Plotting the Tangent Line

I am trying to plot the tangent line to a curve given by the parametric equations x = sin(t), y = cos(t), z = t

I have no idea where to start, but here is the code I have for the equations.

t=linspace(0,2*pi,50);
x=cos(t); y=sin(t); z=t;
plot3(x,y,z, 'r', 'LineWidth', 2);
view([2 2 2]); grid on

Answer 1

[Tushar Sinha]

Hi Jay,

In your case, the parameterized helix curve is given by the equations:

t=linspace(0,2*pi,50);
x=cos(t); y=sin(t); z=t;

The tangent direction < u,v,w > at point < x,y,z > can be calculated as the gradient of the above parameterized equations:

 u = -sin(t)
 v = cos(t)
 w = 1

Now, any tangent line on this curve passing through a point < x,y,z > will be in direction of < u,v,w >. This modifies the above equations as follows:

u = x – y
v = y + x
w = z + 1

In order to draw a line passing through a given point < x,y,z > and in the gradient direction, Let s = linspace(-0.75, 0.75, 50).

The following code will draw the tangents at every 10th point on the helix curve:

function tangentCurve
t=linspace(0,2*pi,50);
x=cos(t); y=sin(t); z=t;
figure;
plot3(x,y,z,'r');hold on;
view([2 2 2]); grid on
s = linspace(-0.75, 0.75, 50);
for i = 1:10:length(x)
    u=@(s)x(i)-y(i)*s;
    v=@(s)y(i)+x(i)*s;
    w=@(s)z(i)+1*s;
    plot3(u(s),v(s),w(s));
end
end

I hope this helps resolve the issue.

Thanks,

Tushar

[Elioth Daniel]

Thank you, This helped me very much!