A Method of Curve - PowerPoint Presentation

Slide1

A Method of Curve Implicitization

Based on:

Curve

Implicitization

Using Moving Lines,

Sederberg

et al

Presented by:

Boris van

SosinSlide2

Implicitization U

sing Moving Lines

Some motivation

Recollection of Projective GeometryPencils on Lines to ConicsGeneralization for Bézier CurvesRemarks on The Generalization for Surfaces

2Slide3

Some Motivation

Why

Implicitization

?3Slide4

Parametric vs Implicit

Parametric

Easy to generate points

For curve/surface

:

choose

and evaluate

.

Hard to determine if point is on curve/surface

Requires solving

for .

Implicit

Easy to determine if point is on curve/surfaceFor curve :Point is on curve iff .Hard to generate points on curve/surfaceRequires solving

4Slide5

Some Motivation

Why

Implicitization

?Easy to determine if point is on curveEasy to solve curve-curve and curve-surface intersectionFor parametric:

and implicit:

Plug parametric into implicit:

Solve for

(analytic, Newton iteration, etc.)

5Slide6

Recollection of Projective Geometry

Points

(or

) in Projective coordinates map to points

in Cartesian coordinates

Cartesian to Projective:

Projective to Cartesian:

6Slide7

Recollection of Projective Geometry

From Perspective Projection (optics):

Pinhole Camera Model

Image Plane

Virtual Image Plane

Focal Point

7Slide8

Recollection of Projective Geometry

Properties:

Points:

, not all zeros

Lines:

, not all zeros

Line goes through point:

(where

is the dot product)

Coordinates

and

are the same point

Same for lines

 8Slide9

Recollection of Projective Geometry

Properties:

Intersection point of two lines:

Cross product

Line that goes through two points:

9Slide10

Recollection of Projective Geometry

Properties:

Line-Point Duality

Equation of line

going through point

:

If

are constants then it’s a constraint on

If

are constants then it’s a constraint on

10Slide11

Pencils of Lines to Conics

Definition:

A Pencil of two

lines

is the collection of all lines:

Intersecting case:

Parallel case:

11Slide12

Pencils of Lines to Conics

Theorem: Given two distinct pencils of lines

and:

,

the locus of all their intersection points

is a conic section

12Slide13

Pencils of Lines to Conics

Theorem: Given two distinct pencils of lines

and

, the locus of all their intersection points is a conic section

Attributed to Jacob Steiner, circa 1830

Can be traced back to Newton

Steiner’s proof was based on projective geometry

We’ll present an algebraic proof

13Slide14

Pencils of Lines to Conics

Theorem: Given two distinct pencils of lines

and

, the locus of all their intersection points is a conic section

Proof outline:

Introduce condition on

Eliminate parameter

14Slide15

Pencils of Lines to Conics

Point

is on the intersection of

and

if:

15Slide16

Pencils of Lines to Conics

16Slide17

Pencils of Lines to Conics

(check later case where

)

17Slide18

Pencils of Lines to Conics

18Slide19

Pencils of Lines to Conics

Substituting

:

19Slide20

Pencils of Lines to Conics

Substituting

and

yields:

Which is the implicit equation of a conic section

▪

20Slide21

Pencils of Lines to Conics

Observation:

The same equation system:

Can be expressed in matrix notation:

And the implicit form is:

21Slide22

Generalization for Bézier

Curves

The Dual of

a Bézier CurveA curve can be regarded as Moving Point (as parameter changes)Bézier

Curve:

Due to point-line duality,

we can talk about Moving Lines:

They have “control lines”…

22Slide23

Generalization for Bézier Curves

Intersection of Moving Lines:

Moving lines

of degrees

intersect at a degree

(*generally) rational

Bézier

curve:

For all values of

:

 23Slide24

Generalization for Bézier

Curves

Intersection of Moving Lines:

Moving lines

intersect at

a

rational

Bézier

curve:This is consistent with the observation on pencils of lines intersecting in a conicA pencil of lines is a degree 1 Bézier lines:

24Slide25

Generalization for Bézier Curves

Intersection of moving lines in equation system notation:

Where

,

are control lines

But given a

Bézier

curve, it’s hard to find two such

Bézier

moving lines … 25Slide26

Generalization for Bézier Curves

Different approach

:

Simplify the equations by eliminating the binomial coefficients:

Where:

26Slide27

Generalization for Bézier Curves

Same in matrix notation:

Each row is a moving line that follows the curve

This can be rewritten as:

27Slide28

Generalization for Bézier Curves

The

Implicitization

Process:Since

are not all zeros, in order to solve the system, we can solve:

This requires calculating the determinant of a large matrix…

28Slide29

Generalization for Bézier Curves

Base Points

For some

Bézier curves (or moving lines) there are values of

for which

Those are called Base points.

For such curves the formula doesn’t work

Can yield “false” zeros

29Slide30

Generalization for Bézier Curves

Base Points

For rational curves:

if

then:

So

Is a curve of lower degree

Eliminates the Base Point

Or a base point with lower multiplicity

30Slide31

Generalization for Bézier Curves

Matrix reduction

In the matrix:

All the elements in the first column are lines that go through

Property of

B

é

zier

curves

So they can be eliminated with row operations

31Slide32

Generalization for Bézier Curves

The matrix

becomes:

32Slide33

Generalization for Bézier Curves

In:

We take only the bottom

lines:

Resulting in a

matrix

33Slide34

Generalization for Bézier Curves

In:

We take only the bottom

lines:

Resulting in a

matrix

34Slide35

Generalization for Bézier Curves

This can be repeated

times (for

or

) until we get 2 lines

Similar

to de

Casteljau

iterations for

In the final matrix the 2 lines are degree

or moving lines which follow the moving point Depending on whether the degree of the original curve is even or odd

 35Slide36

Generalization for Bézier Curves

The resulting system:

is equivalent to finding the common roots of two polynomials

Can be done

using

Bezout's

theorem

36Slide37

Generalization for Surfaces

Using similar reasoning:

Given a parametric surface, find a family of “simpler” implicit surfaces, that when combined using blending functions (i.e. Moving Surfaces) intersect at the given surface

To eliminate the parameters, eliminate the blending functions37Slide38

Conclusion

A

B

ézier curve can be expressed as the intersection of lower degree Bézier moving linesA method for finding such linesA method for replacing the parameter with a constraint on

A method for reducing the computational cost

38Slide39

References

Curve

i

mplicitization using moving lines, Sederberg, Saito, Qi, Klimaszewski – Computer Aided Geometric Design, 1994Implicitization using moving curves and surfaces, Sederberg, Chen – Brigham Young UniversityThe Resultant and Bezout's

Theorem, http://www.mathpages.com/home/kmath544/kmath544.htm

39