Planar Curve Offset Based on Circle Approximation Curve Offset Planar Curve Offset Based on Circle Approximation Lee Kim Elber Concept Circle Approximation Offset Approximation ID: 571234
Download Presentation The PPT/PDF document "Curve Offset" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
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.