Curve implicitization using moving lines. Thomas W. Sederberg , Takafumi Saito, Dongxu Qi, Krzysztof S. Klimaszewski (1993). Presented by: Mira Shalah. Introduction, motivation and definitions. Pencils of lines. Moving lines. Curve implicitization with two moving lines. - PowerPoint PPT Presentation
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Curve implicitization using moving linesThomas W. Sederberg,
Takafumi Saito, Dongxu Qi, Krzysztof S. Klimaszewski
(1993)1Presented by: Mira ShalahIntroduction, motivation and
definitionsPencils of linesMoving lines2Curve implicitization with
two moving linesImplicitization3Implicitization of 2-D curves leads
to many practical algorithms.
For example, a very fast algorithm for computing the
intersection of 2-D curves of low degree is based on
implicitization.
Implicitization reduces the problem of curve intersection to one
of finding the roots of a single polynomial.
Implicitization- why ?4The standard method5Contributions6Some
Definitions78Dualitymoving pointmoving line9101011Pencil of
lines
12
Intersection of two pencils1314
15Pencils on quadratic curves1617
Moving lines: Bernstein form1819Intersection of two moving
linesBase Points and Axial moving lines20Curve representation with
two moving lines2122
Axial moving line on a double point.23In general:24Cubic
curves25
2627Quartic curves
Any quartic rational Bezier curve can be expressed either with
two quadratic28Cont.29
30The general case3132Cont.3334
The zeroing process3536Implicitization373738The conventional
methodThe new methodfaster39?
