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### Title: Computing offsets of NURBS curves and surfaces

### The end

Les A. Piegl

University of South Florida

Wayne Tiller

Geom Ware, Inc

L.H. 2005/10

Abstract of algorithm

- 1. Recognition of special curves and

surfaces;

- 2. Sampling the offset curve or surface

based on bounds on second derivatives;

- 3. Interpolating these points;
- 4. Removing all unwanted knots using the

offset tolerance.

Attractive features

- 1. The error control is based on rigorous

theory;

- 2. The number of points sampled depends

on the curvature and the tolerance;

- 3. The parametrization can be selected ;
- 4. The degree of the offset can be any

number from one to the number of

the sampling points

Shape recognition

- 1. Decompose the NURBS entity into

Bezier pieces;

- 2. Sample each Bezier piece uniformly

at (p+1) points for non-rationals,

and 2(p+1) points for rationals;

- 3. At these points compute geometric

entities such as curvatures and unit

normals;

- 4. See if these entities are within a

certain tolerance.

Example 1

Line: decompose the line into piecewise

Bezier and for each piece at 2(p+1)

sampling points

- 1. Check if
- 2. Check if the projection of Pi onto

line P0Pn lies between the points P0

and Pn

*Graphical Models and Image Processing

1997;59(6):417-433.

Example 2

Circle: decompose the curve into piecewise

Bezier and for each piece at 2(p+1)

sampling points

- 1. Compute the centres (which is the

centre of osculating circle);

- 2. Compute the unit normals to the place

of the curve;

- 3. See if the centres as well as the unit

normals do not deviate more than the

tolerance.

(a) Quadratic circle offset by a

Quadratic circle; (b) quartic

Circle offset by a quartic circle;

And (c) quintic circle offset by

A quadratic circle.

Computing offset of NURBS curves -1

Free-form offsets are approximated by

The following procedure:

- 1. The offset curve is sampled based on the second derivatives;
- 2. The sample points are interpolated

by a non-rational continuous B-

spline curve of given degree p. A

recommended degree is three;

Sampling of offset curve; (b)

- Interpolating sampling points;
- (c) Offset approximation after
- Knot removal.

Estimation of the number of points

To be sampled depends on the second

Derivatives. For parametric curves the

Number of points for equally spaced

Parameters on [0, 1] is computed as:

M is a bound on the second derivative

And is a user defined tolerance.

*CAGD 1986;3(4):295-311.

We did the following -1

- 1. The bound on the second

derivative is computed on each Bezier piece.

- 2. A new formula was introduced to match the growth rate of the control points:

Linear approximation

otherwise

We did the following -2

- 3. The default n=p+1 was introduced.

This ensures that small curve segments are sampled properly even

if the tolerance is large;

How to compute M:

*CAGD 1990;7:83-100

Knot removal

Once M is available, the curve is sampled

Along each Bezier piece, the points are

Offset and interpolated by a non-rational

B-spline curve of a chosen degree, and

The resulting interpolatory curve is

Knot-removed.

The details of knot removal are found in:

*P&T the NURBS book, 2nd ed. Springer

1.Number of control points in comparison with other methods

2.Number of control points

Computing offsets of NURBS surfaces

Most offsetting methods published for

Curves do not extend to surfaces. Our

Method is a exception. In fact, its

Extension is straightforward.

First, special surfaces are recognized and

Offset precisely. Figs.5 and 6 show offsets of toroidal and conical patches,

Respectively.

Fig.5. Offset of a toroidal patch

Fig.6. Offset of a cone patch

Surface of revolution;

- Offset of surface of revolution;
- And (c)control points of surface
- In Fig. (b)

The process of free-form offsetting

Is the same as that for curves, that is:

- 1. Sample the offset surface;
- 2. Interpolate the sampling points;
- 3. Remove all removable knots.

Surface and sampling of its

- Offset;
- (b)Interpolating sampling points
- (c)Offset approximation after
- Knot removal.

Estimation of the number of points -1

As for curves, we have to estimate the

Number of points for each Bezier patch.

The formula for surfaces is:

M1, M2, M3 are bounds on the second

Partial derivatives of the offset surface,

and

*CAGD 1986;3(4):295-311.

Estimation of the number of points -2

Using the same argument as in the case

Of curves, the new formula becomes:

where,

*The detail we can see:

*CAGD 1986;3(1):15-43.

Linear approximation in either direction

Otherwise

Generalized offsets

Generalized, or functional, offsets are defined as:

d(u), d(u,v) are distance functions, V(u),

V(u,v) are direction curves/surfaces.

Two slight modifications :

- 1. The derivatives different;
- 2. For sampling, have to evaluate the

distance function and direction C./S.

Example

- Serving dish;
- Soup dish obtained by
- functional offsetting;
- (c)Dog dish obtained by
- functional offsetting.

Conclusions

this paper presented algorithms for

approximating offsets of NURBS C./S.

The methods rely on point sampling based on derivatives, interpolation and

knot removal.

Attractive feature lies in the choice of

parametrization, low degree, and ease

of generalization to function offsetting.

L.H. 2005/10

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