How to find tangent vector or velocity vector and unit tangent vector of Space curve (3D)?

I have the curvature of a curve, start point P1(x1,y1,z1) and end point P2(x2,y2,z2), radius of curvature, arc length, and a cord length of a curve. Now I want to find the tangent or velocity vector and unit tangent vector of this curve. I am developing a code for continuum robot dynamics.

The orange colour shows the curvature created by spline and while blue line shows the cord or vector from Pi-1 to Pi.

           %Kappas,thetas, phis%
%%--------------------------------------------------------------
        %kappa -- magnitudes (beta and gamma) of segments 
k1 = kappa_vec(1,1);
k2 = kappa_vec(1,2);
k3 = kappa_vec(1,3);

      %theta -- the subsegment bending angle ?i
th1 = theta_vec(1,1);
th2 = theta_vec(1,2);
th3 = theta_vec(1,3);

      %phi -- the bending plane angle ?i
phi1 = phi_vec(1,1);
phi2 = phi_vec(1,2);
phi3 = phi_vec(1,3);

    %epsilon - the subsegment twist angle ?i
e1 = epsilon_list(1,1);
e2 = epsilon_list(1,2);
e3 = epsilon_list(1,3);



%%--------------------------------------------------------------

           %Rotation matrices%
%%--------------------------------------------------------------
    R_1 = rot(phi1, th2, e1);
    R_2 = R_1* rot(phi2, th2, e2);
    R_3 = R_2* rot(phi3, th3, e3);

%%--------------------------------------------------------------



      % POsition vectors%
%%--------------------------------------------------------------
if k1 == 0
    p1 = [0;0;L0];
else
    p1 = p_lcl(k1,th1,phi1);
end
%%-------------------------------------
p_vec = [B p1 p2 p3 p4 p5 p6 p7 p8 p9 p10]; % B is base position, and rest are position eacg disk center.
p_vec(isinf(p_vec)|isnan(p_vec)) = 0; %to remove infinity and NaN

%%-------------------------------------

Answer 1

[Aditya Srikar]

Hi Hussain

Let the position vector be P(t)

Let P′(t) = d[P(t)] / dt

The tangent vector of curve is P′(t)

The unit tangent vector of curve is T(t) = P′(t) / |P′(t)|

Where |P′(t)| represents magnitude of vector P′(t)

Hope it helps !