MapleArcLength3D.mws

Arc Length in Space

Some Approximations

>    with(plots): 

Warning, the name changecoords has been redefined

>    Int(sqrt(1+4*t^2+(3*t^2-6*t+2)^2),t=0..2);

Int((5+52*t^2+9*t^4-36*t^3-24*t)^(1/2),t = 0 .. 2)

>    ArcLength:=evalf(%);

ArcLength := 5.017762950

>    Curve:=spacecurve([t,t^2,t^3-3*t^2+2*t,t=0..2],thickness=3,labels=[x,y,z],axes=boxed,color=red,orientation=[325,75]):

>    segments:=Array(1..4):

>    for i from 1 to 4 do segments[i]:=spacecurve([(i-1)*(2/4)+(2/4)*t,((i-1)*(2/4))^2+(2*i-1)*((2/4)^2)*t,((i-1)*(2/4))^3-3*((i-1)*(2/4))^2+2*(i-1)*(2/4)+((i*2/4)^3-3*(i*2/4)^2+2*i*2/4-(((i-1)*(2/4))^3-3*((i-1)*(2/4))^2+2*(i-1)*(2/4)))*t,t=0..1],thickness=3,color=blue):end do:

>    display(Curve,seq(segments[k],k=1..4));

[Maple Plot]

>    i:='i';

i := 'i'

>    sum(sqrt((2/4)^2+((2*i-1)*((2/4)^2))^2+(((i*2/4)^3-3*(i*2/4)^2+2*i*2/4-(((i-1)*(2/4))^3-3*((i-1)*(2/4))^2+2*(i-1)*(2/4))))^2),i=1..4);

1/8*29^(1/2)+1/8*61^(1/2)+1/8*125^(1/2)+1/8*221^(1/2)

>    evalf(%);

4.905227890

>    ArcLength;

5.017762950

>    segments:=Array(1..8):

>    for i from 1 to 8 do segments[i]:=spacecurve([(i-1)*(2/8)+(2/8)*t,((i-1)*(2/8))^2+(2*i-1)*((2/8)^2)*t,((i-1)*(2/8))^3-3*((i-1)*(2/8))^2+2*(i-1)*(2/8)+((i*2/8)^3-3*(i*2/8)^2+2*i*2/8-(((i-1)*(2/8))^3-3*((i-1)*(2/8))^2+2*(i-1)*(2/8)))*t,t=0..1],thickness=3,color=blue):end do:

>    display(Curve,seq(segments[k],k=1..8));

[Maple Plot]

>    i:='i';

>    sum(sqrt((2/8)^2+((2*i-1)*((2/8)^2))^2+(((i*2/8)^3-3*(i*2/8)^2+2*i*2/8-(((i-1)*(2/8))^3-3*((i-1)*(2/8))^2+2*(i-1)*(2/8))))^2),i=1..8);

1/64*713^(1/2)+1/64*409^(1/2)+1/64*737^(1/2)+1/64*1265^(1/2)+1/64*1777^(1/2)+1/64*2273^(1/2)+1/64*2969^(1/2)+1/64*4297^(1/2)

>    evalf(%);

4.992357744

>    ArcLength;

5.017762950

>    segments:=Array(1..12):

>    for i from 1 to 12 do segments[i]:=spacecurve([(i-1)*(2/12)+(2/12)*t,((i-1)*(2/12))^2+(2*i-1)*((2/12)^2)*t,((i-1)*(2/12))^3-3*((i-1)*(2/12))^2+2*(i-1)*(2/12)+((i*2/12)^3-3*(i*2/12)^2+2*i*2/12-(((i-1)*(2/12))^3-3*((i-1)*(2/12))^2+2*(i-1)*(2/12)))*t,t=0..1],thickness=3,color=blue):end do:

>    display(Curve,seq(segments[k],k=1..12));

[Maple Plot]

>    i:='i':

>    sum(sqrt((2/12)^2+((2*i-1)*((2/12)^2))^2+(((i*2/12)^3-3*(i*2/12)^2+2*i*2/12-(((i-1)*(2/12))^3-3*((i-1)*(2/12))^2+2*(i-1)*(2/12))))^2),i=1..12);

1/216*4357^(1/2)+1/216*2245^(1/2)+1/216*2197^(1/2)+1/216*3349^(1/2)+1/216*5053^(1/2)+1/216*6877^(1/2)+1/216*8605^(1/2)+1/216*10237^(1/2)+1/216*11989^(1/2)+1/216*14293^(1/2)+1/216*17797^(1/2)+1/216*2336...

>    evalf(%);

5.006454725

>    ArcLength;

5.017762950

>    segments:=Array(1..16):

>    for i from 1 to 16 do segments[i]:=spacecurve([(i-1)*(2/16)+(2/16)*t,((i-1)*(2/16))^2+(2*i-1)*((2/16)^2)*t,((i-1)*(2/16))^3-3*((i-1)*(2/16))^2+2*(i-1)*(2/16)+((i*2/16)^3-3*(i*2/16)^2+2*i*2/16-(((i-1)*(2/16))^3-3*((i-1)*(2/16))^2+2*(i-1)*(2/16)))*t,t=0..1],thickness=3,color=blue):end do:

>    display(Curve,seq(segments[k],k=1..16));

[Maple Plot]

>    i:='i':

>    sum(sqrt((2/16)^2+((2*i-1)*((2/16)^2))^2+(((i*2/16)^3-3*(i*2/16)^2+2*i*2/16-(((i-1)*(2/16))^3-3*((i-1)*(2/16))^2+2*(i-1)*(2/16))))^2),i=1..16);

1/512*15185^(1/2)+1/512*8641^(1/2)+1/512*6425^(1/2)+1/512*7241^(1/2)+1/512*10009^(1/2)+1/512*13865^(1/2)+1/512*18161^(1/2)+1/512*22465^(1/2)+1/512*26561^(1/2)+1/512*30449^(1/2)+1/512*34345^(1/2)+1/512*...

>    evalf(%);

5.011405234

>    ArcLength;

5.017762950

>    segments:=Array(1..20):

>    for i from 1 to 20 do segments[i]:=spacecurve([(i-1)*(2/20)+(2/20)*t,((i-1)*(2/20))^2+(2*i-1)*((2/20)^2)*t,((i-1)*(2/20))^3-3*((i-1)*(2/20))^2+2*(i-1)*(2/20)+((i*2/20)^3-3*(i*2/20)^2+2*i*2/20-(((i-1)*(2/20))^3-3*((i-1)*(2/20))^2+2*(i-1)*(2/20)))*t,t=0..1],thickness=3,color=blue):end do:

>    display(Curve,seq(segments[k],k=1..20));

[Maple Plot]

>    i:='i':

>    sum(sqrt((2/20)^2+((2*i-1)*((2/20)^2))^2+(((i*2/20)^3-3*(i*2/20)^2+2*i*2/20-(((i-1)*(2/20))^3-3*((i-1)*(2/20))^2+2*(i-1)*(2/20))))^2),i=1..20);

1/1000*39341^(1/2)+1/1000*24589^(1/2)+1/1000*17261^(1/2)+1/1000*15629^(1/2)+1/1000*18181^(1/2)+1/1000*23621^(1/2)+1/1000*30869^(1/2)+1/1000*39061^(1/2)+1/1000*47549^(1/2)+1/1000*55901^(1/2)+1/1000*6390...
1/1000*39341^(1/2)+1/1000*24589^(1/2)+1/1000*17261^(1/2)+1/1000*15629^(1/2)+1/1000*18181^(1/2)+1/1000*23621^(1/2)+1/1000*30869^(1/2)+1/1000*39061^(1/2)+1/1000*47549^(1/2)+1/1000*55901^(1/2)+1/1000*6390...

>    evalf(%);

5.013695102

>    ArcLength;

5.017762950

>    segments:=Array(1..24):

>    for i from 1 to 24 do segments[i]:=spacecurve([(i-1)*(2/24)+(2/24)*t,((i-1)*(2/24))^2+(2*i-1)*((2/24)^2)*t,((i-1)*(2/24))^3-3*((i-1)*(2/24))^2+2*(i-1)*(2/24)+((i*2/24)^3-3*(i*2/24)^2+2*i*2/24-(((i-1)*(2/24))^3-3*((i-1)*(2/24))^2+2*(i-1)*(2/24)))*t,t=0..1],thickness=3,color=blue):end do:

>    display(Curve,seq(segments[k],k=1..24));

[Maple Plot]

>    i:='i':

>    evalf(sum(sqrt((2/24)^2+((2*i-1)*((2/24)^2))^2+(((i*2/24)^3-3*(i*2/24)^2+2*i*2/24-(((i-1)*(2/24))^3-3*((i-1)*(2/24))^2+2*(i-1)*(2/24))))^2),i=1..24));

5.014938456

>    ArcLength;

5.017762950

>   

>    segments:=Array(1..28):

>    for i from 1 to 28 do segments[i]:=spacecurve([(i-1)*(2/28)+(2/28)*t,((i-1)*(2/28))^2+(2*i-1)*((2/28)^2)*t,((i-1)*(2/28))^3-3*((i-1)*(2/28))^2+2*(i-1)*(2/28)+((i*2/28)^3-3*(i*2/28)^2+2*i*2/28-(((i-1)*(2/28))^3-3*((i-1)*(2/28))^2+2*(i-1)*(2/28)))*t,t=0..1],thickness=3,color=blue):end do:

>    display(Curve,seq(segments[k],k=1..28));

[Maple Plot]

>    i:='i':

>    evalf(sum(sqrt((2/28)^2+((2*i-1)*((2/28)^2))^2+(((i*2/28)^3-3*(i*2/28)^2+2*i*2/28-(((i-1)*(2/28))^3-3*((i-1)*(2/28))^2+2*(i-1)*(2/28))))^2),i=1..28));

5.015687993

>    ArcLength;

5.017762950

>    evalf(sum(sqrt((2/100)^2+((2*i-1)*((2/100)^2))^2+(((i*2/100)^3-3*(i*2/100)^2+2*i*2/100-(((i-1)*(2/100))^3-3*((i-1)*(2/100))^2+2*(i-1)*(2/100))))^2),i=1..100));

5.017600312

>    evalf(sum(sqrt((2/1000)^2+((2*i-1)*((2/1000)^2))^2+(((i*2/1000)^3-3*(i*2/1000)^2+2*i*2/1000-(((i-1)*(2/1000))^3-3*((i-1)*(2/1000))^2+2*(i-1)*(2/1000))))^2),i=1..1000));

5.017761333

>    evalf(sum(sqrt((2/10000)^2+((2*i-1)*((2/10000)^2))^2+(((i*2/10000)^3-3*(i*2/10000)^2+2*i*2/10000-(((i-1)*(2/10000))^3-3*((i-1)*(2/10000))^2+2*(i-1)*(2/10000))))^2),i=1..10000));

5.017762931

>    evalf(sum(sqrt((2/20000)^2+((2*i-1)*((2/20000)^2))^2+(((i*2/20000)^3-3*(i*2/20000)^2+2*i*2/20000-(((i-1)*(2/20000))^3-3*((i-1)*(2/20000))^2+2*(i-1)*(2/20000))))^2),i=1..20000));

5.017762978

>    ArcLength;

5.017762950

>