Tangent line of a parametric curve

How do I program MATLAB to determine the equation of a line tangent to a parametric curve?

I see plenty of videos/threads on plotting parametrics, but I havent found any on using MATLAB to perform the deriving of a tangent equation.

Answer 1

[Star Strider]

The function you have and the result you want are not obvious.

Try this:

t = linspace(0, 2*pi);                                      % Create Curve
y = sin(t);                                                 % Create Curve

Point = randi(numel(t));                                    % Random Point On Curve

dydt = gradient(y) ./ gradient(t);                          % Derivative
b = y(Point) - dydt(Point)*t(Point);                        % Intercept
idxv = max(1,Point-10):min(numel(t),Point+10);              % Index Vector
Tan = dydt(Point)*t(idxv) + b;

figure
plot(t, y)
hold on
plot(t(idxv),Tan, '-r')
hold off
grid

[Christian Zimmerman]

That seems to work. However, I am looking for MATLAB to spit out the equation, not just plot the tangent.

x=sqrt(t) , y=t^2-2*t , where t=4

[Star Strider]

Revised:

t = linspace(0, 9);                                                     % Create Curve
x = @(t) sqrt(t);                                                       % Desired Function
y = @(t) t.^2-2*t;                                                      % Desired Function

tv = 4;                                                                 % Desired Prameter Value
xv = x(tv);
yv = y(tv);
dfx = @(f,x) (f(x+1E-6)-f(x))/1E-6;                                     % Derivative Function
m = dfx(y, tv) / dfx(x, tv);                                            % Slope
b = y(tv) - m*x(tv);                                                    % Intercept
TanEqn = sprintf('y = %.4f x %+.4f', m, b)

Tan = m*x(tv+[-1 1]) + b;
figure
plot(x(t), y(t))
hold on
plot(x(tv+[-1 1]), Tan, '-r')
hold off
grid

and:

TanEqn =
    'y = 24.0000 x -40.0000'

I changed it to include the functions (and a simple derivative calculation replacing the gradient calls). The plot demonstrates that it works and calculates the appropriate tangent line.