My ellipse wont close after running the code .

Question

Panda05 2023-12-2

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2055359-my-ellipse-wont-close-after-running-the-code

评论： Panda05 2023-12-3

Hy guys . In this task , i have to implement periodic boundary conditons to my ellipse . However , the boundary conditions are not being implemented and i dont know what mistake i'm making if you comment them, the whole code still runs and you can see that the ellipse wont close at the end because the boundary conditions are not working at all. Any suggestions and solutions on how to fix this are highly appreciated.

Below is the code and it has to be saved in 3 files to run it since i use 2 functions and the 3rd file i call the 2 functions.

%---------------------------------Example (i use to call both functions)--------------------------------------

% Example with an ellipse

theta = linspace(0, 2 * pi, 8);

t_x = cos(theta);

t_y = sin(theta);

% Interpolate x-coordinate

[b_x, c_x, d_x] = periodic_cubic_spline_interpolation(theta, t_x);

% Interpolate y-coordinate

[b_y, c_y, d_y] = periodic_cubic_spline_interpolation(theta, t_y);

% Evaluate the spline at points

t_eval = linspace(0, 2 * pi, 500);

x_eval = arrayfun(@(t) evaluate_cubic_spline(theta, t_x, b_x, c_x, d_x, t), t_eval);

y_eval = arrayfun(@(t) evaluate_cubic_spline(theta, t_y, b_y, c_y, d_y, t), t_eval);

% Plot the results

figure;

plot(t_x, t_y, 'ko', 'LineWidth', 1.5);

hold on;

grid on;

plot(x_eval, y_eval, 'r-', 'LineWidth', 2);

title('Cubic Spline Interpolation of Ellipse');

legend('True Ellipse', 'Cubic Spline');

xlabel('X-axis');

ylabel('Y-axis');

%---------------------------------------function 1--------------------------------------------------------

function [b, c, d] = periodic_cubic_spline_interpolation(t, x)

n = length(t);

h = diff(t);

alpha = zeros(n, 1);

beta = zeros(n, 1);

for i = 2:n - 1

alpha(i) = 3 * (x(i + 1) - x(i)) / h(i) - 3 * (x(i) - x(i - 1)) / h(i - 1);

beta(i) = 2 * (h(i - 1) + h(i));

end

% Periodic boundary conditions------%(these are not being implemented!!!)%--------------------------------

alpha(1) = 3 * (x(2) - x(1)) / h(1) - 3 * (x(1) - x(n)) / h(n - 1);

alpha(n) = 3 * (x(1) - x(n)) / h(n - 1) - 3 * (x(n) - x(n - 1)) / h(n - 2);

beta(1) = 2 * (h(n - 1) + h(1));

beta(n) = 2 * (h(n - 1) + h(n - 2));

l = zeros(n, 1);

mu = zeros(n, 1);

z = zeros(n, 1);

l(1) = 1;

mu(1) = 0;

z(1) = 0;

for i = 2:n - 1

l(i) = 2 * (t(i + 1) - t(i - 1)) - h(i - 1) * mu(i - 1);

mu(i) = h(i) / l(i);

z(i) = (alpha(i) - h(i - 1) * z(i - 1)) / l(i);

end

l(n) = 1;

z(n) = 0;

c = zeros(n, 1);

b = zeros(n, 1);

d = zeros(n, 1);

for j = n - 1:-1:1

c(j) = z(j) - mu(j) * c(j + 1);

b(j) = (x(j + 1) - x(j)) / h(j) - h(j) * (c(j + 1) + 2 * c(j)) / 3;

d(j) = (c(j + 1) - c(j)) / (3 * h(j));

end

%------------------------------------function 2---------------------------------------------------------

function spline_eval = evaluate_cubic_spline(t, x, b, c, d, t_eval)

n = length(t);

j = max(min(find(t >= t_eval, 1) - 1, n - 2), 1);

tj = t(j);

spline_eval = x(j) + b(j) * (t_eval - tj) + c(j) * (t_eval - tj) ^ 2 + d(j) * (t_eval - tj) ^ 3;

end

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Answer 1

Matt J 2023-12-2

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2055359-my-ellipse-wont-close-after-running-the-code#answer_1364459

编辑：Matt J 2023-12-2

在 MATLAB Online 中打开

Can't you just use the native spline() command?

theta = linspace(0, 2 * pi, 8);

t_x = 2*cos(theta);

t_y = sin(theta);

dt_x=0; %end slope x

dt_y=(t_y(2)-t_y(1))./(theta(2)-theta(1)); %end slope y

theta_eval = linspace(0, 2 * pi, 500);

t_eval=periodicSpline(theta,[t_x;t_y], [dt_x; dt_y], theta_eval);

plot(t_x, t_y, 'ko', 'LineWidth', 1.5);

hold on;

grid on;

plot(t_eval(1,:), t_eval(2,:), 'r-', 'LineWidth', 2); axis equal; axis padded

title('Cubic Spline Interpolation of Ellipse');

legend('True Ellipse', 'Cubic Spline');

xlabel('X-axis');

ylabel('Y-axis');

hold off

function out = periodicSpline(tdata,xydata,xyslope, tquery)

xydata(:,end)=xydata(:,1);

xydata=[xyslope, xydata,xyslope];

out=spline(tdata,xydata,tquery);

end

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Torsten 2023-12-3

编辑：Torsten 2023-12-3

在 MATLAB Online 中打开

@MAKEERA

Make the ansatz

p_j(x) = a_j + b_j*(x-x_j) + c_j*(x-x_j)^2 + d_j*(x-x_j)^3 (j=1,...,N-1)

where N is the number of x-data points.

So in total you have 4*(N-1) unknowns a_j, b_j, c_j and d_j that are to be determined from the 4*(N-1) conditions

p_j(x_j) = y_j (j=1,...,N-1)
p_(N-1)(x_N) = p_1(x_1)
p_j(x_j) = p_(j-1)(x_j) (j=2,...N-1)
p_(N-1)'(x_N) = p_1'(x_1)
p_j'(x_j) = p_(j-1)'(x_j)  (j=2,...,N-1)
p_(N-1)''(x_N) = p_1''(x_1)
p_j''(x_j) = p_(j-1)''(x_j) (j=2,...,N-1)