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BOSL2/nurbs.scad

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add nurbs
2024-11-03 12:10:47 +00:00
/////////////////////////////////////////////////////////////////////
// LibFile: beziers.scad
// B-Splines and Non-uniform Rational B-Splines (NURBS) are a way to represent smooth curves and smoothly curving
// surfaces with a set of control points. The curve or surface is defined by
// the control points and a set of "knot" points. The NURBS can be "clamped" in which case the curve passes through
// the first and last point, or they can be "closed" in which case the first and last point are coincident. Also possible
// are "open" curves which do not necessarily pass through any of their control points. Unlike Bezier curves, a NURBS
// can have an unlimited number of control points and changes to the control points only affect the curve locally.
//
// Includes:
// include <BOSL2/std.scad>
// include <BOSL2/nurbs.scad>
// FileGroup: Advanced Modeling
// FileSummary: NURBS and B-spline curves and surfaces.
//////////////////////////////////////////////////////////////////////
include<BOSL2/std.scad>
include<BOSL2/beziers.scad>
// Section: NURBS Curves
// Function: nurbs_curve()
// Synopsis: Computes one more more points on a NURBS curve.
// SynTags: Path
// Topics: NURBS Curves
// See Also: debug_nurbs()
// Usage:
// pts = nurbs_curve(control, degree, splinesteps, [mult=], [weights=], [type=], [knots=]);
// pts = nurbs_curve(control, degree, u=, [mult=], [weights=], [type=], [knots=]);
// Description:
// Compute the points specified by a NURBS curve. You specify the NURBS by supplying the control points, knots and weights. and knots.
// Only the control points are required. The knots and weights default to uniform, in which case you get a uniform B-spline.
// You can specify endpoint behavior using the `type` parameter. The default, "clamped", gives a curve which starts and
// ends at the first and last control points and moves in the tangent direction to the first and last control point segments.
// If you request an "open" spline you get a curve which starts somewhere in the middle of the control points.
// Finally, a "closed" curve is a one that starts where it ends. Note that each of these types of curve require
// a different number of knots.
// .
// The control points are the most important control over the shape
// of the curve. You must have at least p+1 control points for clamped and open NURBS. Unlike a bezier, there is no maximum
// number of control points. A single NURBS is more like a bezier **path** than like a single bezier spline.
// .
// A NURBS or B-spline is a curve made from a moving average of several Bezier curves. The knots specify when one Bezier fades
// away to be replaced by the next one. At generic points, the curves are differentiable, but by increasing knot multiplicity, you
// can decrease smoothness, or even produce a sharp corner. The knots must be an ascending sequence of values, but repeating values
// is OK and controls the smoothness at the knots. The easiest way to specify the knots is to take the default of uniform knots,
// and simply set the multiplicity to create repeated knots as needed. The total number of knots is then the sum of the multiplicity
// vector. Alternatively you can simply list the knots yourself. Note that regardless of knot values, the domain of evaluation
// for u is always the interval [0,1], and it will be scaled to give the entire valid portion of the curve you have chosen.
// If you give both a knot vector and multiplicity then the multiplicity vector is appled to the provided knots.
// For an open spline the number of knots must be `len(control)+p+1`. For a clamped spline the number of knots is `len(control)-p+1`,
// and for a closed spline you need `len(control)+1` knots. If you are using the default uniform knots then the way to
// ensure that you have the right number is to check that `sum(mult)` is either not set or equal to the correct value.
// .
// You can use this function to evaluate the NURBS at `u`, which can be a single point or a list of points. You can also
// use it to evaluate the NURBS over its entire domain by giving a splinesteps value. This specifies the number of segments
// to use between each knot and guarantees a point exactly at each knot. This may be important if you set the knot multiplicity
// to the degree somewhere in your curve, which creates a corner at the knot, because it guarantees a sharp corner regardless
// of the number of points.
// Arguments:
// control = list of control points in any dimension
// degree = degree of NURBS
// splinesteps = evaluate whole spline with this number of segments between each pair of knots
// ---
// u = list of values or range in the interval [0,1] where the NURBS should be evaluated
// mult = list of multiplicities of the knots. Default: all 1
// weights = vector whose length is the same as control giving weights at each control point. Default: all 1
// type = One of "clamped", "closed" or "open" to define end point handling of the spline. Default: "clamped"
// knots = List of knot values. Default: uniform
// Example(2D,NoAxes): Simple quadratic uniform clamped b-spline with some points computed using splinesteps.
// pts = [[13,43],[30,52],[49,22],[24,3]];
// debug_nurbs(pts,2);
// npts = nurbs_curve(pts, 2, splinesteps=3);
// color("red")move_copies(npts) circle(r=1);
// Example(2D,NoAxes): Simple quadratic uniform clamped b-spline with some points computed using the u parameter. Note that a uniform u parameter doesn't necessarily sample the curve uniformly.
// pts = [[13,43],[30,52],[49,22],[24,3]];
// debug_nurbs(pts,2);
// npts = nurbs_curve(pts, 2, u=[0:.2:1]);
// color("red")move_copies(npts) circle(r=1);
// Example(2D,NoAxes): Same control points, but cubic
// pts = [[13,43],[30,52],[49,22],[24,3]];
// debug_nurbs(pts,3);
// Example(2D,NoAxes): Same control points, quadratic and closed
// pts = [[13,43],[30,52],[49,22],[24,3]];
// debug_nurbs(pts,2,type="closed");
// Example(2D,NoAxes): Same control points, cubic and closed
// pts = [[13,43],[30,52],[49,22],[24,3]];
// debug_nurbs(pts,3,type="closed");
// Example(2D,NoAxes): Ten control points, quadratic, clamped
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// debug_nurbs(pts,2);
// Example(2D,NoAxes): Same thing, degree 4
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// debug_nurbs(pts,4);
// Example(2D,NoAxes): Same control points, degree 2, open. Note it doesn't reach the ends
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// debug_nurbs(pts,2, type="open");
// Example(2D,NoAxes): Same control points, degree 4, open. Note it starts farther from the ends
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// debug_nurbs(pts,4,type="open");
// Example(2D,NoAxes): Same control points, degree 2, closed
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// debug_nurbs(pts,2,type="closed");
// Example(2D,NoAxes): Same control points, degree 4, closed
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// debug_nurbs(pts,4,type="closed");
// Example(2D,NoAxes): Adding weights
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// weights = [1,1,1,3,1,1,3,1,1,1];
// debug_nurbs(pts,4,type="clamped",weights=weights);
// Example(2D,NoAxes): Using knot multiplicity with quadratic clamped case. Knot count is len(control)-degree+1 = 9. The multiplicity 2 knot creates a corner for a quadratic.
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// mult = [1,1,1,2,1,1,1,1];
// debug_nurbs(pts,2,mult=mult,show_knots=true);
// Example(2D,NoAxes): Using knot multiplicity with quadratic clamped case. Two knots of multiplicity 2 gives two corners
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// mult = [1,1,1,2,2,1,1];
// debug_nurbs(pts,2,mult=mult,show_knots=true);
// Example(2D,NoAxes): Using knot multiplicity with cubic clamped case. Knot count is now 8. We need multiplicity equal to degree (3) to create a corner.
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// mult = [1,3,1,1,1,1];
// debug_nurbs(pts,3,mult=mult,show_knots=true);
// Example(2D,NoAxes): Using knot multiplicity with cubic closed case. Knot count is now len(control)+1=11. Here are three corners.
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// mult = [1,3,1,3,3];
// debug_nurbs(pts,3,mult=mult,type="closed",show_knots=true);
// Example(2D,NoAxes): Explicitly specified knots only change the quadratic clamped curve slightly. Knot count is len(control)-degree+1 = 9.
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// knots = [0,1,3,5,9,13,14,19,21];
// debug_nurbs(pts,2);
// Example(2D,NoAxes): Combining explicit knots with mult for the quadratic curve to add a corner
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// knots = [0,1,3,9,13,14,19,21];
// mult = [1,1,1,2,1,1,1,1];
// debug_nurbs(pts,2,knots=knots,mult=mult);
// Example(2D,NoAxes): Directly repeating a knot in the knot list to create a corner for a cubic spline
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// knots = [0,1,3,13,13,13,19,21];
// debug_nurbs(pts,3,knots=knots);
// Example(2D,NoAxes): Open cubic spline with explicit knots
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// knots = [0,1,3,13,13,13,19,21,27,28,29,40,42,44];
// debug_nurbs(pts,3,knots=knots,type="open");
// Example(2D,NoAxes): Closed quintic spline with explicit knots
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// knots = [0,1,3,13,13,13,19,21,27,28,33];
// debug_nurbs(pts,5,knots=knots,type="closed");
// Example(2D,NoAxes): Closed quintic spline with explicit knots and weights
// pts = [[5,0],[0,20],[33,43],[37,88],[60,62],[44,22],[77,44],[79,22],[44,3],[22,7]];
// weights = [1,2,3,4,5,6,7,6,5,4];
// knots = [0,1,3,13,13,13,19,21,27,28,33];
// debug_nurbs(pts,5,knots=knots,weights=weights,type="closed");
// Example(2D,NoAxes): Circular arcs are possible with NURBS. This example gives a semi-circle
// control = [[1,0],[1,2],[-1,2],[-1,0]];
// w = [1,1/3,1/3,1];
// debug_nurbs(control, 3, weights=w, width=0.1, size=.2);
// Example(2D,NoAxes): Gluing two semi-circles together gives a whole circle. Note that this is a clamped not closed NURBS. The interface uses a knot of multiplicity 3 where the clamped ends of the semi-circles meet.
// control = [[1,0],[1,2],[-1,2],[-1,0],[-1,-2],[1,-2],[1,0]];
// w = [1,1/3,1/3,1,1/3,1/3,1];
// debug_nurbs(control, 3, splinesteps=16,weights=w,mult=[1,3,1],width=.1,size=.2);
// Example(2D,NoAxes): Circle constructed with type="closed"
// control = [[1,0],[1,2],[-1,2],[-1,0],[-1,-2],[1,-2]];
// w = [1,1/3,1/3,1,1/3,1/3];
// debug_nurbs(control, 3, splinesteps=16,weights=w,mult=[1,3,3],width=.1,size=.2,type="closed",show_knots=true);
function nurbs_curve(control,degree,splinesteps,u, mult,weights,type="clamped",knots) =
assert(num_defined([splinesteps,u])==1, "Must define exactly one of u and splinesteps")
is_finite(u) ? nurbs_curve(control,degree,[u],mult,weights,type=type)[0]
: assert(is_undef(splinesteps) || (is_int(splinesteps) || splinesteps>0), "splinesteps must be a positive integer")
let(u=is_range(u) ? list(u) : u,f=echo(u=u))
assert(is_undef(u) || (is_vector(u) && min(u)>=0 && max(u)<=1), "u must be a list of points on the interval [0,1] or a range contained in that interval")
is_def(weights) ? assert(is_vector(weights, len(control)), "Weights should be a vector whose length is the number of control points")
let(
dim = len(control[0]),
control = [for(i=idx(control)) [each control[i]*weights[i],weights[i]]],
curve = nurbs_curve(control,degree,u=u,splinesteps=splinesteps, mult=mult,type=type)
)
[for(pt=curve) select(pt,0,-2)/last(pt)]
:
let(
uniform = is_undef(knots),
dum=assert(in_list(type, ["closed","open","clamped"]), str("Unknown nurbs spline type", type))
assert(type=="closed" || len(control)>=degree+1, str(type," nurbs requires at least degree+1 control points"))
assert(is_undef(mult) || is_vector(mult), "mult must be a vector"),
badmult = is_undef(mult) ? []
: [for(i=idx(mult)) if (!(
is_int(mult[i])
&& mult[i]>0
&& (mult[i]<=degree
|| (type!="closed"
&& mult[i]==degree+1
&& (i==0 || i==len(mult)-1)
)
)
)) i],
dummy0 = assert(badmult==[], str("mult vector should contain positive integers no larger than the degree, except at ends of open splines, ",
"where degree+1 is allowed. The mult vector has bad values at indices: ",badmult))
assert(is_undef(knots) || is_undef(mult) || len(mult)==len(knots), "If both mult and knots are given they must be vectors of the same length")
assert(is_undef(mult) || type!="clamped" || sum(mult)==len(control)-degree+1,
str("For ",type," spline knot count (sum of multiplicity vector) must be ",len(control)-degree+1," but is instead ",mult?sum(mult):0))
assert(is_undef(mult) || type!="closed" || sum(mult)==len(control)+1,
str("For closed spline knot count (sum of multiplicity vector) must be ",len(control)+1," but is instead ",mult?sum(mult):0))
assert(is_undef(mult) || type!="open" || sum(mult)==len(control)+degree+1,
str("For closed spline knot count (sum of multiplicity vector) must be ",len(control)+degree+1," but is instead ",mult?sum(mult):0)),
control = type=="open" ? control
: type=="clamped" ? control //concat(repeat(control[0], degree),control, repeat(last(control),degree))
: /*type=="closed"*/ concat(control, select(control,count(degree))),
// n = len(control)-1, // number of control pts -1
// m = n+p+1, // number of knots - 1
mult = !uniform ? mult
: type=="clamped" ? assert(is_undef(mult) || mult[0]==1 && last(mult)==1,"For clamped b-splines, first and last multiplicity must be 1")
[degree+1,each slice(default(mult, repeat(1,len(control)-degree+1)),1,-2),degree+1]
: is_undef(mult) ? repeat(1,len(control)+degree+1)
: type=="open" ? mult
: /* type=="closed" */
let( // Closed spline requires that we identify first and last knots and then step at same
// interval spacing periodically through the knot vector. This means we pick up the first
// multiplicity minus 1 and have to add it to the last multiplicity.
lastmult = last(mult)+mult[0]-1,
dummy=assert(lastmult<=degree, "For closed spline, first and last knot multiplicity cannot total more than the degree+1"),
adjlast = [
each select(mult,0,-2),
lastmult
]
)
_extend_knot_mult(adjlast,1,len(control)+degree+1),
knot = uniform && is_undef(mult) ? lerpn(0,1,len(control)+degree+1)
: uniform ? [for(i=idx(mult)) each repeat(i/(len(mult)-1),mult[i])]
: let(
xknots = is_undef(mult)? knots
: assert(len(mult) == len(knots), "If knot vector and mult vector must be the same length")
[for(i=idx(mult)) each repeat(knots[i], mult[i])]
)
type=="open" ? assert(len(xknots)==len(control)+degree+1, str("For open spline, knot vector with multiplicity must have length ",
len(control)+degree+1," but has length ", len(xknots)))
xknots
: type=="clamped" ? assert(len(xknots) == len(control)+1-degree, str("For clamped spline, knot vector with multiplicity must have length ",
len(control)+1-degree," but has length ", len(xknots)))
assert(xknots[0]!=xknots[1] && last(xknots)!=select(xknots,-2),
"For clamped splint, first and last knots cannot repeat (must have multiplicity one")
concat(repeat(xknots[0],degree), xknots, repeat(last(xknots),degree))
: /*type=="closed"*/ assert(len(xknots) == len(control)+1-degree, str("For closed spline, knot vector (including multiplicity) must have length ",
len(control)+1-degree," but has length ", len(xknots),control))
let(gmult=_calc_mult(xknots))
assert(gmult[0]+last(gmult)<=degree+1, "For closed spline, first and last knot multiplicity together cannot total more than the degree+1")
_extend_knot_vector(xknots,0,len(control)+degree+1),
bound = type=="clamped" ? undef
: [knot[degree], knot[len(control)]],
adjusted_u = !is_undef(splinesteps) ?
[for(i=[degree:1:len(control)-1])
each
if (knot[i]!=knot[i+1])
lerpn(knot[i],knot[i+1],splinesteps, endpoint=false),
if (type!="closed") knot[len(control)]
]
: is_undef(bound) ? u
: add_scalar((bound[1]-bound[0])*u,bound[0])
)
uniform?
let(
msum = cumsum(mult),
)
[for(uval=adjusted_u)
let(
mind = floor(uval*(len(mult)-1)),
knotidxR=msum[mind]-1,
knotidx = knotidxR<len(control) ? knotidxR : knotidxR - mult[mind]
)
bspline_pt_recurse(knot,select(control,knotidx-degree,knotidx),uval,1,degree,0,degree,knotidx)
]
: let(
kmult = _calc_mult(knot),
knotidx =
[for(
kind = kmult[0]-1,
uind=0,
kmultind=1,
output=undef,
done=false
;
!done
;
output = (uind<len(adjusted_u) && approx(adjusted_u[uind],knot[kind]) && ((kmultind>=len(kmult)-1 || kind+kmult[kmultind]>=len(control)))) ? kind-kmult[kmultind-1]
: (uind<len(adjusted_u) && adjusted_u[uind]>=knot[kind] && adjusted_u[uind]>=knot[kind] && adjusted_u[uind]<knot[kind+kmult[kmultind]]) ? kind
: undef,
done = uind==len(adjusted_u),
uind = is_def(output) ? uind+1 : uind,
inc_k = uind<len(adjusted_u) && adjusted_u[uind]>=knot[kind+kmult[kmultind]],
kind = inc_k ? kind+kmult[kmultind] : kind,
kmultind = inc_k ? kmultind+1 : kmultind
)
if (is_def(output)) output]
)
[for(i=idx(adjusted_u))
bspline_pt_recurse(knot,select(control, knotidx[i]-degree,knotidx[i]), adjusted_u[i], 1, degree, 0, degree, knotidx[i])
];
function bspline_pt_recurse(knot, control, u, r, h,s,p,k) =
r>h ? control[0]
:
let(
ctrl_new = [for(i=[k-p+r:1:k-s])
let(
alpha = (u-knot[i]) / (knot[i+p-r+1]-knot[i])
)
(1-alpha) * control[i-1-(k-p)-r+1] + alpha*control[i-(k-p)-r+1]
]
)
bspline_pt_recurse(knot,ctrl_new,u,r+1,h,s,p,k);
function _extend_knot_mult(mult, next, len) =
let(total = sum(mult))
total == len ? mult
: total>len ? [ each select(mult,0,-2), last(mult)-(total-len) ]
: _extend_knot_mult([each mult,mult[next]], next+1, len);
function _extend_knot_vector(knots,next,len) =
len(knots)==len ? knots
: _extend_knot_vector([each knots, last(knots)+knots[next+1]-knots[next]], next+1, len);
function _calc_mult(knots) =
let(
ind=[ 0,
for(i=[1:len(knots)-1])
if (knots[i]!=knots[i-1]) i,
len(knots)
]
)
deltas(ind);
// Module: debug_nurbs()
// Synopsis: Shows a NURBS curve and its control points, knots and weights
// SynTags: Geom
// Topics: NURBS, Debugging
// See Also: nurbs_curve()
// Usage:
// debug_nurbs(control, degree, [width], [splinesteps=], [type=], [mult=], [knots=], [size=], [show_weights=], [show_knots=], [show_idx=]);
// Description:
// Displays a 2D or 3D NURBS and the associated control points to help debug NURBS curves. You can display the
// control point indices and weights, and can also display the knot points.
// Arguments:
// control = control points for NURBS
// degree = degree of NURBS
// splinesteps = number of segments between each pair of knots. Default: 16
// width = width of the line. Default: 1
// size = size of text annotations. Default: 3 times the width
// mult = multiplicity vector for NURBS
// weights = weight vector for NURBS
// type = NURBS type, one of "clamped", "open" or "closed". Default: "clamped"
// show_index = if true then display index of each control point vertex. Default: true
// show_weights = if true then display any non-unity weights. Default: true if weights vector is supplied, false otherwise
// show_knots = If true then show the knots on the spline curve. Default: false
module debug_nurbs(control,degree,splinesteps=16,width=1, size, mult,weights,type="clamped",knots, show_weights, show_knots=false, show_index=true)
{
$fn=8;
size = default(size, 3*width);
show_weights = default(show_weights, is_def(weights));
N=len(control);
twodim = len(control[0])==2;
curve = nurbs_curve(control=control,degree=degree,splinesteps=splinesteps, mult=mult,weights=weights, type=type, knots=knots);
stroke(curve, width=width, closed=type=="closed");//, color="green");
stroke(control, width=width/2, color="lightblue", closed=type=="closed");
if (show_knots){
knotpts = nurbs_curve(control=control, degree=degree, splinesteps=1, mult=mult, weights=weights, type=type, knots=knots);
color("green")
move_copies(knotpts)
if (twodim)circle(r=width);
else sphere(r=width);
}
color("blue")
if (show_index)
move_copies(control){
let(label = str($idx),
anch = show_weights && is_def(weights[$idx]) && weights[$idx]!=1 ? FWD : CENTER)
if (twodim) text(text=label, size=size, anchor=anch);
else rot($vpr) text3d(text=label, size=size, anchor=anch);
}
color("blue")
if ( show_weights)
move_copies(control){
if(is_def(weights[$idx]) && weights[$idx]!=1)
let(label = str("w=",weights[$idx]),
anch = show_index ? BACK : CENTER
)
if (twodim) fwd(size/2*0)text(text=label, size=size, anchor=anch);
else rot($vpr) text3d(text=label, size=size, anchor=anch);
}
}
// Section: NURBS Surfaces
// Function: is_nurbs_patch()
// Synopsis: Returns true if the given item looks like a NURBS patch.
// Topics: NURBS Patches, Type Checking
// Usage:
// bool = is_nurbs_patch(x);
// Description:
// Returns true if the given item looks like a NURBS patch. (a 2D array of 3D points.)
// Arguments:
// x = The value to check the type of.
function is_nurbs_patch(x) =
is_list(x) && is_list(x[0]) && is_vector(x[0][0]) && len(x[0]) == len(x[len(x)-1]);
// Function: nurbs_patch_points()
// Synopsis: Computes specifies point(s) on a NURBS surface patch
// Topics: NURBS Patches
// See Also: nurbs_vnf(), nurbs_curve()
// Usage:
// pointgrid = nurbs_patch_points(patch, degree, [splinesteps], [u=], [v=], [weights=], [type=], [mult=], [knots=]);
// Description:
// Sample a NURBS patch on a point set. If you give splinesteps then it will sampled uniformly in the spline
// parameter between the knots, ensuring that a sample appears at every knot. If you instead give u and v then
// the values at those points in parameter space will be returned. The various NURBS parameters can all be
// single values, if the NURBS has the same parameters in both directions, or pairs listing the value for the
// two directions.
// Arguments:
// patch = rectangular list of control points in any dimension
// degree = a scalar or 2-vector giving the degree of the NURBS in the two directions
// splinesteps = a scalar or 2-vector giving the number of segments between each knot in the two directions
// ---
// u = evaluation points in the u direction of the patch
// v = evaluation points in the v direction of the patch
// mult = a single list or pair of lists giving the knot multiplicity in the two directions. Default: all 1
// knots = a single list of pair of lists giving the knot vector in each of the two directions. Default: uniform
// weights = a single list or pair of lists giving the weight at each control point in the patch. Default: all 1
// type = a single string or pair of strings giving the NURBS type, where each entry is one of "clamped", "open" or "closed". Default: "clamped"
// Example(NoScale): Computing points on a patch using ranges
// patch = [
// [[-50, 50, 0], [-16, 50, 20], [ 16, 50, 20], [50, 50, 0]],
// [[-50, 16, 20], [-16, 16, 40], [ 16, 16, 40], [50, 16, 20]],
// [[-50,-16, 20], [-16,-16, 40], [ 16,-16, 40], [50,-16, 20]],
// [[-50,-50, 0], [-16,-50, 20], [ 16,-50, 20], [50,-50, 0]],
// ];
// pts = nurbs_patch_points(patch, 3, u=[0:.1:1], v=[0:.3:1]);
// move_copies(flatten(pts)) sphere(r=2,$fn=16);
// Example(NoScale): Computing points using splinesteps
// patch = [
// [[-50, 50, 0], [-16, 50, 20], [ 16, 50, 20], [50, 50, 0]],
// [[-50, 16, 20], [-16, 16, 40], [ 16, 16, 40], [50, 16, 20]],
// [[-50,-16, 20], [-16,-16, 40], [ 16,-16, 40], [50,-16, 20]],
// [[-50,-50, 0], [-16,-50, 20], [ 16,-50, 20], [50,-50, 0]],
// ];
// pts = nurbs_patch_points(patch, 3, splinesteps=5);
// move_copies(flatten(pts)) sphere(r=2,$fn=16);
function nurbs_patch_points(patch, degree, splinesteps, u, v, weights, type=["clamped","clamped"], mult=[undef,undef], knots=[undef,undef]) =
assert(is_undef(splinesteps) || !any_defined([u,v]), "Cannot combine splinesteps with u and v")
is_def(weights) ?
assert(is_matrix(weights,len(patch),len(patch[0])), "The weights parameter must be a matrix that matches the size of the patch array")
let(
patch = [for(i=idx(patch)) [for (j=idx(patch[0])) [each patch[i][j]*weights[i][j], weights[i][j]]]],
pts = nurbs_patch_points(patch=patch, degree=degree, splinesteps=splinesteps, u=u, v=v, type=type, mult=mult, knots=knots)
)
[for(row=pts) [for (pt=row) select(pt,0,-2)/last(pt)]]
:
assert(is_undef(u) || is_range(u) || is_vector(u) || is_finite(u), "Input u is invalid")
assert(is_undef(v) || is_range(v) || is_vector(v) || is_finite(v), "Input v is invalid")
assert(num_defined([u,v])!=1, "Must define both u and v (when using)")
let(
u=is_range(u) ? list(u) : u,
v=is_range(v) ? list(v) : v,
degree = force_list(degree,2),
type = force_list(type,2),
splinesteps = is_undef(splinesteps) ? [undef,undef] : force_list(splinesteps,2),
mult = is_vector(mult) || is_undef(mult) ? [mult,mult]
: assert((is_undef(mult[0]) || is_vector(mult[0])) && (is_undef(mult[1]) || is_vector(mult[1])), "mult must be a vector or list of two vectors")
mult,
knots = is_vector(knots) || is_undef(knots) ? [knots,knots]
: assert((is_undef(knots[0]) || is_vector(knots[0])) && (is_undef(knots[1]) || is_vector(knots[1])), "knots must be a vector or list of two vectors")
knots,
)
is_num(u) && is_num(v)? nurbs_curve([for (control=patch) nurbs_curve(control, degree[1], u=v, type=type[1], mult=mult[1], knots=knots[1])],
degree[0], u=u, type=type[0], mult=mult[0], knots=knots[0])
: is_num(u) ? nurbs_patch_points(patch, degree, u=[u], v=v, knots=knots, mult=mult, type=type)[0]
: is_num(v) ? column(nurbs_patch_points(patch, degree, u=u, v=[v], knots=knots, mult=mult, type=type),0)
:
let(
vsplines = [for (i = idx(patch[0])) nurbs_curve(column(patch,i), degree[0], splinesteps=splinesteps[0],u=u, type=type[0],mult=mult[0],knots=knots[0])]
)
[for (i = idx(vsplines[0])) nurbs_curve(column(vsplines,i), degree[1], splinesteps=splinesteps[1], u=v, mult=mult[1], knots=knots[1], type=type[1])];
// Function: nurbs_vnf()
// Synopsis: Generates a (possibly non-manifold) VNF for a single NURBS surface patch.
// SynTags: VNF
// Topics: NURBS Patches
// See Also: nurbs_patch_points()
// Usage:
// vnf = nurbs_vnf(patch, degree, [splinesteps], [mult=], [knots=], [weights=], [type=], [style=]);
// Description:
// Compute a (possibly non-manifold) VNF for a NURBS. The input patch must be an array of control points. If weights is given it
// must be an array of weights that matches the size of the control points. The style parameter
// gives the {{vnf_vertex_array()}} style to use. The other parameters may specify the NURBS parameters in the two directions
// by giving a single value, which applies to both directions, or a list of two values to specify different values in each direction.
// You can specify undef for for a direction to keep the default, such as `mult=[undef,v_multiplicity]`.
// Arguments:
// patch = rectangular list of control points in any dimension
// degree = a scalar or 2-vector giving the degree of the NURBS in the two directions
// splinesteps = a scalar or 2-vector giving the number of segments between each knot in the two directions
// ---
// mult = a single list or pair of lists giving the knot multiplicity in the two directions. Default: all 1
// knots = a single list of pair of lists giving the knot vector in each of the two directions. Default: uniform
// weights = a single list or pair of lists giving the weight at each control point in the. Default: all 1
// type = a single string or pair of strings giving the NURBS type, where each entry is one of "clamped", "open" or "closed". Default: "clamped"
// style = {{vnf_vertex_array ()}} style to use for triangulating the surface. Default: "default"
// Example: Quadratic B-spline surface
// patch = [
// [[-50, 50, 0], [-16, 50, 20], [ 16, 50, 20], [50, 50, 0]],
// [[-50, 16, 20], [-16, 16, 40], [ 16, 16, 40], [50, 16, 20]],
// [[-50,-16, 20], [-16,-16, 40], [ 16,-16, 40], [50,-16, 20]],
// [[-50,-50, 0], [-16,-50, 20], [ 16,-50, 20], [50,-50, 0]],
// ];
// vnf = nurbs_vnf(patch, 2);
// vnf_polyhedron(vnf);
// Example: Cubic B-spline surface
// patch = [
// [[-50, 50, 0], [-16, 50, 20], [ 16, 50, 20], [50, 50, 0]],
// [[-50, 16, 20], [-16, 16, 40], [ 16, 16, 40], [50, 16, 20]],
// [[-50,-16, 20], [-16,-16, 40], [ 16,-16, 40], [50,-16, 20]],
// [[-50,-50, 0], [-16,-50, 20], [ 16,-50, 20], [50,-50, 0]],
// ];
// vnf = nurbs_vnf(patch, 3);
// vnf_polyhedron(vnf);
// Example: Cubic B-spline surface, closed in one direction
// patch = [
// [[-50, 50, 0], [-16, 50, 20], [ 16, 50, 20], [50, 50, 0]],
// [[-50, 16, 20], [-16, 16, 40], [ 16, 16, 40], [50, 16, 20]],
// [[-50,-16, 20], [-16,-16, 40], [ 16,-16, 40], [50,-16, 20]],
// [[-50,-50, 0], [-16,-50, 20], [ 16,-50, 20], [50,-50, 0]],
// ];
// vnf = nurbs_vnf(patch, 3, type=["closed","clamped"]);
// vnf_polyhedron(vnf);
// Example: B-spline surface cubic in one direction, quadratic in the other
// patch = [
// [[-50, 50, 0], [-16, 50, 20], [ 16, 50, 20], [50, 50, 0]],
// [[-50, 16, 20], [-16, 16, 40], [ 16, 16, 40], [50, 16, 20]],
// [[-50,-16, 20], [-16,-16, 40], [ 16,-16, 40], [50,-16, 20]],
// [[-50,-50, 0], [-16,-50, 20], [ 16,-50, 20], [50,-50, 0]],
// ];
// vnf = nurbs_vnf(patch, [3,2],type=["closed","clamped"]);
// vnf_polyhedron(vnf);
// Example: The sphere can be represented using NURBS
// patch = [
// [[0,0,1], [0,0,1], [0,0,1], [0,0,1], [0,0,1], [0,0,1], [0,0,1]],
// [[2,0,1], [2,4,1], [-2,4,1], [-2,0,1], [-2,-4,1], [2,-4,1], [2,0,1]],
// [[2,0,-1],[2,4,-1],[-2,4,-1],[-2,0,-1],[-2,-4,-1], [2,-4,-1],[2,0,-1]],
// [[0,0,-1],[0,0,-1],[0,0,-1], [0,0,-1], [0,0,-1], [0,0,-1], [0,0,-1]]
// ];
// weights = [
// [9,3,3,9,3,3,9],
// [3,1,1,3,1,1,3],
// [3,1,1,3,1,1,3],
// [9,3,3,9,3,3,9],
// ]/9;
// vknots = [0, 1/2, 1/2, 1/2, 1];
// vnf = nurbs_vnf(patch, 3,weights=weights, knots=[undef,vknots]);
// vnf_polyhedron(vnf);
function nurbs_vnf(patch, degree, splinesteps=16, weights, type="clamped", mult, knots, style="default") =
assert(is_nurbs_patch(patch),"Input patch is not a rectangular aray of points")
let(
pts = nurbs_patch_points(patch=patch, degree=degree, splinesteps=splinesteps, type=type, mult=mult, knots=knots, weights=weights),
)
vnf_vertex_array(pts, style=style, row_wrap=type[0]=="closed", col_wrap=type[1]=="closed");
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