* cos(0);
Y3 = R .* sin(P) .* sin(0);
Z3 = R .* cos(P);
mesh(X3,Y3,Z3, ’FaceColor’, ’m’,
%
X4 = R .* sin(P) .* cos(pi/2);
Y4 = R .* sin(P) .* sin(pi/2);
Z4 = R .* cos(P);
mesh(X4,Y4,Z4, ’FaceColor’, ’y’,
%
axis equal
%
echo off; diary off
8. Parametric surface (wedge, 854/36)
Evaluate the integral
Z 3 Z √9−y 2 Z √18−x 2 −y 2
0
√
0
x 2 +y 2
’EdgeColor’, ’k’); hold on
’EdgeColor’, ’k’)
’EdgeColor’, ’k’)
’EdgeColor’, ’k’)
echo off; diary off
x 2 + y 2 + z 2 dz d x d y
9. Parametric surface (the wedge ‘o‘ stilton, 843/18)
by changing to spherical coordinates. Also draw this wedgeshaped region.
Use a triple integral to find the volume of the solid E bounded by
the elliptic cylinder 4x 2 + z 2 = 4 and the planes y = 0 and
y = z + 2.
Solution
Solution
The p
region of integration lies in the first octant, with the cone
z = x 2 + y 2 below and the sphere x 2 + y 2 + z 2 = 18 above.
Here is a picture showing boundary surfaces of the solid.
Here are two 3-D views of the solid’s faces, followed by its
projection onto the x z-plane.
Faces of solid
Stewart 854/36
Projection onto xz−plane
Backside view
2
2
4
Sphere
1
2
0
1
1
z
z
xz−plane
z
3
0
0
−1
−1
z
−1
4
−2
0
2
2
4
0
x
1
y
1
The volume is
Z
ZZZ
1 dV =
V =
Cone
0
0
1
2
x
0
2
1
y
E
p
The cone is ρ cos φ = z = x 2 + y 2 = r = ρ sin φ, whence
tan φ = 1 or φ = π4 . Switching to spherical coordinates we have
Z π/2 Z π/4 Z √18
0
0
0
ρ 2 ·ρ 2 sin φ dρ dφ dθ =
−2
−2
−1
−1
2
x
1
Z
0
1
√
4−4x 2
0
−1
y
Z
√
−1 − 4−4x 2 0
z+2
0
x
1
1 d y dz d x = 4π ≈ 12.57 cm3 .
Here is the code which produced the picture! This is where we
have pushed hvsd beyond its design specs; very subtle. . .
486π √
2 − 1 ≈ 126.485.
5
%
% Stewart 843/18g
%
t = linspace(0, 2*pi, 37);
r = linspace(0, 1, 11);
[R,T] = meshgrid(r,t);
X = R .* cos(T);
Z = 2 * R .* sin(T);
Y = Z + 2;
mesh(X,Y,Z, ’FaceColor’, ’r’, ’EdgeColor’, ’k’)
Here is the code which produced the picture!
%
% Stewart 854/36g
%
r = linspace(0, sqrt(18), 10);
4
%
Y = 0*Y; hold on
mesh(X,Y,Z, ’FaceColor’, ’g’, ’EdgeColor’, ’k’)
%
syms r t y
[W Y T] = hvsd(1, [y, 0, 2*sin(t)+2, 10], [t, 0, 2*pi, 37]);
X = 1 * cos(T);
Z = 2 * sin(T);
mesh(X,Y,Z, ’FaceColor’, ’y’, ’EdgeColor’, ’k’)
axis equal; view(150,10)
%
figure
t = linspace(0, 2*pi, 37);
x = cos(t); y = 2*sin(t);
fill(x,y, ’g’); grid on; axis equal
axis([-1.5 1.5 -2.5 2.5]); hold on
plot([0 0], [-2 2], ’k’)
plot([-1 1], [0 0], ’k’)
%
echo off; diary off
5