How to plot 3D path data as a ribbon in Matlab

The ribbon function can only accept 2D inputs because it uses the 3rd dimension to 'build' the ribbon.

One way to achieve a 3D ribbon is to build series of patch or surface between each point and orient them properly so they look continuous.

The following code will build a ribbon around any arbitrary 3D path defined by an (x,y,z) vector. I will not explain each line of the code but there are plenty of comments and I stopped for intermediate visualisations so you can understand how it is constructed.

%% Input data
a=1; c=1; t=0:.1:100;
x = (a*t/2*pi*c).*sin(t);
y = (a*t/2*pi*c).*cos(t);
z = t/(2*pi*c);

nPts = numel(x) ;

%% display 3D path only
figure;
h.line = plot3(x,y,z,'k','linewidth',2,'Marker','none');
hold on
xlabel('X')
ylabel('Y')
zlabel('Z')

%% Define options
width = ones(size(x)) * .4 ;

% define surface and patch display options (FaceAlpha etc ...), for later
surfoptions  = {'FaceAlpha',0.8 , 'EdgeColor','k' , 'EdgeAlpha',0.8 , 'DiffuseStrength',1 , 'AmbientStrength',1 } ;

%% get the gradient at each point of the curve
Gx = diff([x,x(1)]).' ;
Gy = diff([y,y(1)]).' ;
Gz = diff([z,z(1)]).' ;
% get the middle gradient between 2 segments (optional, just for better rendering if low number of points)
G = [ (Gx+circshift(Gx,1))./2 (Gy+circshift(Gy,1))./2 (Gz+circshift(Gz,1))./2] ;

%% get the angles (azimuth, elevation) of each plane normal to the curve

ux = [1 0 0] ;
uy = [0 1 0] ;
uz = [0 0 1] ;

for k = nPts:-1:1 % running the loop in reverse does automatic preallocation
    a = G(k,:) ./ norm(G(k,:)) ;
    angx(k) =  atan2( norm(cross(a,ux)) , dot(a,ux))  ;
    angy(k) =  atan2( norm(cross(a,uy)) , dot(a,uy))  ;
    angz(k) =  atan2( norm(cross(a,uz)) , dot(a,uz))  ;
    
    [az(k),el(k)] = cart2sph( a(1) , a(2) , a(3) ) ;
end
% compensate for poor choice of initial cross section plane
az = az + pi/2 ; 
el = pi/2 - el ;


%% define basic ribbon element
npRib = 2 ;
xd = [ 0 0] ;
yd = [-1 1] ;
zd = [ 0 0] ;

%% Generate coordinates for each cross section

cRibX = zeros( nPts , npRib ) ;
cRibY = zeros( nPts , npRib ) ;
cRibZ = zeros( nPts , npRib ) ;
cRibC = zeros( nPts , npRib ) ;

for ip = 1:nPts
    % cross section coordinates.
    csTemp = [ ( width(ip) .* xd )  ; ... %// X coordinates
               ( width(ip) .* yd )  ; ... %// Y coordinates
                               zd    ] ;   %// Z coordinates
    
    %// rotate the cross section (around X axis, around origin)
    elev = el(ip) ;
    Rmat = [ 1     0           0     ; ...
             0 cos(elev)  -sin(elev) ; ...
             0 sin(elev)   cos(elev) ] ;
    csTemp = Rmat * csTemp ;
    
    %// do the same again to orient the azimuth (around Z axis)
    azi = az(ip) ;
    Rmat = [ cos(azi)  -sin(azi) 0 ; ...
             sin(azi)   cos(azi) 0 ; ...
               0            0    1 ] ;
    csTemp = Rmat * csTemp ;
    
    %// translate each cross section where it should be and store in global coordinate vector
    cRibX(ip,:) = csTemp(1,:) + x(ip) ;
    cRibY(ip,:) = csTemp(2,:) + y(ip) ;
    cRibZ(ip,:) = csTemp(3,:) + z(ip) ;
end

%% Display the full ribbon
hd.cyl = surf( cRibX , cRibY , cRibZ , cRibC ) ;
set( hd.cyl , surfoptions{:} )

Now you have your graphic object contained in one surface object, you can set the options for the final rendering. For example (only an example, explore the surface object properties to find all te possibilities).

%% Final render
h.line.Visible = 'off' ;
surfoptionsfinal  = {'FaceAlpha',0.8 , 'EdgeColor','none' , 'DiffuseStrength',1 , 'AmbientStrength',1 } ;
set( hd.cyl , surfoptionsfinal{:} )
axis off

Note that this code is an adaptation (simplification) of the code provided in this answer (to that question: Matlab: “X-Ray” plot line through patch).

This method allows to draw an arbitrary cross section (a disc in the answer) and build a surface which will follow a path. For your question I replaced the disc cross section by a simple line. You could also replace it with any arbitrary cross section (a disc, a square, a potatoid ... the sky is the limit).

Edit

Alternative Method:

Well it turns out there is a Matlab function which can do that. I first discarded it because it is meant for 3D volume visualisations, and most ways to call it require gridded input (meshgrid style). Luckily for us, there is also a calling syntax which can work with your data.

% Same input data
a=1; c=1; t=0:.1:100;
x = (a*t/2*pi*c).*sin(t);
y = (a*t/2*pi*c).*cos(t);
z = t/(2*pi*c);

% Define vertices (and place in cell array)
verts = {[x.',y.',z.']};
% Define "twistangle". We do not need to twist it in that direction but the
% function needs this input so filling it with '0'
twistangle = {zeros(size(x.'))} ;
% call 'streamribbon', the 3rd argument is the width of the ribbon.
hs = streamribbon(verts,tw,0.4) ;

% improve rendering
view(25,9)
axis off
shading interp;
camlight
lighting gouraud

Will render the following figure:

For additional graphic control (over the edges of the ribbon), you can refer to this question and my answer: MATLAB streamribbon edge color