Intersection of Parametric Curves Using vpasolve

I'm trying to find the intersections of two parametric curves. I thought it was solved until i zoomed in to see a significant error.

I'm not sure what's going on.

Any help would be appreciated. Thank you.

Problem: Graph the cycloid

and the trochoid

together on the interval [0, 4π]. Find the coordinates of the four points of intersection. (Hint: Solve the equation r(t) = s(u). Note the different independent variables for r and s—the points of intersection need not correspond to the same “time” on each curve. Also, since the coordinate functions are transcendental, you may need to use vpasolve rather than solve.) Use MATLAB to mark the four points on your graph.

From Multivariable Calculus with MATLAB: With Applications to Geometry and Physics 1st ed. 2017 (Lispman, Rosenberg). This is Problem 2.21. Page 29.

Define curves

syms ind t u;
rx(ind)=2*(ind)-2*sin(ind);
ry(ind)=2*(1-cos(ind));
sx(ind)=2*(ind)-sin(ind);
sy(ind)=2-cos(ind);

figure;hold on;xlim([0 25]);
fplot(rx(t),ry(t),[0 4*pi],'b');
fplot(sx(u),sy(u),[0 4*pi],'r');
xlabel('r_x(t) ~ s_x(u)');
ylabel('r_y(t) ~ s_x(u)');

From inspection (see see first plot below), intersections ocurr where r(t) is approximately {1, 12 13 24}. The x component of r(t) is approximately double t. So approximations for t are t=1/2{1,12,13,24}. Also note that and remain approximately close in value as t and u vary.

guesses=[1, 12 13 24]/2; 

Setup the system to solve: (eq)

eq=[rx(t) ry(t)]-[sx(u) sy(u)]==0;

For each guess, solve eq for t and u (using vpasolve).

for guess=guesses
    solution=vpasolve(eq,[t u],guess);

Plot the intersection point Pn. Plot r(t) with a large blue dot and s(u) as a small yellow dot.

    plot(rx(solution.t),ry(solution.t),'b.','markersize',20); % plot R(t) at t_x = Pn_x
    plot(sx(solution.u),sy(solution.u),'y.','markersize',5); % plot S(u) at u_x = Pn_x

Print intersection values of t and u at Pn.

    fprintf('R(t) and S(u) intersect at P%d when',find(guesses==guess));    
    tn=double(solution.t)
    un=double(solution.u)
end

R(t) and S(u) intersect at P1 when

tn = 1.0918

un = 0.3983

R(t) and S(u) intersect at P2 when

tn = 5.1914

un = 5.8849

R(t) and S(u) intersect at P3 when

tn = 7.3750

un = 6.6814

R(t) and S(u) intersect at P4 when

tn = 11.4745

un = 12.1681

hold off;

Zoom-In to see the error

figure; hold on;
fplot(rx(t),ry(t),[0 4*pi],'b');
fplot(sx(u),sy(u),[0 4*pi],'r');
for guess=guesses
    solution=vpasolve(eq,[t u],guess);
    plot(rx(solution.t),ry(solution.t),'k.','markersize',20); % plot R(t) at Pn
    plot(sx(solution.u),sy(solution.u),'g.','markersize',15); % plot S(u) at Pn
end
xlabel('r_x(t) ~ s_x(u)');
ylabel('r_y(t) ~ s_x(u)');
xlim([12.15603 12.15897])
ylim([1.078214 1.078684])

[Simon Chan]

Just wondering why the marker colors are not black and green on the plot for the following code?

plot(rx(solution.t),ry(solution.t),'k.','markersize',20); % plot R(t) at Pn
plot(sx(solution.u),sy(solution.u),'g.','markersize',15); % plot S(u) at Pn

[Simon Le Serve]

hey Simon, thanks you're right!

Not sure though, must have not executed on last edit.

you have a sharp eye!

Answer 1

[John D'Errico]

Your probem is when you zoomed in. The mathematics seems fine.

>> tn
tn =
          11.4745470305524

>> un
un =
          12.1681063167797
          
>> vpa(rx(tn))
ans =
24.724031702281097750999349300293

>> vpa(sx(un))
ans =
24.72403170228109712971250329995

>> vpa(ry(tn))
ans =
1.0782644793999382387376598972632

>> vpa(sy(un))
ans =
1.0782644793999394359320615865557

The MATLAB zoom is not very accurate, not when you zoom in that deeply. It does not re-evaluate those curves, and then give you a hyper-accurate picture. Instead, it gives you a literally microscopic view of the pixels that were plotted.

The numbers are correct. Just that the zoom has confused you.

[Simon Le Serve]

thanks John,

hmm .. definitely feeling confused. If this is true it's a bit concerning. I set the x and y view limits, before drawing the error-plot, i would have thought that would handle precision better, no?

Do you know any way of producing a plot of the intersection with higher precission?

also, thanks a lot for verifying the math!

[John D'Errico]

No. You are mistaken. You set the limits AFTER you created the plot. The very last thing you did was:

xlim([12.15603 12.15897])
ylim([1.078214 1.078684])

Let me be more careful in how the plots are created.

tn
tn =
          11.4745470305524
          
un
un =
          12.1681063167797
          
fplot(rx(t),ry(t),[11.474 11.475],'b');
hold on
fplot(sx(u),sy(u),[12.168 12.169],'r');
plot(rx(solution.t),ry(solution.t),'k.','markersize',20); % plot R(t) at Pn
plot(sx(solution.u),sy(solution.u),'g.','markersize',15); % plot S(u) at Pn

The intersection of the curves is now seen to be accurately placed.

[Simon Le Serve]

I understand now! :)

thanks John