Tomorrow is Groundhog Day!

3-d Plotting

Do not be frightened

Do not be uninterested

Be excited!

Because we will be able to do this

plot3



t = 0:pi/50:10*pi;

st = sin(t);

ct = cos(t);

plot3(st,ct,t)

t = 0:pi/50:pi;

st = exp(t.^2).*sin(t);

ct = (1+2*exp(t)).*cos(t).*sin(t);

plot3(st,ct,t)

scatter3



[X,Y,Z] = sphere(16);

x = [0.5*X(:); 0.75*X(:); X(:)];

y = [0.5*Y(:); 0.75*Y(:); Y(:)];

z = [0.5*Z(:); 0.75*Z(:); Z(:)];

scatter3(x,y,z)

z = linspace(0,4*pi,250);

x = 2*cos(z) + rand(1,250);

y = 2*sin(z) + rand(1,250);

scatter3(x,y,z,'filled')

view(-30,10)

z = linspace(0,4*pi,250);

x = 2*cos(z) + rand(1,250);

y = 2*sin(z) + rand(1,250);

scatter3(x,y,z,'*')

view(-30,10)

Remember the Snowflakes

meshgrid

x = 1:3;

y = 1:5;

[X,Y] = meshgrid(x,y)

x = -2:0.25:2;

[X,Y] = meshgrid(x);

F = X.*exp(-X.^2-Y.^2);

surf(X,Y,F)

x = -2:0.25:2;

y = x;

surf(x,y,x.*exp(-x.^2-(y').^2))

contour



x = linspace(-2*pi,2*pi);

y = linspace(0,4*pi);

[X,Y] = meshgrid(x,y);

Z = sin(X)+cos(Y);

contour(X,Y,Z)

Show 20 lines

[X,Y,Z] = peaks;

contour(X,Y,Z,20)

Show labels on contour

x = -2:0.2:2;

y = -2:0.2:3;

[X,Y] = meshgrid(x,y);

Z = X.*exp(-X.^2-Y.^2);

contour(X,Y,Z,'ShowText','on')

contourf

Z = peaks(20);

contourf(Z,10)

contour3

x = -2:0.25:2;

[X,Y] = meshgrid(x);

Z = X.*exp(-X.^2-Y.^2);

contour3(X,Y,Z,30)

fcontour

f = @(x,y) sin(x) + cos(y);

fcontour(f)

fcontour - multiple

fcontour(@(x,y) sin(x)+cos(y))

hold on

fcontour(@(x,y) x-y)

hold off

Vector Fields

not (sort of)



yes



feather



theta = -pi/2:pi/16:pi/2;

r = 2*ones(size(theta));

[u,v] = pol2cart(theta,r);

feather(u,v)

quiver



[x,y] = meshgrid(0:0.2:2,0:0.2:2);

u = cos(x).*y;

v = sin(x).*y;

quiver(x,y,u,v)

[X,Y] = meshgrid(-2:.2:2);

Z = X.*exp(-X.^2 - Y.^2);

[DX,DY] = gradient(Z,.2,.2);

contour(X,Y,Z)

hold on

quiver(X,Y,DX,DY)

hold off

quiver3

x = -3:0.5:3;

y = -3:0.5:3;

[X,Y] = meshgrid(x, y);

Z = Y.^2 - X.^2;

[U,V,W] = surfnorm(Z);

quiver3(Z,U,V,W)

view(-35,45)

vz = 10; % velocity constant

a = -32; % acceleration constant

t = 0:.1:1;

z = vz*t + 1/2*a*t.^2;

vx = 2;

x = vx*t;

vy = 3;

y = vy*t;

u = gradient(x);

v = gradient(y);

w = gradient(z);

scale = 0;

quiver3(x,y,z,u,v,w,scale)

view([70,18])

streamslice -- not homeslice



load wind

streamslice(x,y,z,u,v,w,[],[],[5])

axis tight

streamline



[x,y] = meshgrid(0:0.1:1,0:0.1:1);

u = x;

v = -y;

quiver(x,y,u,v)

startx = 0.1:0.1:1;

starty = ones(size(startx));

streamline(x,y,u,v,startx,starty)

waterfall



[X,Y,Z] = peaks(30);

waterfall(X,Y,Z)

Intermission



Surface Plots

surf



[X,Y,Z] = peaks(25);

surf(X,Y,Z);

surfc

[X,Y,Z] = peaks(30);

surfc(X,Y,Z);

surface



[X,Y] = meshgrid(-2:0.2:2,-2:0.2:2);

Z = X.*exp(-X.^2 - Y.^2);

surface(X,Y,Z)

view(3)

fsurf

fsurf(@(x,y) sin(x)+cos(y))

r = @(u,v) 2 + sin(7.*u + 5.*v);

funx = @(u,v) r(u,v).*cos(u).*sin(v);

funy = @(u,v) r(u,v).*sin(u).*sin(v);

funz = @(u,v) r(u,v).*cos(v);

fsurf(funx,funy,funz,[0 2*pi 0 pi])

camlight

Mesh Plots

mesh

[X,Y] = meshgrid(-8:.5:8);

R = sqrt(X.^2 + Y.^2) + eps;

Z = sin(R)./R;

mesh(Z)

meshc

[X,Y] = meshgrid(-3:.125:3);

Z = peaks(X,Y);

meshc(Z)

fmesh

r = @(s,t) 2 + sin(7.*s + 5.*t);

x = @(s,t) r(s,t).*cos(s).*sin(t);

y = @(s,t) r(s,t).*sin(s).*sin(t);

z = @(s,t) r(s,t).*cos(t);

fmesh(x,y,z,[0 2*pi 0 pi])

alpha(0.8)

Built-in Shapes

Cylinder



cylinder

t = 0:pi/10:2*pi;

[X,Y,Z] = cylinder(2+cos(t));

surf(X,Y,Z)

axis square

Sphere



sphere

[x,y,z] = sphere;

surf(x,y,z)

hold on

surf(x+3,y-2,z) % centered at (3,-2,0)

surf(x,y+1,z-3) % centered at (0,1,-3)

Ellipsoid



[x, y, z] = ellipsoid(0,0,0,5.9,3.25,3.25,30);

surf(x, y, z)

axis equal

Animation

comet3



t = -10*pi:pi/250:10*pi;

x = (cos(2*t).^2).*sin(t);

y = (sin(2*t).^2).*cos(t);

comet3(x,y,t);


load wind

xmin = min(x(:));

xmax = max(x(:));

ymax = max(y(:));

zmin = min(z(:));

wind_speed = sqrt(u.^2 + v.^2 + w.^2);

hsurfaces = slice(x,y,z,wind_speed,[xmin,100,xmax],ymax,zmin);

set(hsurfaces,'FaceColor','interp','EdgeColor','none')

colormap jet

hcont = ...

contourslice(x,y,z,wind_speed,[xmin,100,xmax],ymax,zmin);

set(hcont,'EdgeColor',[0.7 0.7 0.7],'LineWidth',0.5)

[sx,sy,sz] = meshgrid(80,20:10:50,0:5:15);

hlines = streamline(x,y,z,u,v,w,sx,sy,sz);

set(hlines,'LineWidth',2,'Color','r')

view(3)

daspect([2,2,1])

axis tight

Algorism and Bushism of the Day

"I think war is a dangerous place."

"He is proposing to privatize a big part of Social Security and he's proposing to take $1 trillion, a million billion dollars out of the Social Security trust fund and give it as a tax incentive to young workers."

A trillion is one thousand billion, not a million billion.

Lesson of the Day

OODA Loop



Gödel’s Incompleteness Theorems.

Any logical model of reality is incomplete (and possibly inconsistent) and must be continuously refined/adapted in the face of new observations.

Heisenberg’s Uncertainty Principle.

This principle shows that we cannot simultaneously fix or determine the velocity and position of a particle or body.

2nd Law of Thermodynamics.

Just as a closed system in nature will have increasing entropy, or disorder, so too will a person experience mental entropy or disorder if they’re cut off from the outside world and new information.



Word of the day

Bring the Noise

Why?



The End