MAT331,
Feb 16th, 2010
If a curve fits, wear it (Chapter 2 of the online book)
EXERCISE: Is it possible to find a linear function passing through [1,-1/2], [2,1] and [3,1/3]? Why?
pts:=[ [1,-1/2], [2,1] , [3,1/3]];
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with(plots):
graph1:=pointplot(pts,color=blue):
graph1;
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It is not possible because these points are not colinear.
EXERCISE: Find the linear function passing closest to the points [1,-1/2], [2,1] and [3,1/3]
display(plot(x/3,x=0..4),graph1);
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pt1:=[(pts[1][1]+pts[2][1])/2,(pts[1][2]+pts[2][2])/2 ];
pt2:=[(pts[3][1]+pts[2][1])/2,(pts[3][2]+pts[2][2])/2 ];
line:=(x-pt1[1])*(pt2[2]-pt1[2])/(pt2[1]-pt1[1])+pt1[2];
NyQjIiIkIiIjIyIiIiIiJQ==
NyQjIiImIiIjI0YlIiIk
LCZJInhHNiIjIiImIiM3IyEiJCIiKSIiIg==
display(plot([x/3,line],x=0..4,thickness=3),graph1,plot([pt1,pt2],style=point));
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EXERCISE: Find the quadratic function passing closest to the points [1,-1/2], [2,1] and [3,1/3]
EXERCISE: Find the quadratic function passing closest to the points [1,2], [2.4,1.5] , [3,1] and [-0.4,3]
EXERCISE: Find the linear function passing closest to the points [1,-1/2], [2,1], [2.4,1] and [3,1/3] . You can use the graph below to study the distance.
pts:=[ [1,-1/2], [2,1], [2.4,1] , [3,1/3] ];
NyY3JCIiIiMhIiIiIiM3JEYnRiQ3JCQiI0NGJkYkNyQiIiQjRiRGLQ==
Note: The graph below works only if all the vertical blue segments have length larger than 0.
L:=x->(x-1);
display(plot(pts, x = 0 .. 5, style = point, symbol = box, color = green), plot([L(x)], x = 0 .. 5, color = red), seq(plot([pts[i][1], t, t = pts[i][2] .. L(pts[i][1])], thickness = 3, color = blue), i = 1 .. nops(pts)));
Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCY5JCIiIiEiIkYrRiVGJUYl
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distancePointLine:=P->(m*P[1]+b-P[2])^2;
Zio2I0kiUEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQsKComSSJtR0YlIiIiJjkkNiNGLUYtRi1JImJHRiVGLSZGLzYjIiIjISIiRjRGJUYlRiU=
distancePointsLine:=pts->sum(distancePointLine(pts[i]),i=1..nops(pts));
Zio2I0kkcHRzRzYiRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtSSRzdW1HRiU2JC1JMmRpc3RhbmNlUG9pbnRMaW5lR0YlNiMmOSQ2I0kiaUdGJS9GMjsiIiItSSVub3BzRyUqcHJvdGVjdGVkRzYjRjBGJUYlRiU=
DD:=distancePointsLine(pts);
LCoqJCwoSSJtRzYiIiIiSSJiR0YmRicjRiciIiNGJ0YqRicqJCwoRiVGKkYoRichIiJGJ0YqRicqJCwoRiUkIiNDRi1GKEYnRi1GJ0YqRicqJCwoRiUiIiRGKEYnI0YtRjRGJ0YqRic=
simplify(DD);
LC4qJEkibUc2IiIiIyQiKysrK3c+ISIpKiZGJCIiIkkiYkdGJUYrJCIrKysrIW8iRilGJCQhKysrKyspKiEiKiokRixGJiQiIiUiIiFGLCQhK25tbW1PRjEkIis2NjZoQkYxRis=
eq1:=diff(DD,m);
eq2:=diff(DD,b);
LChJIm1HNiIkIiVfUiEiI0kiYkdGJCQiJG8iISIiJCEjKSpGKyIiIg==
LChJIm1HNiIkIiRvIiEiIkkiYkdGJCIiKSMhIzYiIiQiIiI=
sol:=solve({eq1,eq2},{m,b});
PCQvSSJiRzYiJCEraitoPGUhIzUvSSJtR0YlJCIrKj1JRyZcRig=
foundedLine:=x->subs(sol, m*x+b);
Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUklc3Vic0clKnByb3RlY3RlZEc2JEkkc29sR0YlLCYqJkkibUdGJSIiIjkkRjFGMUkiYkdGJUYxRiVGJUYl
L:=foundedLine;
display(plot(pts, x = 0 .. 5, style = point, symbol = box, color = green), plot([L(x)], x = 0 .. 5, color = red), seq(plot([pts[i][1], t, t = pts[i][2] .. L(pts[i][1])], thickness = 3, color = blue), i = 1 .. nops(pts)));
SSxmb3VuZGVkTGluZUc2Ig==
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plot3d(DD,m=-10..10,b=-10..10);
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EXERCISE: Consider the points
newpts := [seq([i, 1.5*i+1], i = -1 .. 5), [8, 1.5*8+.5], seq([i, 1.5*i+1], i = 9 .. 11)];
a. Find a polynomial passing through those points.
b. Fit the points with a line using the least squares method. Use the graph of the previous exercise to visualize the line you found, and see how "close" the line and the points are.
c. Fit the points with a quadratic function using the least squares method.
d. Find a piecewise linear function passing through all the points.
e. Plot the points and all the functions together.
f. Compare the results. In particular, compute the value of the function you found in each case when x=12.3
Problems 11 and 13
Fitting points to a circle
Here is a program that produces a list of points approximately in a circle. You do not have to worry about the details of this program (but you are wellcome to do it if you want), just run it to obtain the points.
Seed:=readlib(randomize)():
with(stats):
myrand := (a,b) -> evalf(a+(b-a)*(rand()/10^12)):
points_wanted := 50:
circlePts := proc()
local fuzz, a, b, r,s,x,i;
a:= myrand(-3,3);
b:= myrand(-3,3);
r:= myrand(1,2);
fuzz := [ random[normald[0,0.1]](points_wanted)];
x := [ random[uniform[a-r,a+r]](points_wanted)];
for i from 1 to points_wanted do if (myrand(-10,10) > 0) then s[i]:=1 else s[i]:=-1 fi:od:
return [seq([x[i],b+s[i]*sqrt(r^2-(x[i]-a)^2)+fuzz[i]],i=1..points_wanted)];
end:
myCirclePts:=circlePts():
JSFH