Curve Offset - PowerPoint Presentation

Slide1

Curve Offset

Planar Curve Offset

Based on Circle

ApproximationSlide2

Curve Offset

Planar Curve

Offset Based

on Circle

Approximation – Lee, Kim,

Elber

.

Concept

Circle Approximation

Offset Approximation

Eliminating Self Intersecting Loops

Results

Comparison of offset approximation methodsSlide3

Concept

Given a planar regular parametric curve

with normal

the offset curve is

.

The offset curve is generally not rational, and cannot be described as a rational B-spline.

 Slide4

Concept

The article suggests an approximation

, calculated as the envelope of a convolution of the original curve and an approximated circle

.

This is achieved by adding to each point on the curve a specific point on the approximated circle of radius

: is a reparameterization that keeps and at the same direction.This entails that is normal to , and therefore an approximation of . Slide5

Circle Approximation

A quadratic Bezier curve is given by:

+

Assuming

,

because of symmetry:

 Slide6

Circle Approximation

An alternative measure for the error, instead of

:

Requiring

for

extremal error, there are five solutions:

 Slide7

Method 1: Tangent to the circle at both ends

If each quadratic Bezier curve is tangent to the circle at its endpoints, the whole piecewise curve is of

continuity.

This means that the middle control point of

should be

, and therefore .The resulting error has extremal values at the endpoints, minimal with value 0. The maximum error is in the middle at :

 Slide8

Method 2: Uniform scaling of Method 1

The error of

is always positive, and it is outside of the unit circle. It is possible to minimize the maximal error by uniform scaling of the curve by some constant

:

The choice of

affects the error function:The

extrema

are at the same values of

, so by setting

the minimal

is achieved.

 Slide9

Method 2: Uniform scaling of Method 1

the value of

is determined:

The size of the error

is slightly less than half of

:

Since the scaling factor

depends on

, the piecewise curve will only remain continuous (and preserve

continuity) if all

segments share the same

.

 Slide10

Method 3: Interpolating three circle points

If the continuity restriction is lifted, The middle point can be positioned at the mid-point of the arc:

The error at

is now

, and the maximum error is

 Slide11

Method 4: Interpolation with

e

qui

-oscillating Error

Requiring the

same magnitude for maximal and minimal error: This determines the value of and the magnitude:

=

 Slide12

Method 5: Uniform scaling of Method 3

The error

of

is always positive, and it is inside the unit circle. It is possible to minimize the maximal error by uniform scaling of the curve by some constant

:

The choice of affects the error function:

Again, the

extrema

are at the same values of

, so by setting

the minimal

is achieved.

 Slide13

Method 5: Uniform scaling of Method 3

the value of

is determined:

The size of the error

is slightly more than half of

: Since the scaling factor depends on , the piecewise curve will only remain continuous if all

segments share the same

.

 Slide14

Circle ApproximationSlide15

Offset Approximation

The purpose of defining the approximated arc segments was providing a quadratic equation to be re-parameterized in order to add it to the original curve

.

Different segments are relevant at different values of

, so the adding is done separately for each continuous part of

that has a single corresponding approximated arc. Slide16

Hodograph

Definitions:

is a planar regular parametric curve.

The hodograph curve

is the locus of

.The tangential angular map of is  Slide17

Hodograph

Lemma 1:

Let

be the hodograph of

.

If the tangential angular map of is one-to-one, any ray from the origin intersects with at no more than one point.Proof:If intersects with

at two different

points

and

,

, then

and

have the same ratio as and , implying

=

 Slide18

Hodograph

Lemma 2:

is the hodograph of

is the hodograph of

If

and

are intersection points of a ray

starting from the origin, Then

and

have the same tangent direction at

and

. Slide19

Hodograph

Proof:

The direction of

is the direction of

vetors

and , therefore and have the same direction. Slide20

Approximated Offset Curve

is one-to-one.

is the ray from origin through

By the first lemma, intersects with at the point , This defines a mapping from to . By the second lemma,

and

are in the same direction, and therefore the curve

is indeed the well-defined convolution curve needed.

 Slide21

Approximated Offset Curve

is quadratic, and it’s hodograph curve is linear:

By demanding the same direction for

and

:

 Slide22

Approximated Offset Curve

+

If

is a polynomial of degree

:

is a rational polynomial of degree

is a rational polynomial of degree

is a rational curve of degree

If

is a rational polynomial of degree : is a rational polynomial of degree

is a rational polynomial of degree

is a rational curve of degree

 Slide23

Subdivision of

Until now it was assumed that

,

and

was an approximation of an arc from angle 0 to .Several can be connected to approximate a whole circle. is subdivided into , where each part satisfies

 Slide24

Subdivision of

Hodograph of

:

Hodograph of

: Offset curve: Slide25

Subdivision of

If the circle was approximated in methods 3, 4 or 5, the hodograph is not continuous. This can be solved by adding a zero-radius arcs between the intervals, which become arcs in the hodograph.

The discontinuity

of

tangent angles at the Endpoints of segmentsof Q(s): Slide26

Subdivision of

The error of the approximation is determined by the choice of circle approximation method and by the choice of

.

 Slide27

Eliminating self intersecting loops

Original curve

Sampled points

Offset of sampled points

Offset only of segments with curvature

Intersections by Plane SweepValid offset curve Slide28

resultsSlide29

resultsSlide30

resultsSlide31

Comparisons

Control-polygon based methods

:

Cobb – translation of control-points in normal direction

.

Tiller and Hanson – translation of control segmentsCoquillart – translation of control points using closest normal to curveElber and Cohen – error minimization of new control pointsSlide32

Comparisons

Interpolation methods

:

Hoschek

– least squares of errors, parallel endpoints.Slide33

Comparisons

Quadratic PolynomialSlide34

Comparisons

Cubic PolynomialSlide35

Comparisons

Cubic PolynomialSlide36

Comparisons

Quadratic PolynomialSlide37

Comparisons

Results

:

Least Square Error performs well on general curves.

Tiller and Hanson is

very good for quadratic curves.The best-performing geometrical method for general curve is circle approximation.