# Revolution 2D spline to 3D surface

4 views (last 30 days)
Daan Van Gorp on 4 Oct 2016
Answered: Unai San Miguel on 16 Mar 2017
Hi all,
I would like to revolute a 2D spline that I constructed (to describe a train wheel). How do I perform this revolution so the description in 3D remains a analytical (thus spline) expression?
Thank you

Unai San Miguel on 16 Mar 2017
It depends on what you mean by analytical expression. For example, it is not possible to express a circle and thus a revolution surface with polynomial splines and you should use rational splines if you wanted that.
But usually a polynomial spline with enough degrees of freedom can approximate well a circle.
Let su = spmak(knotsu, coefsu) be your 2D (2-valued, 1-variate) spline and size(coefsu) = [2, Nu]. The first component represents the radial coordinate r and the second the axial z.
Let sv = spmak(knotsv, coefsv) be the 2-valued, 1-variate spline approximation to a circle of radius 1 and centered at the origin of coordinates.
A general revolution surface is of the form [r(u) * cos(v), r(u) * sin(v), z(u)]', with the u=u0=const curves being circles of radius r(u0) at z = z(u0), and the v=v0=const curves rotations of the generatrix at angle v0 along the z-axis.
As sv is an approximation of [cos(v), sin(v)]' you can construct the tensor product spline
s = spmak({knotsu, knotsv}, coefs);
with size(coefs) = [3, su.number, sv.number] and the following definitions
coefs(1,:,:) = su.coefs(1,:)' * sv.coefs(1,:); % r(u) * cos(v)
coefs(2,:,:) = su.coefs(1,:)' * sv.coefs(2,:); % r(u) * sin(v)
coefs(3,:,:) = su.coefs(2,:)' * ones(size(sv.coefs(1,:))); % z(u)
taking advantage of the affine-conservation properties of splines.
I don't use rational splines, but in The NURBS Book by Piegl and Tiller this solution with NURBS is explained.