Very Computer

could you please help me?

i'm at the stage of writing a surface interrogation program where everything
is in place and it should all work.

but, for some reason my unit normal calculator is giving me incorrect
results.

I think the problem lies in my derivatives of the B-Spline basis function
(i'm using NURBS).

could anyone point me in the direction of a correct, working version of the
b-spline derivative?

thanks
STU

p.s. just to make sure i'm not being stupid:
to calculat the normal at a point on a NURBS surface, do the cross product
of the two tangent vectors.
to get these tangent vectors (one for u and one for v(the paameters)) you
apply the same NURBS formula as to find a point but replace either the u or
v basis functions with its derivative.

correct????

Quote:> i'm at the stage of writing a surface
> interrogation program where everything
> is in place and it should all work.

> but, for some reason my unit normal
> calculator is giving me incorrect
> results.

> I think the problem lies in my derivatives
> of the B-Spline basis function (i'm using
> NURBS).

> could anyone point me in the direction of
> a correct, working version of the
> b-spline derivative?

Quote:> p.s. just to make sure i'm not being stupid:
> to calculat the normal at a point on a NURBS
> surface, do the cross product of the two
> tangent vectors.

Quote:> to get these tangent vectors (one for u and
> one for v(the paameters)) you apply the same
> NURBS formula as to find a point but replace
> either the u or v basis functions with its
> derivative.

--
Dave Eberly

http://www.magic-software.com
http://www.wild-magic.com

sorry, where exactly are they??
i can't seen to find what i'm looking for

thanks
STU

Quote:> sorry, where exactly are they??
> i can't seen to find what i'm looking for

There are links to files.  The files
MgcBSplineBasis.{h,inl,cpp} implement the
B-spline basis functions and their derivatives.

The files MgcBSplineRectangle.{h,inl,cpp} have
an implementation of a B-spline rectangle patch
including member functions to compute position
and derivatives.

The files MgcNURBSRectangle.{h,inl,cpp} have
an implementation of a NURBS rectangle patch
including member functions to compute position
and derivatives.

--
Dave Eberly

http://www.magic-software.com
http://www.wild-magic.com

yes i've got those, but i don't understand the code!!

i was looking for the actual equation.

thanks

STU

Quote:> yes i've got those, but i don't understand
> the code!!

> i was looking for the actual equation.

I implemented this code by reading Dave Roger's
book "An Introduction to NURBS:  with Historical
Perspective".  The key observation is that rather
than apply the recursive B-spline basis function
formula from the top-down, you work from the
bottom-up (Roger's "dependency" picture for the
bottom-up is not quite right).  By doing this
you gain some efficiency in the calculations.
The function BSplineBasis::Compute(...) is what
implements this.

I wish I could say that the implementation is a
simple coding of an "equation", but in fact that
is not the case.  Getting it implemented and
getting it right do take a bit of effort.

--
Dave Eberly

http://www.magic-software.com
http://www.wild-magic.com

surly the equation is what SHOULD be implemented?!?!?

for example, the B-Spline basis function is defined by:

N_i_p(t)    =    N_i_p-1(t)    *    (t - T_i)  /  (T_i+p - T_i)

     +    N_i+1_p-1(t)    *    (T_i+p+1  - t)  /  (T_i+p+1  -  T_i+1)

where T is the knot vector, p is the degree and t is the parameter

is the derivative basis function not just the derivative of this? (ignoring
computational efficiency!)

thanks again
STU

Quote:> surly the equation is what SHOULD be implemented?!?!?

> for example, the B-Spline basis function is defined by:

> N_i_p(t)    =    N_i_p-1(t)    *    (t - T_i)  /  (T_i+p - T_i)

>      +    N_i+1_p-1(t)    *    (T_i+p+1  - t)  /  (T_i+p+1  -  T_i+1)

> where T is the knot vector, p is the degree and t is the parameter

> is the derivative basis function not just the derivative of this?
(ignoring
> computational efficiency!)

--
Dave Eberly

http://www.magic-software.com
http://www.wild-magic.com

don't you just calculate the 4 coordinates (x, y, z & w) then devide the
first three by the fourth?

do this for partial derivetive in u and v then take the cross product?

thanks
STU

Let's do a first quadrant circular arc, with classic p = [x/w,y/w]:

  x(t) = 1 - t^2
  y(t) = 2t
  w(t) = 1 + t^2

Without the weights, w, no problem. The quadratic Bezier curve with
control points Q0, Q1, Q2 has derivative with linear Bezier difference
control points L0 = 2(Q1-Q0) and L1 = 2(Q2-Q1).

  Q0 = [1,0]    Q1 = [1,1]    Q2 = [0,2]
  L0 = [0,2]    L1 = [-2,2]

Our w control values are 1, 1, and 2; so our rational Bezier curve has
control points

  C0 = [1,0;1]    C1 = [1,1;1]    C2 = [0,1;2]

We *cannot* simply difference these to get derivative controls! For,
the horizontal values and vertical values are given by

  h = (1-t^2) / (1+t^2)
  v = 2t / (1+t^2)

with derivatives

  h' = -4t / (1+2t^2+t^4)
  v' = (2-2t^2) / (1+2t^2+t^4)

Clearly we will need a quartic rational Bezier curve just to handle
the weights.

Does this fit your idea? If not, you need to get the right formulae;
otherwise, it's probably just a coding typo.

Quote:> what do you mean by taking the derivative of ratios?

> don't you just calculate the 4 coordinates (x, y, z & w) then devide the
> first three by the fourth?

> do this for partial derivetive in u and v then take the cross product?

If you have computed the B-spline (x,y,z,w), then
the 3D surface is X = x/w, Y = y/w, and Z = z/w.
The partial derivative of X with respect to u is
  dX/du = (w*dx/du - x*dw/du)/w^2
Similar for the other two components.

--
Dave Eberly

http://www.magic-software.com
http://www.wild-magic.com

i think that might solve the problem.  just a slight oversight!!!

thanks again
STU