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Involute of an Ellipse Univ.-Prof. Dr.-Ing. habil. J. BETTEN RWTH University Aachen Mathematical Models in Materials Science and Continuum Mechanics Augustinerbach 4-20 D-52056 A a c h e n , Germany betten@ mmw.rwth-aachen.de Abstract Using Maple 11, Ellipse-Evolvents have been constructed. This can be done by solving elliptic integrals with Maple. Furthermore, the author proposes a simple approximation, which is nearly identical to the elliptic-integral-solution. Keywords: Elliptic Integrals; Involute; Approximation; Fourier-Series Elliptic-Integral-Solution In the following the construction of the involute of an ellipse is discussed by utilzing the software MAPLE V, Release 10. restart: x(t)[ellipse]:=a*cos(t); y(t)[ellipse]:=b*sin(t);

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The involute of the ellipse has the following parameter form: xi(t):=a*cos(t)+a*S(t)*sin(t)/ sqrt(a^2*(sin(t))^2+b^2*(cos(t))^2); eta(t):=b*sin(t)-b*S(t)*cos(t)/ sqrt(a^2*(sin(t))^2+b^2*(cos(t))^2);

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where S(t) is an elliptic integral of the second kind expressing the arclength between two points t = 0 and t = t of the ellipse: S(t):=Int(sqrt(a^2*(sin(x))^2+b^2*(cos(x))^2),x=0..t)= b*Int(sqrt(k^2*(sin(x))^2+(cos(x))^2),x=0..t);

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The integration can be splitted into two parts: S(t,k)[0..Pi]:= b*simplify(int(sqrt(k^2*(sin(x))^2+(cos(x))^2),x=0..t)) assuming t>0 and t<Pi, k>=0 and k<infinity;

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S(Pi,k):=simplify(subs(t=Pi,%));

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S(t,k)[Pi..2*Pi]:= simplify(%+b*int(sqrt(k^2*(sin(x))^2+(cos(x))^2),x=Pi..t)) assuming t>Pi and t<2*Pi, k>=0 and k<infinity;

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For the parameter k = a/b with a = 3/2 and b = 1 we arrive at the arclength: S(t)[0..Pi]:=simplify(subs({b=1,k=3/2},%%%));

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S(t)[Pi..2*Pi]:=simplify(subs({b=1,k=3/2},%%));

NiM+Ji0lIlNHNiMlInRHNiM7JSNQaUcsJComIiIjIiIiJSNQaUciIiIiIiIsJiomIyIiKiIiIyIiIi0lKkVsbGlwdGljRUc2IywkKiYiIiQhIiIpIiImIyIiIiIiIyIiIiIiIiIiIiIiIiomIyIiJCIiIyIiIi0lKkVsbGlwdGljRUc2JC0lJGNvc0c2IyUidEcsJComIiIkISIiKSIiJiMiIiIiIiMiIiIiIiIiIiIiIiI=

Some parts of the arclength of the ellipse are given as follows: for i in [Pi/2,Pi] do S(i):=evalf(simplify(subs(t=i,S(t)[0..Pi]))) od;

NiM+LSUiU0c2IywkKiYiIiMhIiIlI1BpRyIiIiIiIiQiK1wqekopPiEiKg== NiM+LSUiU0c2IyUjUGlHJCIrKSopZmonUiEiKg==

for i in [3*Pi/2,2*Pi] do S(i):=evalf(simplify(subs(t=i,S(t)[Pi..2*Pi]))) od;

NiM+LSUiU0c2IywkKigiIiQiIiIiIiMhIiIlI1BpRyIiIiIiIiQiK1opUiZcZiEiKg== NiM+LSUiU0c2IywkKiYiIiMiIiIlI1BpRyIiIiIiIiQiKyd6PkYkeiEiKg==

Approximation Instead of the elliptic integral S(t) let us introduce the following FOURIER series as a suitable approximation, which has been deduced in a later chapter of this article. s(t):=1.2625*t-0.12437*sin(2*t)-0.0030935*sin(4*t); # FOURIER series

NiM+LSUic0c2IyUidEcsKComJCImREUiISIlIiIiJSJ0RyIiIiIiIiomJCImUEMiISImIiIiLSUkc2luRzYjLCQqJiIiIyIiIiUidEciIiIiIiIiIiIhIiIqJiQiJk40JCEiKCIiIi0lJHNpbkc2IywkKiYiIiUiIiIlInRHIiIiIiIiIiIiISIi

for i in [Pi/2,Pi,3*Pi/2,2*Pi] do s(i):=evalf(subs(t=i,s(t))) od;

NiM+LSUic0c2IywkKiYiIiMhIiIlI1BpRyIiIiIiIiQiK2ouOCQpPiEiKg== NiM+LSUic0c2IyUjUGlHJCIrRTJFbVIhIio= NiM+LSUic0c2IywkKigiIiQiIiIiIiMhIiIlI1BpRyIiIiIiIiQiKyozIlJcZiEiKg== NiM+LSUic0c2IywkKiYiIiMiIiIlI1BpRyIiIiIiIiQiK105X0t6ISIq

We see, the approximation s(t) is nearly identical to the elliptic integral S(t). Both functions, s(t) and S(t) are plotted in the next Figure. alias(H=Heaviside,th=thickness,c=color): p[1]:=plot(S(t)[0..Pi],t=0..Pi,scaling=constrained,c=black): p[2]:=plot(S(t)[Pi..2*Pi],t=Pi..2*Pi,c=black): p[3]:=plot(s(t),t=0..2*Pi,th=3,c=black): p[4]:=plot({S(Pi),s(t)*H(t-Pi)},t=0..1.001*Pi,c=black): p[5]:=plot({Pi,2*Pi,t,S(2*Pi),S(2*Pi)*H(t-2*Pi)}, t=0..2.001*Pi,c=black): plots[display]({seq(p[k],k=1..5)});

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In this Figure, the arclength S(t) and its approximation s(t) are nearly congruent. The 45-degrees-line is valid for the unit circle. Representations of the Involutes In representing the involute of an ellipse we need its coordinates [X(t), Y(t)]: X(t):=subs({a=3/2,b=1},xi(t)); Y(t):=subs({a=3/2,b=1},eta(t));

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X(t)[0..Pi]:=subs(S(t)=S(t)[0..Pi],%%); Y(t)[0..Pi]:=subs(S(t)=S(t)[0..Pi],%%);

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X(t)[Pi..2*Pi]:=(3/2)*cos(t)+(3/2)*S(t)[Pi..2*Pi]*sin(t)/ sqrt((9/4)*(sin(t))^2+(cos(t))^2); Y(t)[Pi..2*Pi]:=sin(t)-S(t)[Pi..2*Pi]*cos(t)/ sqrt((9/4)*(sin(t))^2+(cos(t))^2);

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The coordinates [x(t), y(t)] of the approximated involute can be expressed as: x(t):=(3/2)*cos(t)+(3/2)*s(t)*sin(t)/ sqrt((9/4)*(sin(t))^2+(cos(t))^2); y(t):=sin(t)-s(t)*cos(t)/ sqrt((9/4)*(sin(t))^2+(cos(t))^2);

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In the next Figure the involute [X(t), Y(t)] and its approximation [x(t), y(t)] is represented. with(plots,implicitplot): alias(H=Heaviside,th=thickness,c=color): p[1]:=plot([(3/2)*cos(t),sin(t),t=0..2*Pi], scaling=constrained,c=black): p[2]:=plot([X(t)[0..Pi],Y(t)[0..Pi],t=0..Pi],c=black): p[3]:=plot([X(t)[Pi..2*Pi],Y(t)[Pi..2*Pi],t=Pi..2*Pi], c=black): p[4]:=plot([x(t),y(t),t=0..2*Pi],th=3,c=black): p[5]:=plot(1,x=0..1.9832,c=black): p[6]:=plot(-1,x=-5.94954..0,c=black): p[7]:=plot({-7.93272,3.96636},x=-5.94954..1.9832,c=black): p[8]:=plot({-7.93272*H(x+5.94954),3.96636*H(x+5.94954), -7.93272*H(x-1.9832),3.96636*H(x-1.9832)}, x=-5.9496..1.9835,c=black): p[9]:=plot(3.96636*H(x+3/2),x=-3.001/2..-2.999/2,c=black): p[10]:=plot(-7.93272*H(x-3/2),x=2.999/2..3.001/2,c=black): p[11]:=plot([[1.9832,1],[-3/2,3.96636],[-5.94954,-1], [3/2,-7.93272]],style=point,symbol=circle, symbolsize=30,c=black): p[12]:=plot([[1.5178,2.426088],[-5.0826096,1.9741928], [-4.8180232,-4.62623]],style=point,symbol=cross, symbolsize=60,c=black): p[13]:=plots[textplot]({[1.65,1.25,`A`],[-1.25,3.6,`B`], [-5.6,-1.2,`C`],[1.22,-7.5,`D`]},c=black): plots[display]({seq(p[k],k=1..13)});

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In this Figure the involute [X(t), Y(t)] and its approximation [x(t), y(t)] are nearly congruent. The points [A, B, C, D] and ( + ) are points calculated from [X(t), Y(t)]. FOURIER Series The approximation s(t) is based upon the FOURIER analysis as mentioned before. FOURIER_series:= a[0]/2+sum(a[k]*cos(k*x)+b[k]*sin(k*x),k=1..infinity);

NiM+JS9GT1VSSUVSX3Nlcmllc0csJiomIyIiIiIiIyIiIiYlImFHNiMiIiEiIiIiIiItJSRzdW1HNiQsJiomJiUiYUc2IyUia0ciIiItJSRjb3NHNiMqJiUia0ciIiIlInhHIiIiIiIiIiIiKiYmJSJiRzYjJSJrRyIiIi0lJHNpbkc2IyomJSJrRyIiIiUieEciIiIiIiIiIiIvJSJrRzsiIiIlKWluZmluaXR5RyIiIg==

a[k]:=(1/Pi)*Int(f(x)*cos(k*x),x=-Pi..Pi); # k=0,1,2,...

NiM+JiUiYUc2IyUia0csJComKiQlI1BpRyEiIiIiIi0lJEludEc2JComLSUiZkc2IyUieEciIiItJSRjb3NHNiMqJiUia0ciIiIlInhHIiIiIiIiLyUieEc7LCQlI1BpRyEiIiUjUGlHIiIiIiIi

a[0]:=simplify(subs(k=0,%));

NiM+JiUiYUc2IyIiISwkKiYqJCUjUGlHISIiIiIiLSUkSW50RzYkLSUiZkc2IyUieEcvJSJ4RzssJCUjUGlHISIiJSNQaUciIiIiIiI=

b[k]:=(1/Pi)*Int(f(x)*sin(k*x),x=-Pi..Pi); # k=1,2,...

NiM+JiUiYkc2IyUia0csJComKiQlI1BpRyEiIiIiIi0lJEludEc2JComLSUiZkc2IyUieEciIiItJSRzaW5HNiMqJiUia0ciIiIlInhHIiIiIiIiLyUieEc7LCQlI1BpRyEiIiUjUGlHIiIiIiIi

The integrand in S(t) is: w(x):=sqrt((9/4)*(sin(x))^2+(cos(x))^2);

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A[0]:=value(subs(f(x)=w(x),a[0]));

NiM+JiUiQUc2IyIiISwkKigiIiciIiItJSpFbGxpcHRpY0VHNiMsJComIiIkISIiKSIiJiMiIiIiIiMiIiIiIiIiIiIlI1BpRyEiIiIiIg==

A[0]:=evalf(%,6);

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A[k]:=value(subs(f(x)=w(x),a[k]));

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for i in [0,2,4,6] do A[i]:=evalf(value(subs(k=i,A[k])),6) od;

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B[k]:=value(subs(f(x)=w(x),b[k]));

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The FOURIER series represents a Cosinus series, since the integrand in S(t) is an even function: FOURIER_series[k=4]:=A[0]/2+sum(A[2*k]*cos(2*k*x),k=1..2);

NiM+JiUvRk9VUklFUl9zZXJpZXNHNiMvJSJrRyIiJSwoJCIrKytgaTchIioiIiIqJiQiJ1coWyMhIiciIiItJSRjb3NHNiMsJComIiIjIiIiJSJ4RyIiIiIiIiIiIiEiIiomJCInVVA3ISIoIiIiLSUkY29zRzYjLCQqJiIiJSIiIiUieEciIiIiIiIiIiIhIiI=

F(x):=evalf(%,5);

NiM+LSUiRkc2IyUieEcsKCQiJkRFIiEiJSIiIiomJCImdVsjISImIiIiLSUkY29zRzYjLCQqJiQiIiMiIiEiIiIlInhHIiIiIiIiIiIiISIiKiYkIiZ1QiIhIiciIiItJSRjb3NHNiMsJComJCIiJSIiISIiIiUieEciIiIiIiIiIiIhIiI=

for i in [0,Pi/2,Pi] do F(i):=evalf(simplify(subs(x=i,F(x))),7) od;

NiM+LSUiRkc2IyIiISQiKCdRLDUhIic= NiM+LSUiRkc2IywkKiYiIiMhIiIlI1BpRyIiIiIiIiQiKG0pKVwiISIn NiM+LSUiRkc2IyUjUGlHJCIoJ1EsNSEiJw==

The exact values of the integrand are F(0) = F(Pi) = 1 and F(Pi/2) = 3/2. The quality of the approximation F(x) in comparison with the integrand w(x) can be expressed by the L-two-nor L[2]:=sqrt((1/T)*Int((w(tau)-F(tau))^2,tau=0..T));

NiM+JiUiTEc2IyIiIyokKSwkKiYqJCUiVEchIiIiIiItJSRJbnRHNiQqJCksJi0lIndHNiMlJHRhdUciIiItJSJGRzYjJSR0YXVHISIiIiIjIiIiLyUkdGF1RzsiIiElIlRHIiIiIiIiIyIiIiIiIyIiIg==

for i in [Pi/2,Pi] do L_two[0..i]:= evalf(subs(t=i,sqrt((1/i)*int((w(x)-F(x))^2,x=0..t))),5) od;

NiM+JiUmTF90d29HNiM7IiIhLCQqJiIiIyEiIiUjUGlHIiIiIiIiJCImQyEpKSEiKQ== NiM+JiUmTF90d29HNiM7IiIhJSNQaUckIiZCISkpISIp

These values demonstrate the quality of the approximation. Thus, the elementary function s(t) yields an ellipse-involute [x(t), y(t)], which is nearly congruent with the elliptic-integral- solution [X(t), Y(t)] , as discussed before. For a = b = 1 we arrive immediately at the involute of the unit-circle. xi(t)[circle]:=cos(t)+t*sin(t); eta(t)[circle]:=sin(t)-t*cos(t);

NiM+Ji0lI3hpRzYjJSJ0RzYjJSdjaXJjbGVHLCYtJSRjb3NHNiMlInRHIiIiKiYlInRHIiIiLSUkc2luRzYjJSJ0RyIiIiIiIg== NiM+Ji0lJGV0YUc2IyUidEc2IyUnY2lyY2xlRywmLSUkc2luRzYjJSJ0RyIiIiomJSJ0RyIiIi0lJGNvc0c2IyUidEciIiIhIiI=

The comparison with the involute of the ellipse is illustrated in the next Figure: with(plots,implicitplot): alias(H=Heaviside,th=thickness,c=color): p[1]:=plot([cos(t),sin(t),t=0..2*Pi], scaling=constrained,c=black): p[2]:=plot([(3/2)*cos(t),sin(t),t=0..2*Pi],c=black): p[3]:=plot([xi(t)[circle],eta(t)[circle],t=0..2*Pi], th=3,c=black): p[4]:=plot([x(t),y(t),t=0..2*Pi],th=3,c=black): p[5]:=plot(1,x=0..1.9832,c=black): p[6]:=plot(-1,x=-5.94954..0,c=black): p[7]:=plot(Pi*H(x+1),x=-1.001..-0.999,c=black): p[8]:=plot(-2*Pi*H(x-1),x=0.999..1.001,c=black): p[9]:=plot(3.9664*H(x+3/2),x=-3.001/2..-2.999/2,c=black): p[10]:=plot(-7.933*H(x-3/2),x=2.9997/2..3.001/2,c=black): plots[display]({seq(p[k],k=1..10)});

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