Manipulating NURBS objects Non-uniform rational B-spline

1 manipulating nurbs objects

1.1 knot insertion
1.2 knot removal
1.3 degree elevation
1.4 curvature

manipulating nurbs objects

motoryacht design.

a number of transformations can applied nurbs object. instance, if curve defined using degree , n control points, same curve can expressed using same degree , n+1 control points. in process number of control points change position , knot inserted in knot vector. these manipulations used extensively during interactive design. when adding control point, shape of curve should stay same, forming starting point further adjustments. number of these operations discussed below.

knot insertion

as term suggests, knot insertion inserts knot knot vector. if degree of curve



n


{\displaystyle n}

,



n
−
1


{\displaystyle n-1}

control points replaced



n


{\displaystyle n}

new ones. shape of curve stays same.

a knot can inserted multiple times, maximum multiplicity of knot. referred knot refinement , can achieved algorithm more efficient repeated knot insertion.

knot removal

knot removal reverse of knot insertion. purpose remove knots , associated control points in order more compact representation. obviously, not possible while retaining exact shape of curve. in practice, tolerance in accuracy used determine whether knot can removed. process used clean after interactive session in control points may have been added manually, or after importing curve different representation, straightforward conversion process leads redundant control points.

degree elevation

a nurbs curve of particular degree can represented nurbs curve of higher degree. used when combining separate nurbs curves, e.g., when creating nurbs surface interpolating between set of nurbs curves or when unifying adjacent curves. in process, different curves should brought same degree, maximum degree of set of curves. process known degree elevation.

curvature

the important property in differential geometry curvature



κ


{\displaystyle \kappa }

. describes local properties (edges, corners, etc.) , relations between first , second derivative, , thus, precise curve shape. having determined derivatives easy compute



κ
=




|


r
′

(
t
)
×

r
″

(
t
)

|




|


r
′

(
t
)


|


3







{\displaystyle \kappa ={\frac {|r (t)\times r (t)|}{|r (t)|^{3}}}}

or approximated arclength second derivate



κ
=

|


r
″

(

s

o


)

|



{\displaystyle \kappa =|r (s_{o})|}

. direct computation of curvature



κ


{\displaystyle \kappa }

these equations big advantage of parameterized curves against polygonal representations.